Simulation of 2D rarefied gas flows based on the numerical solution of the Boltzmann equation

NASA Astrophysics Data System (ADS)

Poleshkin, Sergey O.; Malkov, Ewgenij A.; Kudryavtsev, Alexey N.; Shershnev, Anton A.; Bondar, Yevgeniy A.; Kohanchik, A. A.

2017-10-01

There are various methods for calculating rarefied gas flows, in particular, statistical methods and deterministic methods based on the finite-difference solutions of the Boltzmann nonlinear kinetic equation and on the solutions of model kinetic equations. There is no universal method; each has its disadvantages in terms of efficiency or accuracy. The choice of the method depends on the problem to be solved and on parameters of calculated flows. Qualitative theoretical arguments help to determine the range of parameters of effectively solved problems for each method; however, it is advisable to perform comparative tests of calculations of the classical problems performed by different methods and with different parameters to have quantitative confirmation of this reasoning. The paper provides the results of the calculations performed by the authors with the help of the Direct Simulation Monte Carlo method and finite-difference methods of solving the Boltzmann equation and model kinetic equations. Based on this comparison, conclusions are made on selecting a particular method for flow simulations in various ranges of flow parameters.

Lattice Boltzmann model for the compressible Navier-Stokes equations with flexible specific-heat ratio.

PubMed

Kataoka, Takeshi; Tsutahara, Michihisa

2004-03-01

We have developed a lattice Boltzmann model for the compressible Navier-Stokes equations with a flexible specific-heat ratio. Several numerical results are presented, and they agree well with the corresponding solutions of the Navier-Stokes equations. In addition, an explicit finite-difference scheme is proposed for the numerical calculation that can make a stable calculation with a large Courant number.

Finite-volume method with lattice Boltzmann flux scheme for incompressible porous media flow at the representative-elementary-volume scale.

PubMed

Hu, Yang; Li, Decai; Shu, Shi; Niu, Xiaodong

2016-02-01

Based on the Darcy-Brinkman-Forchheimer equation, a finite-volume computational model with lattice Boltzmann flux scheme is proposed for incompressible porous media flow in this paper. The fluxes across the cell interface are calculated by reconstructing the local solution of the generalized lattice Boltzmann equation for porous media flow. The time-scaled midpoint integration rule is adopted to discretize the governing equation, which makes the time step become limited by the Courant-Friedricks-Lewy condition. The force term which evaluates the effect of the porous medium is added to the discretized governing equation directly. The numerical simulations of the steady Poiseuille flow, the unsteady Womersley flow, the circular Couette flow, and the lid-driven flow are carried out to verify the present computational model. The obtained results show good agreement with the analytical, finite-difference, and/or previously published solutions.

Comment on ``Alternative approach to the solution of the dispersion relation for a generalized lattice Boltzmann equation''

NASA Astrophysics Data System (ADS)

Lallemand, Pierre; Luo, Li-Shi

2008-12-01

Recently Reis and Phillips [Phys. Rev. E 77, 026702 (2008)] proposed a perturbative method to solve the dispersion equation derived from the linearized lattice Boltzmann equation. We will demonstrate that the method proposed by Reis and Phillips is a reinvention of an existing method. We would also like to refute a number of claims made by Reis and Phillips.

Reply to ``Comment on `Alternative approach to the solution of the dispersion relation for a generalized lattice Boltzmann equation' ''

NASA Astrophysics Data System (ADS)

Reis, T.; Phillips, T. N.

2008-12-01

In this reply to the comment by Lallemand and Luo, we defend our assertion that the alternative approach for the solution of the dispersion relation for a generalized lattice Boltzmann dispersion equation [T. Reis and T. N. Phillips, Phys. Rev. E 77, 026702 (2008)] is mathematically transparent, elegant, and easily justified. Furthermore, the rigorous perturbation analysis used by Reis and Phillips does not require the reciprocals of the relaxation parameters to be small.

Dissipative Relativistic Fluid Dynamics: A New Way to Derive the Equations of Motion from Kinetic Theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Denicol, G. S.; Koide, T.; Rischke, D. H.

2010-10-15

We rederive the equations of motion of dissipative relativistic fluid dynamics from kinetic theory. In contrast with the derivation of Israel and Stewart, which considered the second moment of the Boltzmann equation to obtain equations of motion for the dissipative currents, we directly use the latter's definition. Although the equations of motion obtained via the two approaches are formally identical, the coefficients are different. We show that, for the one-dimensional scaling expansion, our method is in better agreement with the solution obtained from the Boltzmann equation.

Global Well-Posedness of the Boltzmann Equation with Large Amplitude Initial Data

NASA Astrophysics Data System (ADS)

Duan, Renjun; Huang, Feimin; Wang, Yong; Yang, Tong

2017-07-01

The global well-posedness of the Boltzmann equation with initial data of large amplitude has remained a long-standing open problem. In this paper, by developing a new {L^∞_xL^1v\\cap L^∞_{x,v}} approach, we prove the global existence and uniqueness of mild solutions to the Boltzmann equation in the whole space or torus for a class of initial data with bounded velocity-weighted {L^∞} norm under some smallness condition on the {L^1_xL^∞_v} norm as well as defect mass, energy and entropy so that the initial data allow large amplitude oscillations. Both the hard and soft potentials with angular cut-off are considered, and the large time behavior of solutions in the {L^∞_{x,v}} norm with explicit rates of convergence are also studied.

Diffuse sorption modeling.

PubMed

Pivovarov, Sergey

2009-04-01

This work presents a simple solution for the diffuse double layer model, applicable to calculation of surface speciation as well as to simulation of ionic adsorption within the diffuse layer of solution in arbitrary salt media. Based on Poisson-Boltzmann equation, the Gaines-Thomas selectivity coefficient for uni-bivalent exchange on clay, K(GT)(Me(2+)/M(+))=(Q(Me)(0.5)/Q(M)){M(+)}/{Me(2+)}(0.5), (Q is the equivalent fraction of cation in the exchange capacity, and {M(+)} and {Me(2+)} are the ionic activities in solution) may be calculated as [surface charge, mueq/m(2)]/0.61. The obtained solution of the Poisson-Boltzmann equation was applied to calculation of ionic exchange on clays and to simulation of the surface charge of ferrihydrite in 0.01-6 M NaCl solutions. In addition, a new model of acid-base properties was developed. This model is based on assumption that the net proton charge is not located on the mathematical surface plane but diffusely distributed within the subsurface layer of the lattice. It is shown that the obtained solution of the Poisson-Boltzmann equation makes such calculations possible, and that this approach is more efficient than the original diffuse double layer model.

Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gamba, Irene M.; Tharkabhushanam, Sri Harsha

We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation (BTE) in d-dimensions for variable hard sphere (VHS) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computation is reduced to a separate integral over the unit sphere S{sup d-1}. The conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibilitymore » (inelastic interactions) or elastic models of slowing down process. Our simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation, both for elastic and inelastic VHS interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in Bobylev et al. [A.V. Bobylev, C. Cercignani, G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, Journal of Statistical Physics 111 (2003) 403-417] and generalized to a wide range of related models in Bobylev et al. [A.V. Bobylev, C. Cercignani, I.M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Communication in Mathematical Physics, in press. URL: ()]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods [T.V. Noije, M. Ernst, Velocity distributions in homogeneously cooling and heated granular fluids, Granular Matter 1(57) (1998); M.H. Ernst, R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, Journal of Statistical Physics 109 (2002) 407-432; S.J. Moon, M.D. Shattuck, J. Swift, Velocity distributions and correlations in homogeneously heated granular media, Physical Review E 64 (2001) 031303; I.M. Gamba, S. Rjasanow, W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Mathematical and Computer Modelling 42 (2005) 683-700] and rigorously proven in Gamba et al. [I.M. Gamba, V. Panferov, C. Villani, On the Boltzmann equation for diffusively excited granular media, Communications in Mathematical Physics 246 (2004) 503-541(39)] and [A.V. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, Journal of Statistical Physics 116 (2004) 1651-1682].« less

Numerical study of radiometric forces via the direct solution of the Boltzmann kinetic equation

NASA Astrophysics Data System (ADS)

Anikin, Yu. A.

2011-07-01

The two-dimensional rarefied gas motion in a Crookes radiometer and the resulting radiometric forces are studied by numerically solving the Boltzmann kinetic equation. The collision integral is directly evaluated using a projection method, and second-order accurate TVD schemes are used to solve the advection equation. The radiometric forces are found as functions of the Knudsen number and the temperatures, and their spatial distribution is analyzed.

Solution to the Boltzmann equation for layered systems for current perpendicular to the planes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Butler, W. H.; Zhang, X.-G.; MacLaren, J. M.

2000-05-01

Present theories of giant magnetoresistance (GMR) for current perpendicular to the planes (CPP) are based on an extremely restricted solution to the Boltzmann equation that assumes a single free electron band structure for all layers and all spin channels. Within this model only the scattering rate changes from one layer to the next. This model leads to the remarkable result that the resistance of a layered material is simply the sum of the resistances of each layer. We present a solution to the Boltzmann equation for CPP for the case in which the electronic structure can be different for differentmore » layers. The problem of matching boundary conditions between layers is much more complicated than in the current in the planes (CIP) geometry because it is necessary to include the scattering-in term of the Boltzmann equation even for the case of isotropic scattering. This term couples different values of the momentum parallel to the planes. When the electronic structure is different in different layers there is an interface resistance even in the absence of intermixing of the layers. The size of this interface resistance is affected by the electronic structure, scattering rates, and thicknesses of nearby layers. For Co-Cu, the calculated interface resistance and its spin asymmetry is comparable to that measured at low temperature in sputtered samples. (c) 2000 American Institute of Physics.« less

Nonequilibrium Entropy in a Shock

DOE PAGES

Margolin, Len G.

2017-07-19

In a classic paper, Morduchow and Libby use an analytic solution for the profile of a Navier–Stokes shock to show that the equilibrium thermodynamic entropy has a maximum inside the shock. There is no general nonequilibrium thermodynamic formulation of entropy; the extension of equilibrium theory to nonequililbrium processes is usually made through the assumption of local thermodynamic equilibrium (LTE). However, gas kinetic theory provides a perfectly general formulation of a nonequilibrium entropy in terms of the probability distribution function (PDF) solutions of the Boltzmann equation. In this paper I will evaluate the Boltzmann entropy for the PDF that underlies themore » Navier–Stokes equations and also for the PDF of the Mott–Smith shock solution. I will show that both monotonically increase in the shock. As a result, I will propose a new nonequilibrium thermodynamic entropy and show that it is also monotone and closely approximates the Boltzmann entropy.« less

Nonequilibrium Entropy in a Shock

DOE Office of Scientific and Technical Information (OSTI.GOV)

Margolin, Len G.

In a classic paper, Morduchow and Libby use an analytic solution for the profile of a Navier–Stokes shock to show that the equilibrium thermodynamic entropy has a maximum inside the shock. There is no general nonequilibrium thermodynamic formulation of entropy; the extension of equilibrium theory to nonequililbrium processes is usually made through the assumption of local thermodynamic equilibrium (LTE). However, gas kinetic theory provides a perfectly general formulation of a nonequilibrium entropy in terms of the probability distribution function (PDF) solutions of the Boltzmann equation. In this paper I will evaluate the Boltzmann entropy for the PDF that underlies themore » Navier–Stokes equations and also for the PDF of the Mott–Smith shock solution. I will show that both monotonically increase in the shock. As a result, I will propose a new nonequilibrium thermodynamic entropy and show that it is also monotone and closely approximates the Boltzmann entropy.« less

Comparison of approximate solutions to the phonon Boltzmann transport equation with the relaxation time approximation: Spherical harmonics expansions and the discrete ordinates method

NASA Astrophysics Data System (ADS)

Christenson, J. G.; Austin, R. A.; Phillips, R. J.

2018-05-01

The phonon Boltzmann transport equation is used to analyze model problems in one and two spatial dimensions, under transient and steady-state conditions. New, explicit solutions are obtained by using the P1 and P3 approximations, based on expansions in spherical harmonics, and are compared with solutions from the discrete ordinates method. For steady-state energy transfer, it is shown that analytic expressions derived using the P1 and P3 approximations agree quantitatively with the discrete ordinates method, in some cases for large Knudsen numbers, and always for Knudsen numbers less than unity. However, for time-dependent energy transfer, the PN solutions differ qualitatively from converged solutions obtained by the discrete ordinates method. Although they correctly capture the wave-like behavior of energy transfer at short times, the P1 and P3 approximations rely on one or two wave velocities, respectively, yielding abrupt, step-changes in temperature profiles that are absent when the angular dependence of the phonon velocities is captured more completely. It is shown that, with the gray approximation, the P1 approximation is formally equivalent to the so-called "hyperbolic heat equation." Overall, these results support the use of the PN approximation to find solutions to the phonon Boltzmann transport equation for steady-state conditions. Such solutions can be useful in the design and analysis of devices that involve heat transfer at nanometer length scales, where continuum-scale approaches become inaccurate.

Lattice Boltzmann simulation of antiplane shear loading of a stationary crack

NASA Astrophysics Data System (ADS)

Schlüter, Alexander; Kuhn, Charlotte; Müller, Ralf

2018-01-01

In this work, the lattice Boltzmann method is applied to study the dynamic behaviour of linear elastic solids under antiplane shear deformation. In this case, the governing set of partial differential equations reduces to a scalar wave equation for the out of plane displacement in a two dimensional domain. The lattice Boltzmann approach developed by Guangwu (J Comput Phys 161(1):61-69, 2000) in 2006 is used to solve the problem numerically. Some aspects of the scheme are highlighted, including the treatment of the boundary conditions. Subsequently, the performance of the lattice Boltzmann scheme is tested for a stationary crack problem for which an analytic solution exists. The treatment of cracks is new compared to the examples that are discussed in Guangwu's work. Furthermore, the lattice Boltzmann simulations are compared to finite element computations. Finally, the influence of the lattice Boltzmann relaxation parameter on the stability of the scheme is illustrated.

Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations.

PubMed

Li, Q; He, Y L; Wang, Y; Tao, W Q

2007-11-01

A coupled double-distribution-function lattice Boltzmann method is developed for the compressible Navier-Stokes equations. Different from existing thermal lattice Boltzmann methods, this method can recover the compressible Navier-Stokes equations with a flexible specific-heat ratio and Prandtl number. In the method, a density distribution function based on a multispeed lattice is used to recover the compressible continuity and momentum equations, while the compressible energy equation is recovered by an energy distribution function. The energy distribution function is then coupled to the density distribution function via the thermal equation of state. In order to obtain an adjustable specific-heat ratio, a constant related to the specific-heat ratio is introduced into the equilibrium energy distribution function. Two different coupled double-distribution-function lattice Boltzmann models are also proposed in the paper. Numerical simulations are performed for the Riemann problem, the double-Mach-reflection problem, and the Couette flow with a range of specific-heat ratios and Prandtl numbers. The numerical results are found to be in excellent agreement with analytical and/or other solutions.

Stochastic and Boltzmann-like models for behavioral changes, and their relation to game theory

NASA Astrophysics Data System (ADS)

Helbing, Dirk

1993-03-01

In the last decade, stochastic models have shown to be very useful for quantitative modelling of social processes. Here, a configurational master equation for the description of behavioral changes by pair interactions of individuals is developed. Three kinds of social pair interactions are distinguished: Avoidance processes, compromising processes, and imitative processes. Computational results are presented for a special case of imitative processes: the competition of two equivalent strategies. They show a phase transition that describes the self-organization of a behavioral convention. This phase transition is further analyzed by examining the equations for the most probable behavioral distribution, which are Boltzmann-like equations. Special cases of Boltzmann-like equations do not obey the H-theorem and have oscillatory or even chaotic solutions. A suitable Taylor approximation leads to the so-called game dynamical equations (also known as selection-mutation equations in the theory of evolution).

On the use temperature parameterized rate coefficients in the estimation of non-equilibrium reaction rates

NASA Astrophysics Data System (ADS)

Shizgal, Bernie D.; Chikhaoui, Aziz

2006-06-01

The present paper considers a detailed analysis of the nonequilibrium effects for a model reactive system with the Chapman-Eskog (CE) solution of the Boltzmann equation as well as an explicit time dependent solution. The elastic cross sections employed are a hard sphere cross section and the Maxwell molecule cross section. Reactive cross sections which model reactions with and without activation energy are used. A detailed comparison is carried out with these solutions of the Boltzmann equation and the approximation introduced by Cukrowski and coworkers [J. Chem. Phys. 97 (1992) 9086; Chem. Phys. 89 (1992) 159; Physica A 188 (1992) 344; Chem. Phys. Lett. A 297 (1998) 402; Physica A 275 (2000) 134; Chem. Phys. Lett. 341 (2001) 585; Acta Phys. Polonica B 334 (2003) 3607.] based on the temperature of the reactive particles. We show that the Cukrowski approximation has limited applicability for the large class of reactive systems studied in this paper. The explicit time dependent solutions of the Boltzmann equation demonstrate that the CE approach is valid only for very slow reactions for which the corrections to the equilibrium rate coefficient are very small.

Modeling mass transfer and reaction of dilute solutes in a ternary phase system by the lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Fu, Yu-Hang; Bai, Lin; Luo, Kai-Hong; Jin, Yong; Cheng, Yi

2017-04-01

In this work, we propose a general approach for modeling mass transfer and reaction of dilute solute(s) in incompressible three-phase flows by introducing a collision operator in lattice Boltzmann (LB) method. An LB equation was used to simulate the solute dynamics among three different fluids, in which the newly expanded collision operator was used to depict the interface behavior of dilute solute(s). The multiscale analysis showed that the presented model can recover the macroscopic transport equations derived from the Maxwell-Stefan equation for dilute solutes in three-phase systems. Compared with the analytical equation of state of solute and dynamic behavior, these results are proven to constitute a generalized framework to simulate solute distributions in three-phase flows, including compound soluble in one phase, compound adsorbed on single-interface, compound in two phases, and solute soluble in three phases. Moreover, numerical simulations of benchmark cases, such as phase decomposition, multilayered planar interfaces, and liquid lens, were performed to test the stability and efficiency of the model. Finally, the multiphase mass transfer and reaction in Janus droplet transport in a straight microchannel were well reproduced.

A fast iterative scheme for the linearized Boltzmann equation

NASA Astrophysics Data System (ADS)

Wu, Lei; Zhang, Jun; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.

2017-06-01

Iterative schemes to find steady-state solutions to the Boltzmann equation are efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator L into the form L = (L + Nδh) - Nδh, where δ is the gas rarefaction parameter, h is the velocity distribution function, and N is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when Nδ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference between these results and those using the hard-sphere potential is discussed.

A lattice Boltzmann model for the Burgers-Fisher equation.

PubMed

Zhang, Jianying; Yan, Guangwu

2010-06-01

A lattice Boltzmann model is developed for the one- and two-dimensional Burgers-Fisher equation based on the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. In order to obtain the two-dimensional Burgers-Fisher equation, vector sigma(j) has been used. And in order to overcome the drawbacks of "error rebound," a new assumption of additional distribution is presented, where two additional terms, in first order and second order separately, are used. Comparisons with the results obtained by other methods reveal that the numerical solutions obtained by the proposed method converge to exact solutions. The model under new assumption gives better results than that with second order assumption. (c) 2010 American Institute of Physics.

A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments.

PubMed

Fisicaro, G; Genovese, L; Andreussi, O; Marzari, N; Goedecker, S

2016-01-07

The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.

A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fisicaro, G., E-mail: giuseppe.fisicaro@unibas.ch; Goedecker, S.; Genovese, L.

2016-01-07

The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and themore » linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.« less

Slip Boundary Conditions for the Compressible Navier-Stokes Equations

NASA Astrophysics Data System (ADS)

Aoki, Kazuo; Baranger, Céline; Hattori, Masanari; Kosuge, Shingo; Martalò, Giorgio; Mathiaud, Julien; Mieussens, Luc

2017-11-01

The slip boundary conditions for the compressible Navier-Stokes equations are derived systematically from the Boltzmann equation on the basis of the Chapman-Enskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar-Gross-Krook model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.

Maxwell iteration for the lattice Boltzmann method with diffusive scaling

NASA Astrophysics Data System (ADS)

Zhao, Weifeng; Yong, Wen-An

2017-03-01

In this work, we present an alternative derivation of the Navier-Stokes equations from Bhatnagar-Gross-Krook models of the lattice Boltzmann method with diffusive scaling. This derivation is based on the Maxwell iteration and can expose certain important features of the lattice Boltzmann solutions. Moreover, it will be seen to be much more straightforward and logically clearer than the existing approaches including the Chapman-Enskog expansion.

A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Liu, Chang, E-mail: cliuaa@ust.hk; Xu, Kun, E-mail: makxu@ust.hk; Sun, Quanhua, E-mail: qsun@imech.ac.cn

Fluid dynamic equations are valid in their respective modeling scales, such as the particle mean free path scale of the Boltzmann equation and the hydrodynamic scale of the Navier–Stokes (NS) equations. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. Even though the Boltzmann equation is claimed to be valid in all scales, many Boltzmann solvers, including direct simulation Monte Carlo method, require the cell resolution to the order of particle mean free path scale. Therefore, they are still single scale methods. In order to study multiscale flow evolution efficiently, themore » dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. Even though the UGKS is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time, it still has space to develop more accurate flow solver in the region, where the time step is comparable with the local particle mean free time. In such a scale, there is dynamic difference from the full Boltzmann collision term and the model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region where it is needed. The central ingredient of the UGKS is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface, where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled. The newly developed UGKS has the asymptotic preserving (AP) property of recovering the NS solutions in the continuum flow regime, and the full Boltzmann solution in the rarefied regime. In the mostly unexplored transition regime, the UGKS itself provides a valuable tool for the non-equilibrium flow study. The mathematical properties of the scheme, such as stability, accuracy, and the asymptotic preserving, will be analyzed in this paper as well.« less

The structure of shock wave in a gas consisting of ideally elastic, rigid spherical molecules

NASA Technical Reports Server (NTRS)

Cheremisin, F. G.

1972-01-01

Principal approaches are examined to the theoretical study of the shock layer structure. The choice of a molecular model is discussed and three procedures are formulated. These include a numerical calculation method, solution of the kinetic relaxation equation, and solution of the Boltzmann equation.

Lattice Boltzmann model for simulation of magnetohydrodynamics

NASA Technical Reports Server (NTRS)

Chen, Shiyi; Chen, Hudong; Martinez, Daniel; Matthaeus, William

1991-01-01

A numerical method, based on a discrete Boltzmann equation, is presented for solving the equations of magnetohydrodynamics (MHD). The algorithm provides advantages similar to the cellular automaton method in that it is local and easily adapted to parallel computing environments. Because of much lower noise levels and less stringent requirements on lattice size, the method appears to be more competitive with traditional solution methods. Examples show that the model accurately reproduces both linear and nonlinear MHD phenomena.

Hilbert's sixth problem and the failure of the Boltzmann to Euler limit

NASA Astrophysics Data System (ADS)

Slemrod, Marshall

2018-04-01

This paper addresses the main issue of Hilbert's sixth problem, namely the rigorous passage of solutions to the mesoscopic Boltzmann equation to macroscopic solutions of the Euler equations of compressible gas dynamics. The results of the paper are that (i) in general Hilbert's program will fail because of the appearance of van der Waals-Korteweg capillarity terms in a macroscopic description of motion of a gas, and (ii) the van der Waals-Korteweg theory itself might satisfy Hilbert's quest for a map from the `atomistic view' to the laws of motion of continua. This article is part of the theme issue `Hilbert's sixth problem'.

Unified solution of the Boltzmann equation for electron and ion velocity distribution functions and transport coefficients in weakly ionized plasmas

NASA Astrophysics Data System (ADS)

Konovalov, Dmitry A.; Cocks, Daniel G.; White, Ronald D.

2017-10-01

The velocity distribution function and transport coefficients for charged particles in weakly ionized plasmas are calculated via a multi-term solution of Boltzmann's equation and benchmarked using a Monte-Carlo simulation. A unified framework for the solution of the original full Boltzmann's equation is presented which is valid for ions and electrons, avoiding any recourse to approximate forms of the collision operator in various limiting mass ratio cases. This direct method using Lebedev quadratures over the velocity and scattering angles avoids the need to represent the ion mass dependence in the collision operator through an expansion in terms of the charged particle to neutral mass ratio. For the two-temperature Burnett function method considered in this study, this amounts to avoiding the need for the complex Talmi-transformation methods and associated mass-ratio expansions. More generally, we highlight the deficiencies in the two-temperature Burnett function method for heavy ions at high electric fields to calculate the ion velocity distribution function, even though the transport coefficients have converged. Contribution to the Topical Issue "Physics of Ionized Gases (SPIG 2016)", edited by Goran Poparic, Bratislav Obradovic, Dragana Maric and Aleksandar Milosavljevic.

Perfect fluidity of a dissipative system: Analytical solution for the Boltzmann equation in AdS 2 Ⓧ S 2

DOE PAGES

Noronha, Jorge; Denicol, Gabriel S.

2015-12-30

In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime AdS 2 Ⓧ S 2. We further derive explicit analytic expressions for the momentum dependence of the single-particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density doesmore » not match the equilibrium form. The nonequilibrium contribution to the entropy density is shown to be due to higher-order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Furthermore, in this system the slowly moving hydrodynamic degrees of freedom can exhibit true perfect fluidity while being totally decoupled from the fast moving, nonhydrodynamical microscopic degrees of freedom that lead to entropy production.« less

New solution decomposition and minimization schemes for Poisson-Boltzmann equation in calculation of biomolecular electrostatics

NASA Astrophysics Data System (ADS)

Xie, Dexuan

2014-10-01

The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model in the calculation of electrostatic potential energy for biomolecules in ionic solvent, but its numerical solution remains a challenge due to its strong singularity and nonlinearity caused by its singular distribution source terms and exponential nonlinear terms. To effectively deal with such a challenge, in this paper, new solution decomposition and minimization schemes are proposed, together with a new PBE analysis on solution existence and uniqueness. Moreover, a PBE finite element program package is developed in Python based on the FEniCS program library and GAMer, a molecular surface and volumetric mesh generation program package. Numerical tests on proteins and a nonlinear Born ball model with an analytical solution validate the new solution decomposition and minimization schemes, and demonstrate the effectiveness and efficiency of the new PBE finite element program package.

Lattice Boltzmann approach for complex nonequilibrium flows.

PubMed

Montessori, A; Prestininzi, P; La Rocca, M; Succi, S

2015-10-01

We present a lattice Boltzmann realization of Grad's extended hydrodynamic approach to nonequilibrium flows. This is achieved by using higher-order isotropic lattices coupled with a higher-order regularization procedure. The method is assessed for flow across parallel plates and three-dimensional flows in porous media, showing excellent agreement of the mass flow with analytical and numerical solutions of the Boltzmann equation across the full range of Knudsen numbers, from the hydrodynamic regime to ballistic motion.

A discontinuous Poisson-Boltzmann equation with interfacial jump: homogenisation and residual error estimate.

PubMed

Fellner, Klemens; Kovtunenko, Victor A

2016-01-01

A nonlinear Poisson-Boltzmann equation with inhomogeneous Robin type boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic multiphase medium with dilute solid particles. The key issue stems from interfacial jumps, which necessitate discontinuous solutions to the problem. Based on variational techniques, we derive the homogenisation of the discontinuous problem and establish a rigorous residual error estimate up to the first-order correction.

Decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio.

PubMed

Hu, Kainan; Zhang, Hongwu; Geng, Shaojuan

2016-10-01

A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e., the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion, so it is easy to construct new schemes of higher-order accuracy. To validate the proposed scheme, a one-dimensional shock tube simulation is performed. The numerical results agree with the analytical solutions very well.

Function-Space-Based Solution Scheme for the Size-Modified Poisson-Boltzmann Equation in Full-Potential DFT.

PubMed

Ringe, Stefan; Oberhofer, Harald; Hille, Christoph; Matera, Sebastian; Reuter, Karsten

2016-08-09

The size-modified Poisson-Boltzmann (MPB) equation is an efficient implicit solvation model which also captures electrolytic solvent effects. It combines an account of the dielectric solvent response with a mean-field description of solvated finite-sized ions. We present a general solution scheme for the MPB equation based on a fast function-space-oriented Newton method and a Green's function preconditioned iterative linear solver. In contrast to popular multigrid solvers, this approach allows us to fully exploit specialized integration grids and optimized integration schemes. We describe a corresponding numerically efficient implementation for the full-potential density-functional theory (DFT) code FHI-aims. We show that together with an additional Stern layer correction the DFT+MPB approach can describe the mean activity coefficient of a KCl aqueous solution over a wide range of concentrations. The high sensitivity of the calculated activity coefficient on the employed ionic parameters thereby suggests to use extensively tabulated experimental activity coefficients of salt solutions for a systematic parametrization protocol.

A review of recent advances in the spherical harmonics expansion method for semiconductor device simulation.

PubMed

Rupp, K; Jungemann, C; Hong, S-M; Bina, M; Grasser, T; Jüngel, A

The Boltzmann transport equation is commonly considered to be the best semi-classical description of carrier transport in semiconductors, providing precise information about the distribution of carriers with respect to time (one dimension), location (three dimensions), and momentum (three dimensions). However, numerical solutions for the seven-dimensional carrier distribution functions are very demanding. The most common solution approach is the stochastic Monte Carlo method, because the gigabytes of memory requirements of deterministic direct solution approaches has not been available until recently. As a remedy, the higher accuracy provided by solutions of the Boltzmann transport equation is often exchanged for lower computational expense by using simpler models based on macroscopic quantities such as carrier density and mean carrier velocity. Recent developments for the deterministic spherical harmonics expansion method have reduced the computational cost for solving the Boltzmann transport equation, enabling the computation of carrier distribution functions even for spatially three-dimensional device simulations within minutes to hours. We summarize recent progress for the spherical harmonics expansion method and show that small currents, reasonable execution times, and rare events such as low-frequency noise, which are all hard or even impossible to simulate with the established Monte Carlo method, can be handled in a straight-forward manner. The applicability of the method for important practical applications is demonstrated for noise simulation, small-signal analysis, hot-carrier degradation, and avalanche breakdown.

Non-Boltzmann Modeling for Air Shock-Layer Radiation at Lunar-Return Conditions

NASA Technical Reports Server (NTRS)

Johnston, Christopher O.; Hollis, Brian R.; Sutton, Kenneth

2008-01-01

This paper investigates the non-Boltzmann modeling of the radiating atomic and molecular electronic states present in lunar-return shock-layers. The Master Equation is derived for a general atom or molecule while accounting for a variety of excitation and de-excitation mechanisms. A new set of electronic-impact excitation rates is compiled for N, O, and N2+, which are the main radiating species for most lunar-return shock-layers. Based on these new rates, a novel approach of curve-fitting the non-Boltzmann populations of the radiating atomic and molecular states is developed. This new approach provides a simple and accurate method for calculating the atomic and molecular non-Boltzmann populations while avoiding the matrix inversion procedure required for the detailed solution of the Master Equation. The radiative flux values predicted by the present detailed non-Boltzmann model and the approximate curve-fitting approach are shown to agree within 5% for the Fire 1634 s case.

Linearized lattice Boltzmann method for micro- and nanoscale flow and heat transfer.

PubMed

Shi, Yong; Yap, Ying Wan; Sader, John E

2015-07-01

Ability to characterize the heat transfer in flowing gases is important for a wide range of applications involving micro- and nanoscale devices. Gas flows away from the continuum limit can be captured using the Boltzmann equation, whose analytical solution poses a formidable challenge. An efficient and accurate numerical simulation of the Boltzmann equation is thus highly desirable. In this article, the linearized Boltzmann Bhatnagar-Gross-Krook equation is used to develop a hierarchy of thermal lattice Boltzmann (LB) models based on half-space Gaussian-Hermite (GH) quadrature ranging from low to high algebraic precision, using double distribution functions. Simplified versions of the LB models in the continuum limit are also derived, and are shown to be consistent with existing thermal LB models for noncontinuum heat transfer reported in the literature. Accuracy of the proposed LB hierarchy is assessed by simulating thermal Couette flows for a wide range of Knudsen numbers. Effects of the underlying quadrature schemes (half-space GH vs full-space GH) and continuum-limit simplifications on computational accuracy are also elaborated. The numerical findings in this article provide direct evidence of improved computational capability of the proposed LB models for modeling noncontinuum flows and heat transfer at small length scales.

Challenge for more precise e- and ion-transport in gases and liquids

NASA Astrophysics Data System (ADS)

White, Ron

2016-09-01

The full potential of technologies driven by non-equilibrium electron and ion processes in gases, liquids and soft-matter can only be realised once the basic physics has been mastered. The central component in this pursuit is an ever increasing need for the precise determination of electron and ion transport in such media. Over the last few decades, the group at James Cook University and collaborators have developed a suite of multi-term Boltzmann equation solutions to treat temporal and spatial non-locality for electrons and ions in electric and magnetic fields in gaseous systems. In this presentation, we will highlight recent developments including (i) a space-time multi-term solution of Boltzmann's equation; (ii) a unified treatment of electron and ion solutions of Boltzmann's equation which avoids mass ratio expansions; (iii) the treatment dense gases and liquids, including coherent scattering, screened potentials and (self) trapped bubble state effects, the latter of which can give rise to fractional transport behaviour, and (iv) the application to consider the self-consistency of cross-sections for electrons in biomolecules. Contributors: G. Boyle, P. Stokes, M. Casey, N. Garland, D. Cocks, D. Konovalov, S. Dujko, R. E. Robson, K. F. Ness, M. Brunger, S. Buckman, J. de Urquijo and Z. Lj. Petrovic. Support: Australian Research Council.

The Poisson-Helmholtz-Boltzmann model.

PubMed

Bohinc, K; Shrestha, A; May, S

2011-10-01

We present a mean-field model of a one-component electrolyte solution where the mobile ions interact not only via Coulomb interactions but also through a repulsive non-electrostatic Yukawa potential. Our choice of the Yukawa potential represents a simple model for solvent-mediated interactions between ions. We employ a local formulation of the mean-field free energy through the use of two auxiliary potentials, an electrostatic and a non-electrostatic potential. Functional minimization of the mean-field free energy leads to two coupled local differential equations, the Poisson-Boltzmann equation and the Helmholtz-Boltzmann equation. Their boundary conditions account for the sources of both the electrostatic and non-electrostatic interactions on the surface of all macroions that reside in the solution. We analyze a specific example, two like-charged planar surfaces with their mobile counterions forming the electrolyte solution. For this system we calculate the pressure between the two surfaces, and we analyze its dependence on the strength of the Yukawa potential and on the non-electrostatic interactions of the mobile ions with the planar macroion surfaces. In addition, we demonstrate that our mean-field model is consistent with the contact theorem, and we outline its generalization to arbitrary interaction potentials through the use of a Laplace transformation. © EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2011

Transport of Multivalent Electrolyte Mixtures in Micro- and Nanochannels

DTIC Science & Technology

2013-11-08

equations for this process are the unsteady Navier-Stokes equations along with continuity and the Poisson- Nernst -Planck system for the electro- static part...about five times the Debye screening length D (the 1/e lengthscale for the potential from the solution of the linearized Poisson- Boltzmann equation

Immersed boundary method for Boltzmann model kinetic equations

NASA Astrophysics Data System (ADS)

Pekardan, Cem; Chigullapalli, Sruti; Sun, Lin; Alexeenko, Alina

2012-11-01

Three different immersed boundary method formulations are presented for Boltzmann model kinetic equations such as Bhatnagar-Gross-Krook (BGK) and Ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) model equations. 1D unsteady IBM solution for a moving piston is compared with the DSMC results and 2D quasi-steady microscale gas damping solutions are verified by a conformal finite volume method solver. Transient analysis for a sinusoidally moving beam is also carried out for the different pressure conditions (1 atm, 0.1 atm and 0.01 atm) corresponding to Kn=0.05,0.5 and 5. Interrelaxation method (Method 2) is shown to provide a faster convergence as compared to the traditional interpolation scheme used in continuum IBM formulations. Unsteady damping in rarefied regime is characterized by a significant phase-lag which is not captured by quasi-steady approximations.

Transport coefficients in ultrarelativistic kinetic theory

NASA Astrophysics Data System (ADS)

Ambruş, Victor E.

2018-02-01

A spatially periodic longitudinal wave is considered in relativistic dissipative hydrodynamics. At sufficiently small wave amplitudes, an analytic solution is obtained in the linearized limit of the macroscopic conservation equations within the first- and second-order relativistic hydrodynamics formulations. A kinetic solver is used to obtain the numerical solution of the relativistic Boltzmann equation for massless particles in the Anderson-Witting approximation for the collision term. It is found that, at small values of the Anderson-Witting relaxation time τ , the transport coefficients emerging from the relativistic Boltzmann equation agree with those predicted through the Chapman-Enskog procedure, while the relaxation times of the heat flux and shear pressure are equal to τ . These claims are further strengthened by considering a moment-type approximation based on orthogonal polynomials under which the Chapman-Enskog results for the transport coefficients are exactly recovered.

Electrostatic forces in the Poisson-Boltzmann systems

NASA Astrophysics Data System (ADS)

Xiao, Li; Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray

2013-09-01

Continuum modeling of electrostatic interactions based upon numerical solutions of the Poisson-Boltzmann equation has been widely used in structural and functional analyses of biomolecules. A limitation of the numerical strategies is that it is conceptually difficult to incorporate these types of models into molecular mechanics simulations, mainly because of the issue in assigning atomic forces. In this theoretical study, we first derived the Maxwell stress tensor for molecular systems obeying the full nonlinear Poisson-Boltzmann equation. We further derived formulations of analytical electrostatic forces given the Maxwell stress tensor and discussed the relations of the formulations with those published in the literature. We showed that the formulations derived from the Maxwell stress tensor require a weaker condition for its validity, applicable to nonlinear Poisson-Boltzmann systems with a finite number of singularities such as atomic point charges and the existence of discontinuous dielectric as in the widely used classical piece-wise constant dielectric models.

An Improved Green's Function for Ion Beam Transport

NASA Technical Reports Server (NTRS)

Tweed, J.; Wilson, J. W.; Tripathi, R. K.

2003-01-01

Ion beam transport theory allows testing of material transmission properties in the laboratory environment generated by particle accelerators. This is a necessary step in materials development and evaluation for space use. The approximations used in solving the Boltzmann transport equation for the space setting are often not sufficient for laboratory work and those issues are the main emphasis of the present work. In consequence, an analytic solution of the linear Boltzmann equation is pursued in the form of a Green's function allowing flexibility in application to a broad range of boundary value problems. It has been established that simple solutions can be found for the high charge and energy (HZE) by ignoring nuclear energy downshifts and dispersion. Such solutions were found to be supported by experimental evidence with HZE ion beams when multiple scattering was added. Lacking from the prior solutions were range and energy straggling and energy downshift with dispersion associated with nuclear events. Recently, we have found global solutions including these effects providing a broader class of HZE ion solutions.

Numerical Treatment of the Boltzmann Equation for Self-Propelled Particle Systems

NASA Astrophysics Data System (ADS)

Thüroff, Florian; Weber, Christoph A.; Frey, Erwin

2014-10-01

Kinetic theories constitute one of the most promising tools to decipher the characteristic spatiotemporal dynamics in systems of actively propelled particles. In this context, the Boltzmann equation plays a pivotal role, since it provides a natural translation between a particle-level description of the system's dynamics and the corresponding hydrodynamic fields. Yet, the intricate mathematical structure of the Boltzmann equation substantially limits the progress toward a full understanding of this equation by solely analytical means. Here, we propose a general framework to numerically solve the Boltzmann equation for self-propelled particle systems in two spatial dimensions and with arbitrary boundary conditions. We discuss potential applications of this numerical framework to active matter systems and use the algorithm to give a detailed analysis to a model system of self-propelled particles with polar interactions. In accordance with previous studies, we find that spatially homogeneous isotropic and broken-symmetry states populate two distinct regions in parameter space, which are separated by a narrow region of spatially inhomogeneous, density-segregated moving patterns. We find clear evidence that these three regions in parameter space are connected by first-order phase transitions and that the transition between the spatially homogeneous isotropic and polar ordered phases bears striking similarities to liquid-gas phase transitions in equilibrium systems. Within the density-segregated parameter regime, we find a novel stable limit-cycle solution of the Boltzmann equation, which consists of parallel lanes of polar clusters moving in opposite directions, so as to render the overall symmetry of the system's ordered state nematic, despite purely polar interactions on the level of single particles.

The Approach to Equilibrium: Detailed Balance and the Master Equation

ERIC Educational Resources Information Center

Alexander, Millard H.; Hall, Gregory E.; Dagdigian, Paul J.

2011-01-01

The approach to the equilibrium (Boltzmann) distribution of populations of internal states of a molecule is governed by inelastic collisions in the gas phase and with surfaces. The set of differential equations governing the time evolution of the internal state populations is commonly called the master equation. An analytic solution to the master…

Kinetic equation and nonequilibrium entropy for a quasi-two-dimensional gas.

PubMed

Brey, J Javier; Maynar, Pablo; García de Soria, M I

2016-10-01

A kinetic equation for a dilute gas of hard spheres confined between two parallel plates separated a distance smaller than two particle diameters is derived. It is a Boltzmann-like equation, which incorporates the effect of the confinement on the particle collisions. A function S(t) is constructed by adding to the Boltzmann expression a confinement contribution. Then it is shown that for the solutions of the kinetic equation, S(t) increases monotonically in time, until the system reaches a stationary inhomogeneous state, when S becomes the equilibrium entropy of the confined system as derived from equilibrium statistical mechanics. From the entropy, other equilibrium properties are obtained, and molecular dynamics simulations are used to verify some of the theoretical predictions.

almaBTE : A solver of the space-time dependent Boltzmann transport equation for phonons in structured materials

NASA Astrophysics Data System (ADS)

Carrete, Jesús; Vermeersch, Bjorn; Katre, Ankita; van Roekeghem, Ambroise; Wang, Tao; Madsen, Georg K. H.; Mingo, Natalio

2017-11-01

almaBTE is a software package that solves the space- and time-dependent Boltzmann transport equation for phonons, using only ab-initio calculated quantities as inputs. The program can predictively tackle phonon transport in bulk crystals and alloys, thin films, superlattices, and multiscale structures with size features in the nm- μm range. Among many other quantities, the program can output thermal conductances and effective thermal conductivities, space-resolved average temperature profiles, and heat-current distributions resolved in frequency and space. Its first-principles character makes almaBTE especially well suited to investigate novel materials and structures. This article gives an overview of the program structure and presents illustrative examples for some of its uses. PROGRAM SUMMARY Program Title:almaBTE Program Files doi:http://dx.doi.org/10.17632/8tfzwgtp73.1 Licensing provisions: Apache License, version 2.0 Programming language: C++ External routines/libraries: BOOST, MPI, Eigen, HDF5, spglib Nature of problem: Calculation of temperature profiles, thermal flux distributions and effective thermal conductivities in structured systems where heat is carried by phonons Solution method: Solution of linearized phonon Boltzmann transport equation, Variance-reduced Monte Carlo

An efficient numerical method for solving the Boltzmann equation in multidimensions

NASA Astrophysics Data System (ADS)

Dimarco, Giacomo; Loubère, Raphaël; Narski, Jacek; Rey, Thomas

2018-01-01

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) (Dimarco and Loubère, 2013 [26]) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3 D × 3 D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.

Anisotropic hydrodynamics for conformal Gubser flow

NASA Astrophysics Data System (ADS)

Nopoush, Mohammad; Ryblewski, Radoslaw; Strickland, Michael

2015-02-01

We derive the equations of motion for a system undergoing boost-invariant longitudinal and azimuthally symmetric transverse "Gubser flow" using leading-order anisotropic hydrodynamics. This is accomplished by assuming that the one-particle distribution function is ellipsoidally symmetric in the momenta conjugate to the de Sitter coordinates used to parametrize the Gubser flow. We then demonstrate that the S O (3 )q symmetry in de Sitter space further constrains the anisotropy tensor to be of spheroidal form. The resulting system of two coupled ordinary differential equations for the de Sitter-space momentum scale and anisotropy parameter are solved numerically and compared to a recently obtained exact solution of the relaxation-time-approximation Boltzmann equation subject to the same flow. We show that anisotropic hydrodynamics describes the spatiotemporal evolution of the system better than all currently known dissipative hydrodynamics approaches. In addition, we prove that anisotropic hydrodynamics gives the exact solution of the relaxation-time approximation Boltzmann equation in the ideal, η /s →0 , and free-streaming, η /s →∞, limits.

Maximum-entropy reconstruction method for moment-based solution of the Boltzmann equation

NASA Astrophysics Data System (ADS)

Summy, Dustin; Pullin, Dale

2013-11-01

We describe a method for a moment-based solution of the Boltzmann equation. This starts with moment equations for a 10 + 9 N , N = 0 , 1 , 2 . . . -moment representation. The partial-differential equations (PDEs) for these moments are unclosed, containing both higher-order moments and molecular-collision terms. These are evaluated using a maximum-entropy construction of the velocity distribution function f (c , x , t) , using the known moments, within a finite-box domain of single-particle-velocity (c) space. Use of a finite-domain alleviates known problems (Junk and Unterreiter, Continuum Mech. Thermodyn., 2002) concerning existence and uniqueness of the reconstruction. Unclosed moments are evaluated with quadrature while collision terms are calculated using a Monte-Carlo method. This allows integration of the moment PDEs in time. Illustrative examples will include zero-space- dimensional relaxation of f (c , t) from a Mott-Smith-like initial condition toward equilibrium and one-space dimensional, finite Knudsen number, planar Couette flow. Comparison with results using the direct-simulation Monte-Carlo method will be presented.

The Poisson-Boltzmann theory for the two-plates problem: some exact results.

PubMed

Xing, Xiang-Jun

2011-12-01

The general solution to the nonlinear Poisson-Boltzmann equation for two parallel charged plates, either inside a symmetric electrolyte, or inside a 2q:-q asymmetric electrolyte, is found in terms of Weierstrass elliptic functions. From this we derive some exact asymptotic results for the interaction between charged plates, as well as the exact form of the renormalized surface charge density.

Mesoscopic electrohydrodynamic simulations of binary colloidal suspensions.

PubMed

Rivas, Nicolas; Frijters, Stefan; Pagonabarraga, Ignacio; Harting, Jens

2018-04-14

A model is presented for the solution of electrokinetic phenomena of colloidal suspensions in fluid mixtures. We solve the discrete Boltzmann equation with a Bhatnagar-Gross-Krook collision operator using the lattice Boltzmann method to simulate binary fluid flows. Solvent-solvent and solvent-solute interactions are implemented using a pseudopotential model. The Nernst-Planck equation, describing the kinetics of dissolved ion species, is solved using a finite difference discretization based on the link-flux method. The colloids are resolved on the lattice and coupled to the hydrodynamics and electrokinetics through appropriate boundary conditions. We present the first full integration of these three elements. The model is validated by comparing with known analytic solutions of ionic distributions at fluid interfaces, dielectric droplet deformations, and the electrophoretic mobility of colloidal suspensions. Its possibilities are explored by considering various physical systems, such as breakup of charged and neutral droplets and colloidal dynamics at either planar or spherical fluid interfaces.

Mesoscopic electrohydrodynamic simulations of binary colloidal suspensions

NASA Astrophysics Data System (ADS)

Rivas, Nicolas; Frijters, Stefan; Pagonabarraga, Ignacio; Harting, Jens

2018-04-01

A model is presented for the solution of electrokinetic phenomena of colloidal suspensions in fluid mixtures. We solve the discrete Boltzmann equation with a Bhatnagar-Gross-Krook collision operator using the lattice Boltzmann method to simulate binary fluid flows. Solvent-solvent and solvent-solute interactions are implemented using a pseudopotential model. The Nernst-Planck equation, describing the kinetics of dissolved ion species, is solved using a finite difference discretization based on the link-flux method. The colloids are resolved on the lattice and coupled to the hydrodynamics and electrokinetics through appropriate boundary conditions. We present the first full integration of these three elements. The model is validated by comparing with known analytic solutions of ionic distributions at fluid interfaces, dielectric droplet deformations, and the electrophoretic mobility of colloidal suspensions. Its possibilities are explored by considering various physical systems, such as breakup of charged and neutral droplets and colloidal dynamics at either planar or spherical fluid interfaces.

Hypersonic Shock Wave Computations Using the Generalized Boltzmann Equation

NASA Astrophysics Data System (ADS)

Agarwal, Ramesh; Chen, Rui; Cheremisin, Felix G.

2006-11-01

Hypersonic shock structure in diatomic gases is computed by solving the Generalized Boltzmann Equation (GBE), where the internal and translational degrees of freedom are considered in the framework of quantum and classical mechanics respectively [1]. The computational framework available for the standard Boltzmann equation [2] is extended by including both the rotational and vibrational degrees of freedom in the GBE. There are two main difficulties encountered in computation of high Mach number flows of diatomic gases with internal degrees of freedom: (1) a large velocity domain is needed for accurate numerical description of the distribution function resulting in enormous computational effort in calculation of the collision integral, and (2) about 50 energy levels are needed for accurate representation of the rotational spectrum of the gas. Our methodology addresses these problems, and as a result the efficiency of calculations has increased by several orders of magnitude. The code has been validated by computing the shock structure in Nitrogen for Mach numbers up to 25 including the translational and rotational degrees of freedom. [1] Beylich, A., ``An Interlaced System for Nitrogen Gas,'' Proc. of CECAM Workshop, ENS de Lyon, France, 2000. [2] Cheremisin, F., ``Solution of the Boltzmann Kinetic Equation for High Speed Flows of a Rarefied Gas,'' Proc. of the 24th Int. Symp. on Rarefied Gas Dynamics, Bari, Italy, 2004.

Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation

NASA Astrophysics Data System (ADS)

Lin, Yu-Chu; Wang, Haitao; Wu, Kung-Chien

2018-04-01

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside the finite Mach number region (time like region), we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region (space like region), we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results extend the classical results of Liu and Yu (Commun Pure Appl Math 57:1543-1608, 2004), (Bull Inst Math Acad Sin 1:1-78, 2006), (Bull Inst Math Acad Sin 6:151-243, 2011) and Lee et al. (Commun Math Phys 269:17-37, 2007) to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.

Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation

NASA Astrophysics Data System (ADS)

Lin, Yu-Chu; Wang, Haitao; Wu, Kung-Chien

2018-06-01

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside the finite Mach number region (time like region), we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region (space like region), we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results extend the classical results of Liu and Yu (Commun Pure Appl Math 57:1543-1608, 2004), (Bull Inst Math Acad Sin 1:1-78, 2006), (Bull Inst Math Acad Sin 6:151-243, 2011) and Lee et al. (Commun Math Phys 269:17-37, 2007) to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.

An improved Green's function for ion beam transport

NASA Technical Reports Server (NTRS)

Tweed, J.; Wilson, J. W.; Tripathi, R. K.

2004-01-01

Ion beam transport theory allows testing of material transmission properties in the laboratory environment generated by particle accelerators. This is a necessary step in materials development and evaluation for space use. The approximations used in solving the Boltzmann transport equation for the space setting are often not sufficient for laboratory work and those issues are the main emphasis of the present work. In consequence, an analytic solution of the linear Boltzmann equation is pursued in the form of a Green's function allowing flexibility in application to a broad range of boundary value problems. It has been established that simple solutions can be found for high charge and energy (HZE) ions by ignoring nuclear energy downshifts and dispersion. Such solutions were found to be supported by experimental evidence with HZE ion beams when multiple scattering was added. Lacking from the prior solutions were range and energy straggling and energy downshift with dispersion associated with nuclear events. Recently, we have found global solutions including these effects providing a broader class of HZE ion solutions. c2004 COSPAR. Published by Elsevier Ltd. All rights reserved.

An analysis of numerical convergence in discrete velocity gas dynamics for internal flows

NASA Astrophysics Data System (ADS)

Sekaran, Aarthi; Varghese, Philip; Goldstein, David

2018-07-01

The Discrete Velocity Method (DVM) for solving the Boltzmann equation has significant advantages in the modeling of non-equilibrium and near equilibrium flows as compared to other methods in terms of reduced statistical noise, faster solutions and the ability to handle transient flows. Yet the DVM performance for rarefied flow in complex, small-scale geometries, in microelectromechanical (MEMS) devices for instance, is yet to be studied in detail. The present study focuses on the performance of the DVM for locally large Knudsen number flows of argon around sharp corners and other sources for discontinuities in the distribution function. Our analysis details the nature of the solution for some benchmark cases and introduces the concept of solution convergence for the transport terms in the discrete velocity Boltzmann equation. The limiting effects of the velocity space discretization are also investigated and the constraints on obtaining a robust, consistent solution are derived. We propose techniques to maintain solution convergence and demonstrate the implementation of a specific strategy and its effect on the fidelity of the solution for some benchmark cases.

Unsteady electroosmosis in a microchannel with Poisson-Boltzmann charge distribution.

PubMed

Chang, Chien C; Kuo, Chih-Yu; Wang, Chang-Yi

2011-11-01

The present study is concerned with unsteady electroosmotic flow (EOF) in a microchannel with the electric charge distribution described by the Poisson-Boltzmann (PB) equation. The nonlinear PB equation is solved by a systematic perturbation with respect to the parameter λ which measures the strength of the wall zeta potential relative to the thermal potential. In the small λ limits (λ<1), we recover the linearized PB equation - the Debye-Hückel approximation. The solutions obtained by using only three terms in the perturbation series are shown to be accurate with errors <1% for λ up to 2. The accurate solution to the PB equation is then used to solve the electrokinetic fluid transport equation for two types of unsteady flow: transient flow driven by a suddenly applied voltage and oscillatory flow driven by a time-harmonic voltage. The solution for the transient flow has important implications on EOF as an effective means for transporting electrolytes in microchannels with various electrokinetic widths. On the other hand, the solution for the oscillatory flow is shown to have important physical implications on EOF in mixing electrolytes in terms of the amplitude and phase of the resulting time-harmonic EOF rate, which depends on the applied frequency and the electrokinetic width of the microchannel as well as on the parameter λ. Copyright © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The Fourier transforms for the spatially homogeneous Boltzmann equation and Landau equation

NASA Astrophysics Data System (ADS)

Meng, Fei; Liu, Fang

2018-03-01

In this paper, we study the Fourier transforms for two equations arising in the kinetic theory. The first equation is the spatially homogeneous Boltzmann equation. The Fourier transform of the spatially homogeneous Boltzmann equation has been first addressed by Bobylev (Sov Sci Rev C Math Phys 7:111-233, 1988) in the Maxwellian case. Alexandre et al. (Arch Ration Mech Anal 152(4):327-355, 2000) investigated the Fourier transform of the gain operator for the Boltzmann operator in the cut-off case. Recently, the Fourier transform of the Boltzmann equation is extended to hard or soft potential with cut-off by Kirsch and Rjasanow (J Stat Phys 129:483-492, 2007). We shall first establish the relation between the results in Alexandre et al. (2000) and Kirsch and Rjasanow (2007) for the Fourier transform of the Boltzmann operator in the cut-off case. Then we give the Fourier transform of the spatially homogeneous Boltzmann equation in the non cut-off case. It is shown that our results cover previous works (Bobylev 1988; Kirsch and Rjasanow 2007). The second equation is the spatially homogeneous Landau equation, which can be obtained as a limit of the Boltzmann equation when grazing collisions prevail. Following the method in Kirsch and Rjasanow (2007), we can also derive the Fourier transform for Landau equation.

Two-relaxation-time lattice Boltzmann method for the anisotropic dispersive Henry problem

NASA Astrophysics Data System (ADS)

Servan-Camas, Borja; Tsai, Frank T.-C.

2010-02-01

This study develops a lattice Boltzmann method (LBM) with a two-relaxation-time collision operator (TRT) to cope with anisotropic heterogeneous hydraulic conductivity and anisotropic velocity-dependent hydrodynamic dispersion in the saltwater intrusion problem. The directional-speed-of-sound technique is further developed to address anisotropic hydraulic conductivity and dispersion tensors. Forcing terms are introduced in the LBM to correct numerical errors that arise during the recovery procedure and to describe the sink/source terms in the flow and transport equations. In order to facilitate the LBM implementation, the forcing terms are combined with the equilibrium distribution functions (EDFs) to create pseudo-EDFs. This study performs linear stability analysis and derives LBM stability domains to solve the anisotropic advection-dispersion equation. The stability domains are used to select the time step at which the lattice Boltzmann method provides stable solutions to the numerical examples. The LBM was implemented for the anisotropic dispersive Henry problem with high ratios of longitudinal to transverse dispersivities, and the results compared well to the solutions in the work of Abarca et al. (2007).

Applications of MMPBSA to Membrane Proteins I: Efficient Numerical Solutions of Periodic Poisson-Boltzmann Equation

PubMed Central

Botello-Smith, Wesley M.; Luo, Ray

2016-01-01

Continuum solvent models have been widely used in biomolecular modeling applications. Recently much attention has been given to inclusion of implicit membrane into existing continuum Poisson-Boltzmann solvent models to extend their applications to membrane systems. Inclusion of an implicit membrane complicates numerical solutions of the underlining Poisson-Boltzmann equation due to the dielectric inhomogeneity on the boundary surfaces of a computation grid. This can be alleviated by the use of the periodic boundary condition, a common practice in electrostatic computations in particle simulations. The conjugate gradient and successive over-relaxation methods are relatively straightforward to be adapted to periodic calculations, but their convergence rates are quite low, limiting their applications to free energy simulations that require a large number of conformations to be processed. To accelerate convergence, the Incomplete Cholesky preconditioning and the geometric multi-grid methods have been extended to incorporate periodicity for biomolecular applications. Impressive convergence behaviors were found as in the previous applications of these numerical methods to tested biomolecules and MMPBSA calculations. PMID:26389966

Condensation of monovalent and divalent metal ions on a Langmuir monolayer

NASA Astrophysics Data System (ADS)

Bloch, J. Mati; Yun, Wenbing

1990-01-01

A system that consists of a monolayer spread on a solution containing a monovalent and a divalent ion is investigated. The solution of the Poisson-Boltzmann-Stern equation for this system indicates that the metal ions segregating to the surface can be found in two distinct states. Divalent ions are chemically condensed on the monolayer, while monovalent ions are electrically attracted to it. We derive simple expressions for the charge left on the surfactant monolayer and the amount of metal ions condensed on the monolayer. These formulas reproduce very accurately (to within pro milles) the values obtained using the nonlinear Grahame equation and eliminate the need to solve that equation. That permits a simple identification of the state of the surfactant monolayer and we propose a universal condensation chart that characterizes the state of the surfactant. We further derive a chemical equilibrium equation for the surface components that has considerable range of validity. This equation requires a knowledge of the bulk concentrations only, and thus allows in many cases the identification of the state of the monolayer, avoiding the need to solve the full nonlinear Poisson-Boltzmann equation. All existing experimental results on Langmuir systems are in good agreement with the one-dimensional Poisson-Boltzmann-Stern model with no adjustable parameters. Several of these fits are presented in this work and are also mapped on the condensation chart. Our calculations point to some characteristic differences between the monovalent and the divalent ions that explain why it is possible to build Langmuir-Blodgett multilayers from divalent compensated surfactants but not from monovalent ones.

Renormalization of the fragmentation equation: exact self-similar solutions and turbulent cascades.

PubMed

Saveliev, V L; Gorokhovski, M A

2012-12-01

Using an approach developed earlier for renormalization of the Boltzmann collision integral [Saveliev and Nanbu, Phys. Rev. E 65, 051205 (2002)], we derive an exact divergence form for the fragmentation operator. Then we reduce the fragmentation equation to the continuity equation in size space, with the flux given explicitly. This allows us to obtain self-similar solutions and to find the integral of motion for these solutions (we call it the bare flux). We show how these solutions can be applied as a description of cascade processes in three- and two-dimensional turbulence. We also suggested an empirical cascade model of impact fragmentation of brittle materials.

Kadomtsev−Petviashvili equation for a flow of highly nonisothermal collisionless plasma

DOE Office of Scientific and Technical Information (OSTI.GOV)

Movsesyants, Yu. B., E-mail: yumovsesyants@gmail.com; Rukhadze, A. A., E-mail: rukh@fpl.gpi.ru; Tyuryukanov, P. M.

2016-01-15

It is shown that the equations of two-fluid electrodynamics for a cold ions flow and Boltzmann electrons in the vicinity of the ion-sound point can be reduced to the Kadomtsev−Petviashvili equation. Examples of two-dimensional equilibria with pole singularities obtained by exactly solving the equations are presented. An exact self-similar solution describing a two-dimensional transonic flow and having no pole singularities is found.

Kadomtsev-Petviashvili equation for a flow of highly nonisothermal collisionless plasma

NASA Astrophysics Data System (ADS)

Movsesyants, Yu. B.; Rukhadze, A. A.; Tyuryukanov, P. M.

2016-01-01

It is shown that the equations of two-fluid electrodynamics for a cold ions flow and Boltzmann electrons in the vicinity of the ion-sound point can be reduced to the Kadomtsev-Petviashvili equation. Examples of two-dimensional equilibria with pole singularities obtained by exactly solving the equations are presented. An exact self-similar solution describing a two-dimensional transonic flow and having no pole singularities is found.

On the Maxwellian distribution, symmetric form, and entropy conservation for the Euler equations

NASA Technical Reports Server (NTRS)

Deshpande, S. M.

1986-01-01

The Euler equations of gas dynamics have some very interesting properties in that the flux vector is a homogeneous function of the unknowns and the equations can be cast in symmetric hyperbolic form and satisfy the entropy conservation. The Euler equations are the moments of the Boltzmann equation of the kinetic theory of gases when the velocity distribution function is a Maxwellian. The present paper shows the relationship between the symmetrizability and the Maxwellian velocity distribution. The entropy conservation is in terms of the H-function, which is a slight modification of the H-function first introduced by Boltzmann in his famous H-theorem. In view of the H-theorem, it is suggested that the development of total H-diminishing (THD) numerical methods may be more profitable than the usual total variation diminishing (TVD) methods for obtaining wiggle-free solutions.

Foundations of radiation hydrodynamics

NASA Astrophysics Data System (ADS)

Mihalas, D.; Mihalas, B. W.

This book is the result of an attempt, over the past few years, to gather the basic tools required to do research on radiating flows in astrophysics. The microphysics of gases is discussed, taking into account the equation of state of a perfect gas, the first and second law of thermodynamics, the thermal properties of a perfect gas, the distribution function and Boltzmann's equation, the collision integral, the Maxwellian velocity distribution, Boltzmann's H-theorem, the time of relaxation, and aspects of classical statistical mechanics. Other subjects explored are related to the dynamics of ideal fluids, the dynamics of viscous and heat-conducting fluids, relativistic fluid flow, waves, shocks, winds, radiation and radiative transfer, the equations of radiation hydrodynamics, and radiating flows. Attention is given to small-amplitude disturbances, nonlinear flows, the interaction of radiation and matter, the solution of the transfer equation, acoustic waves, acoustic-gravity waves, basic concepts of special relativity, and equations of motion and energy.

Moving charged particles in lattice Boltzmann-based electrokinetics

NASA Astrophysics Data System (ADS)

Kuron, Michael; Rempfer, Georg; Schornbaum, Florian; Bauer, Martin; Godenschwager, Christian; Holm, Christian; de Graaf, Joost

2016-12-01

The motion of ionic solutes and charged particles under the influence of an electric field and the ensuing hydrodynamic flow of the underlying solvent is ubiquitous in aqueous colloidal suspensions. The physics of such systems is described by a coupled set of differential equations, along with boundary conditions, collectively referred to as the electrokinetic equations. Capuani et al. [J. Chem. Phys. 121, 973 (2004)] introduced a lattice-based method for solving this system of equations, which builds upon the lattice Boltzmann algorithm for the simulation of hydrodynamic flow and exploits computational locality. However, thus far, a description of how to incorporate moving boundary conditions into the Capuani scheme has been lacking. Moving boundary conditions are needed to simulate multiple arbitrarily moving colloids. In this paper, we detail how to introduce such a particle coupling scheme, based on an analogue to the moving boundary method for the pure lattice Boltzmann solver. The key ingredients in our method are mass and charge conservation for the solute species and a partial-volume smoothing of the solute fluxes to minimize discretization artifacts. We demonstrate our algorithm's effectiveness by simulating the electrophoresis of charged spheres in an external field; for a single sphere we compare to the equivalent electro-osmotic (co-moving) problem. Our method's efficiency and ease of implementation should prove beneficial to future simulations of the dynamics in a wide range of complex nanoscopic and colloidal systems that were previously inaccessible to lattice-based continuum algorithms.

Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow

NASA Astrophysics Data System (ADS)

Zheng, Lin; Zheng, Song; Zhai, Qinglan

2016-02-01

In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface force (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter via Cahn-Hilliard equation which is solved in the frame work of LBE. The scalar convection-diffusion equation for temperature field is resolved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two/three dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results.

The optimal modified variational iteration method for the Lane-Emden equations with Neumann and Robin boundary conditions

NASA Astrophysics Data System (ADS)

Singh, Randhir; Das, Nilima; Kumar, Jitendra

2017-06-01

An effective analytical technique is proposed for the solution of the Lane-Emden equations. The proposed technique is based on the variational iteration method (VIM) and the convergence control parameter h . In order to avoid solving a sequence of nonlinear algebraic or complicated integrals for the derivation of unknown constant, the boundary conditions are used before designing the recursive scheme for solution. The series solutions are found which converges rapidly to the exact solution. Convergence analysis and error bounds are discussed. Accuracy, applicability of the method is examined by solving three singular problems: i) nonlinear Poisson-Boltzmann equation, ii) distribution of heat sources in the human head, iii) second-kind Lane-Emden equation.

New variational principles for locating periodic orbits of differential equations.

PubMed

Boghosian, Bruce M; Fazendeiro, Luis M; Lätt, Jonas; Tang, Hui; Coveney, Peter V

2011-06-13

We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier-Stokes equations, simulated using the lattice Boltzmann equation.

Theory of the Lattice Boltzmann Equation: Symmetry properties of Discrete Velocity Sets

NASA Technical Reports Server (NTRS)

Rubinstein, Robert; Luo, Li-Shi

2007-01-01

In the lattice Boltzmann equation, continuous particle velocity space is replaced by a finite dimensional discrete set. The number of linearly independent velocity moments in a lattice Boltzmann model cannot exceed the number of discrete velocities. Thus, finite dimensionality introduces linear dependencies among the moments that do not exist in the exact continuous theory. Given a discrete velocity set, it is important to know to exactly what order moments are free of these dependencies. Elementary group theory is applied to the solution of this problem. It is found that by decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group, it becomes relatively straightforward to assess the behavior of moments in the theory. The construction of some standard two- and three-dimensional models is reviewed from this viewpoint, and procedures for constructing some new higher dimensional models are suggested.

Renormalization of the fragmentation equation: Exact self-similar solutions and turbulent cascades

NASA Astrophysics Data System (ADS)

Saveliev, V. L.; Gorokhovski, M. A.

2012-12-01

Using an approach developed earlier for renormalization of the Boltzmann collision integral [Saveliev and Nanbu, Phys. Rev. E1539-375510.1103/PhysRevE.65.051205 65, 051205 (2002)], we derive an exact divergence form for the fragmentation operator. Then we reduce the fragmentation equation to the continuity equation in size space, with the flux given explicitly. This allows us to obtain self-similar solutions and to find the integral of motion for these solutions (we call it the bare flux). We show how these solutions can be applied as a description of cascade processes in three- and two-dimensional turbulence. We also suggested an empirical cascade model of impact fragmentation of brittle materials.

A paradigm for modeling and computation of gas dynamics

NASA Astrophysics Data System (ADS)

Xu, Kun; Liu, Chang

2017-02-01

In the continuum flow regime, the Navier-Stokes (NS) equations are usually used for the description of gas dynamics. On the other hand, the Boltzmann equation is applied for the rarefied flow. These two equations are based on distinguishable modeling scales for flow physics. Fortunately, due to the scale separation, i.e., the hydrodynamic and kinetic ones, both the Navier-Stokes equations and the Boltzmann equation are applicable in their respective domains. However, in real science and engineering applications, they may not have such a distinctive scale separation. For example, around a hypersonic flying vehicle, the flow physics at different regions may correspond to different regimes, where the local Knudsen number can be changed significantly in several orders of magnitude. With a variation of flow physics, theoretically a continuous governing equation from the kinetic Boltzmann modeling to the hydrodynamic Navier-Stokes dynamics should be used for its efficient description. However, due to the difficulties of a direct modeling of flow physics in the scale between the kinetic and hydrodynamic ones, there is basically no reliable theory or valid governing equations to cover the whole transition regime, except resolving flow physics always down to the mean free path scale, such as the direct Boltzmann solver and the Direct Simulation Monte Carlo (DSMC) method. In fact, it is an unresolved problem about the exact scale for the validity of the NS equations, especially in the small Reynolds number cases. The computational fluid dynamics (CFD) is usually based on the numerical solution of partial differential equations (PDEs), and it targets on the recovering of the exact solution of the PDEs as mesh size and time step converging to zero. This methodology can be hardly applied to solve the multiple scale problem efficiently because there is no such a complete PDE for flow physics through a continuous variation of scales. For the non-equilibrium flow study, the direct modeling methods, such as DSMC, particle in cell, and smooth particle hydrodynamics, play a dominant role to incorporate the flow physics into the algorithm construction directly. It is fully legitimate to combine the modeling and computation together without going through the process of constructing PDEs. In other words, the CFD research is not only to obtain the numerical solution of governing equations but to model flow dynamics as well. This methodology leads to the unified gas-kinetic scheme (UGKS) for flow simulation in all flow regimes. Based on UGKS, the boundary for the validation of the Navier-Stokes equations can be quantitatively evaluated. The combination of modeling and computation provides a paradigm for the description of multiscale transport process.

Assessment of Linear Finite-Difference Poisson-Boltzmann Solvers

PubMed Central

Wang, Jun; Luo, Ray

2009-01-01

CPU time and memory usage are two vital issues that any numerical solvers for the Poisson-Boltzmann equation have to face in biomolecular applications. In this study we systematically analyzed the CPU time and memory usage of five commonly used finite-difference solvers with a large and diversified set of biomolecular structures. Our comparative analysis shows that modified incomplete Cholesky conjugate gradient and geometric multigrid are the most efficient in the diversified test set. For the two efficient solvers, our test shows that their CPU times increase approximately linearly with the numbers of grids. Their CPU times also increase almost linearly with the negative logarithm of the convergence criterion at very similar rate. Our comparison further shows that geometric multigrid performs better in the large set of tested biomolecules. However, modified incomplete Cholesky conjugate gradient is superior to geometric multigrid in molecular dynamics simulations of tested molecules. We also investigated other significant components in numerical solutions of the Poisson-Boltzmann equation. It turns out that the time-limiting step is the free boundary condition setup for the linear systems for the selected proteins if the electrostatic focusing is not used. Thus, development of future numerical solvers for the Poisson-Boltzmann equation should balance all aspects of the numerical procedures in realistic biomolecular applications. PMID:20063271

Boltzmann-type control of opinion consensus through leaders

PubMed Central

Albi, G.; Pareschi, L.; Zanella, M.

2014-01-01

The study of formations and dynamics of opinions leading to the so-called opinion consensus is one of the most important areas in mathematical modelling of social sciences. Following the Boltzmann-type control approach recently introduced by the first two authors, we consider a group of opinion leaders who modify their strategy accordingly to an objective functional with the aim of achieving opinion consensus. The main feature of the Boltzmann-type control is that, owing to an instantaneous binary control formulation, it permits the minimization of the cost functional to be embedded into the microscopic leaders’ interactions of the corresponding Boltzmann equation. The related Fokker–Planck asymptotic limits are also derived, which allow one to give explicit expressions of stationary solutions. The results demonstrate the validity of the Boltzmann-type control approach and the capability of the leaders’ control to strategically lead the followers’ opinion. PMID:25288820

Comparison of fluid neutral models for one-dimensional plasma edge modeling with a finite volume solution of the Boltzmann equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Horsten, N., E-mail: niels.horsten@kuleuven.be; Baelmans, M.; Dekeyser, W.

2016-01-15

We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assumingmore » equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation.« less

Brownian motion from Boltzmann's equation.

NASA Technical Reports Server (NTRS)

Montgomery, D.

1971-01-01

Two apparently disparate lines of inquiry in kinetic theory are shown to be equivalent: (1) Brownian motion as treated by the (stochastic) Langevin equation and Fokker-Planck equation; and (2) Boltzmann's equation. The method is to derive the kinetic equation for Brownian motion from the Boltzmann equation for a two-component neutral gas by a simultaneous expansion in the density and mass ratios.

Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mishra, Subhash C.; Roy, Hillol K.

2007-04-10

The lattice Boltzmann method (LBM) was used to solve the energy equation of a transient conduction-radiation heat transfer problem. The finite volume method (FVM) was used to compute the radiative information. To study the compatibility of the LBM for the energy equation and the FVM for the radiative transfer equation, transient conduction and radiation heat transfer problems in 1-D planar and 2-D rectangular geometries were considered. In order to establish the suitability of the LBM, the energy equations of the two problems were also solved using the FVM of the computational fluid dynamics. The FVM used in the radiative heatmore » transfer was employed to compute the radiative information required for the solution of the energy equation using the LBM or the FVM (of the CFD). To study the compatibility and suitability of the LBM for the solution of energy equation and the FVM for the radiative information, results were analyzed for the effects of various parameters such as the scattering albedo, the conduction-radiation parameter and the boundary emissivity. The results of the LBM-FVM combination were found to be in excellent agreement with the FVM-FVM combination. The number of iterations and CPU times in both the combinations were found comparable.« less

The linear Boltzmann equation in slab geometry - Development and verification of a reliable and efficient solution

NASA Technical Reports Server (NTRS)

Stamnes, K.; Lie-Svendsen, O.; Rees, M. H.

1991-01-01

The linear Boltzmann equation can be cast in a form mathematically identical to the radiation-transport equation. A multigroup procedure is used to reduce the energy (or velocity) dependence of the transport equation to a series of one-speed problems. Each of these one-speed problems is equivalent to the monochromatic radiative-transfer problem, and existing software is used to solve this problem in slab geometry. The numerical code conserves particles in elastic collisions. Generic examples are provided to illustrate the applicability of this approach. Although this formalism can, in principle, be applied to a variety of test particle or linearized gas dynamics problems, it is particularly well-suited to study the thermalization of suprathermal particles interacting with a background medium when the thermal motion of the background cannot be ignored. Extensions of the formalism to include external forces and spherical geometry are also feasible.

Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics

NASA Astrophysics Data System (ADS)

Farrell, Patricio; Koprucki, Thomas; Fuhrmann, Jürgen

2017-10-01

We compare three thermodynamically consistent numerical fluxes known in the literature, appearing in a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics. Our discussion includes an extension of the Scharfetter-Gummel scheme to non-Boltzmann (e.g. Fermi-Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

DL_MG: A Parallel Multigrid Poisson and Poisson-Boltzmann Solver for Electronic Structure Calculations in Vacuum and Solution.

PubMed

Womack, James C; Anton, Lucian; Dziedzic, Jacek; Hasnip, Phil J; Probert, Matt I J; Skylaris, Chris-Kriton

2018-03-13

The solution of the Poisson equation is a crucial step in electronic structure calculations, yielding the electrostatic potential-a key component of the quantum mechanical Hamiltonian. In recent decades, theoretical advances and increases in computer performance have made it possible to simulate the electronic structure of extended systems in complex environments. This requires the solution of more complicated variants of the Poisson equation, featuring nonhomogeneous dielectric permittivities, ionic concentrations with nonlinear dependencies, and diverse boundary conditions. The analytic solutions generally used to solve the Poisson equation in vacuum (or with homogeneous permittivity) are not applicable in these circumstances, and numerical methods must be used. In this work, we present DL_MG, a flexible, scalable, and accurate solver library, developed specifically to tackle the challenges of solving the Poisson equation in modern large-scale electronic structure calculations on parallel computers. Our solver is based on the multigrid approach and uses an iterative high-order defect correction method to improve the accuracy of solutions. Using two chemically relevant model systems, we tested the accuracy and computational performance of DL_MG when solving the generalized Poisson and Poisson-Boltzmann equations, demonstrating excellent agreement with analytic solutions and efficient scaling to ∼10 9 unknowns and 100s of CPU cores. We also applied DL_MG in actual large-scale electronic structure calculations, using the ONETEP linear-scaling electronic structure package to study a 2615 atom protein-ligand complex with routinely available computational resources. In these calculations, the overall execution time with DL_MG was not significantly greater than the time required for calculations using a conventional FFT-based solver.

AQUEOUS PROTONATION PROPERTIES OF AMPHOTERIC NANOPARTICLES

EPA Science Inventory

A divergence is predicted between the acidity behavior of charged sites on micron sized colloidal particles and nanoparticles. Utilizing the approximate analytical solution to the Poisson-Boltzmann equation published by Ohshima et al. (1982), findings from the work included: 1):...

A new Green's function Monte Carlo algorithm for the solution of the two-dimensional nonlinear Poisson–Boltzmann equation: Application to the modeling of the communication breakdown problem in space vehicles during re-entry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chatterjee, Kausik, E-mail: kausik.chatterjee@aggiemail.usu.edu; Center for Atmospheric and Space Sciences, Utah State University, Logan, UT 84322; Roadcap, John R., E-mail: john.roadcap@us.af.mil

The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals ofmore » the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.« less

A new Green's function Monte Carlo algorithm for the solution of the two-dimensional nonlinear Poisson-Boltzmann equation: Application to the modeling of the communication breakdown problem in space vehicles during re-entry

NASA Astrophysics Data System (ADS)

Chatterjee, Kausik; Roadcap, John R.; Singh, Surendra

2014-11-01

The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson-Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.

Comparison of the Nernst-Planck model and the Poisson-Boltzmann model for electroosmotic flows in microchannels.

PubMed

Park, H M; Lee, J S; Kim, T W

2007-11-15

In the analysis of electroosmotic flows, the internal electric potential is usually modeled by the Poisson-Boltzmann equation. The Poisson-Boltzmann equation is derived from the assumption of thermodynamic equilibrium where the ionic distributions are not affected by fluid flows. Although this is a reasonable assumption for steady electroosmotic flows through straight microchannels, there are some important cases where convective transport of ions has nontrivial effects. In these cases, it is necessary to adopt the Nernst-Planck equation instead of the Poisson-Boltzmann equation to model the internal electric field. In the present work, the predictions of the Nernst-Planck equation are compared with those of the Poisson-Boltzmann equation for electroosmotic flows in various microchannels where the convective transport of ions is not negligible.

Stability: Conservation laws, Painlevé analysis and exact solutions for S-KP equation in coupled dusty plasma

NASA Astrophysics Data System (ADS)

EL-Kalaawy, O. H.; Moawad, S. M.; Wael, Shrouk

The propagation of nonlinear waves in unmagnetized strongly coupled dusty plasma with Boltzmann distributed electrons, iso-nonthermal distributed ions and negatively charged dust grains is considered. The basic set of fluid equations is reduced to the Schamel Kadomtsev-Petviashvili (S-KP) equation by using the reductive perturbation method. The variational principle and conservation laws of S-KP equation are obtained. It is shown that the S-KP equation is non-integrable using Painlevé analysis. A set of new exact solutions are obtained by auto-Bäcklund transformations. The stability analysis is discussed for the existence of dust acoustic solitary waves (DASWs) and it is found that the physical parameters have strong effects on the stability criterion. In additional to, the electric field and the true Mach number of this solution are investigated. Finally, we will study the physical meanings of solutions.

The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, III: The 3-D Boltzmann equation

NASA Astrophysics Data System (ADS)

Yin, Huicheng; Zhao, Wenbin

2018-01-01

This paper is a continuation of the works in [35] and [37], where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In the present paper, we will confirm this physical phenomenon for the Boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function.

Numerical study of active control of mixing in electro-osmotic flows by temperature difference using lattice Boltzmann methods.

PubMed

Alizadeh, A; Wang, J K; Pooyan, S; Mirbozorgi, S A; Wang, M

2013-10-01

In this paper, the effect of temperature difference between inlet flow and walls on the electro-osmotic flow through a two-dimensional microchannel is investigated. The main objective is to study the effect of temperature variations on the distribution of ions and consequently internal electric potential field, electric body force, and velocity fields in an electro-osmotic flow. We assume constant temperature and zeta potential on walls and use the mean temperature of each cross section to characterize the Boltzmann ion distribution across the channel. Based on these assumptions, the multiphysical transports are still able to be described by the classical Poisson-Boltzmann model. In this work, the Navier-Stokes equation for fluid flow, the Poisson-Boltzmann equation for ion distribution, and the energy equation for heat transfer are solved by a couple lattice Boltzmann method. The modeling results indicate that the temperature difference between walls and the inlet solution may lead to two symmetrical vortices at the entrance region of the microchannel which is appropriate for mixing enhancements. The advantage of this phenomenon for active control of mixing in electro-osmotic flow is the manageability of the vortex scale without extra efforts. For instance, the effective domain of this pattern could broaden by the following modulations: decreasing the external electric potential field, decreasing the electric double layer thickness, or increasing the temperature difference between inlet flow and walls. This work may provide a novel strategy for design or optimization of microsystems. Copyright © 2013 Elsevier Inc. All rights reserved.

Hybrid simulation combining two space-time discretization of the discrete-velocity Boltzmann equation

NASA Astrophysics Data System (ADS)

Horstmann, Jan Tobias; Le Garrec, Thomas; Mincu, Daniel-Ciprian; Lévêque, Emmanuel

2017-11-01

Despite the efficiency and low dissipation of the stream-collide scheme of the discrete-velocity Boltzmann equation, which is nowadays implemented in many lattice Boltzmann solvers, a major drawback exists over alternative discretization schemes, i.e. finite-volume or finite-difference, that is the limitation to Cartesian uniform grids. In this paper, an algorithm is presented that combines the positive features of each scheme in a hybrid lattice Boltzmann method. In particular, the node-based streaming of the distribution functions is coupled with a second-order finite-volume discretization of the advection term of the Boltzmann equation under the Bhatnagar-Gross-Krook approximation. The algorithm is established on a multi-domain configuration, with the individual schemes being solved on separate sub-domains and connected by an overlapping interface of at least 2 grid cells. A critical parameter in the coupling is the CFL number equal to unity, which is imposed by the stream-collide algorithm. Nevertheless, a semi-implicit treatment of the collision term in the finite-volume formulation allows us to obtain a stable solution for this condition. The algorithm is validated in the scope of three different test cases on a 2D periodic mesh. It is shown that the accuracy of the combined discretization schemes agrees with the order of each separate scheme involved. The overall numerical error of the hybrid algorithm in the macroscopic quantities is contained between the error of the two individual algorithms. Finally, we demonstrate how such a coupling can be used to adapt to anisotropic flows with some gradual mesh refinement in the FV domain.

Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation

PubMed Central

Yang, Jaw-Yen; Yan, Chin-Yuan; Huang, Juan-Chen; Li, Zhihui

2014-01-01

Computations of rarefied gas dynamical flows governed by the semiclassical Boltzmann ellipsoidal-statistical (ES) kinetic model equation using an accurate numerical method are presented. The semiclassical ES model was derived through the maximum entropy principle and conserves not only the mass, momentum and energy, but also contains additional higher order moments that differ from the standard quantum distributions. A different decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. The numerical method in phase space combines the discrete-ordinate method in momentum space and the high-resolution shock capturing method in physical space. Numerical solutions of two-dimensional Riemann problems for two configurations covering various degrees of rarefaction are presented and various contours of the quantities unique to this new model are illustrated. When the relaxation time becomes very small, the main flow features a display similar to that of ideal quantum gas dynamics, and the present solutions are found to be consistent with existing calculations for classical gas. The effect of a parameter that permits an adjustable Prandtl number in the flow is also studied. PMID:25104904

A quantum mechanical-Poisson-Boltzmann equation approach for studying charge flow between ions and a dielectric continuum

NASA Astrophysics Data System (ADS)

Gogonea, Valentin; Merz, Kenneth M.

2000-02-01

This paper presents a theoretical model for the investigation of charge transfer between ions and a solvent treated as a dielectric continuum media. The method is a combination of a semiempirical effective Hamiltonian with a modified Poisson-Boltzmann equation which includes charge transfer in the form of a surface charge density positioned at the dielectric interface. The new Poisson-Boltzmann equation together with new boundary conditions results in a new set of equations for the electrostatic potential (or polarization charge densities). Charge transfer adds a new free energy component to the solvation free energy term, which accounts for all interactions between the transferred charge at the dielectric interface, the solute wave function and the solvent polarization charges. Practical calculations on a set of 19 anions and 17 cations demonstrate that charge exchange with a dielectric is present and it is in the range of 0.06-0.4 eu. Furthermore, the pattern of the magnitudes of charge transfer can be related to the acid-base properties of the ions in many cases, but exceptions are also found. Finally, we show that the method leads to an energy decomposition scheme of the total electrostatic energy, which can be used in mechanistic studies on protein and DNA interaction with water.

Non-Lorentzian ion cyclotron resonance line shapes arising from velocity-dependent ion-neutral collision frequencies

NASA Technical Reports Server (NTRS)

Whealton, J. H.; Mason, E. A.

1973-01-01

An asymptotic solution of the Boltzmann equation is developed for ICR absorption, without restrictions on the ion-neutral collision frequency or mass ratio. Velocity dependence of the collision frequency causes deviations from Lorentzian line shape.

An Improved Neutron Transport Algorithm for Space Radiation

NASA Technical Reports Server (NTRS)

Heinbockel, John H.; Clowdsley, Martha S.; Wilson, John W.

2000-01-01

A low-energy neutron transport algorithm for use in space radiation protection is developed. The algorithm is based upon a multigroup analysis of the straight-ahead Boltzmann equation by using a mean value theorem for integrals. This analysis is accomplished by solving a realistic but simplified neutron transport test problem. The test problem is analyzed by using numerical and analytical procedures to obtain an accurate solution within specified error bounds. Results from the test problem are then used for determining mean values associated with rescattering terms that are associated with a multigroup solution of the straight-ahead Boltzmann equation. The algorithm is then coupled to the Langley HZETRN code through the evaporation source term. Evaluation of the neutron fluence generated by the solar particle event of February 23, 1956, for a water and an aluminum-water shield-target configuration is then compared with LAHET and MCNPX Monte Carlo code calculations for the same shield-target configuration. The algorithm developed showed a great improvement in results over the unmodified HZETRN solution. In addition, a two-directional solution of the evaporation source showed even further improvement of the fluence near the front of the water target where diffusion from the front surface is important.

Simulations of reactive transport and precipitation with smoothed particle hydrodynamics

NASA Astrophysics Data System (ADS)

Tartakovsky, Alexandre M.; Meakin, Paul; Scheibe, Timothy D.; Eichler West, Rogene M.

2007-03-01

A numerical model based on smoothed particle hydrodynamics (SPH) was developed for reactive transport and mineral precipitation in fractured and porous materials. Because of its Lagrangian particle nature, SPH has several advantages for modeling Navier-Stokes flow and reactive transport including: (1) in a Lagrangian framework there is no non-linear term in the momentum conservation equation, so that accurate solutions can be obtained for momentum dominated flows and; (2) complicated physical and chemical processes such as surface growth due to precipitation/dissolution and chemical reactions are easy to implement. In addition, SPH simulations explicitly conserve mass and linear momentum. The SPH solution of the diffusion equation with fixed and moving reactive solid-fluid boundaries was compared with analytical solutions, Lattice Boltzmann [Q. Kang, D. Zhang, P. Lichtner, I. Tsimpanogiannis, Lattice Boltzmann model for crystal growth from supersaturated solution, Geophysical Research Letters, 31 (2004) L21604] simulations and diffusion limited aggregation (DLA) [P. Meakin, Fractals, scaling and far from equilibrium. Cambridge University Press, Cambridge, UK, 1998] model simulations. To illustrate the capabilities of the model, coupled three-dimensional flow, reactive transport and precipitation in a fracture aperture with a complex geometry were simulated.

Variational method enabling simplified solutions to the linearized Boltzmann equation for oscillatory gas flows

NASA Astrophysics Data System (ADS)

Ladiges, Daniel R.; Sader, John E.

2018-05-01

Nanomechanical resonators and sensors, operated in ambient conditions, often generate low-Mach-number oscillating rarefied gas flows. Cercignani [C. Cercignani, J. Stat. Phys. 1, 297 (1969), 10.1007/BF01007482] proposed a variational principle for the linearized Boltzmann equation, which can be used to derive approximate analytical solutions of steady (time-independent) flows. Here we extend and generalize this principle to unsteady oscillatory rarefied flows and thus accommodate resonating nanomechanical devices. This includes a mathematical approach that facilitates its general use and allows for systematic improvements in accuracy. This formulation is demonstrated for two canonical flow problems: oscillatory Couette flow and Stokes' second problem. Approximate analytical formulas giving the bulk velocity and shear stress, valid for arbitrary oscillation frequency, are obtained for Couette flow. For Stokes' second problem, a simple system of ordinary differential equations is derived which may be solved to obtain the desired flow fields. Using this framework, a simple and accurate formula is provided for the shear stress at the oscillating boundary, again for arbitrary frequency, which may prove useful in application. These solutions are easily implemented on any symbolic or numerical package, such as Mathematica or matlab, facilitating the characterization of flows produced by nanomechanical devices and providing insight into the underlying flow physics.

Benchmark calculations of nonconservative charged-particle swarms in dc electric and magnetic fields crossed at arbitrary angles.

PubMed

Dujko, S; White, R D; Petrović, Z Lj; Robson, R E

2010-04-01

A multiterm solution of the Boltzmann equation has been developed and used to calculate transport coefficients of charged-particle swarms in gases under the influence of electric and magnetic fields crossed at arbitrary angles when nonconservative collisions are present. The hierarchy resulting from a spherical-harmonic decomposition of the Boltzmann equation in the hydrodynamic regime is solved numerically by representing the speed dependence of the phase-space distribution function in terms of an expansion in Sonine polynomials about a Maxwellian velocity distribution at an internally determined temperature. Results are given for electron swarms in certain collisional models for ionization and attachment over a range of angles between the fields and field strengths. The implicit and explicit effects of ionization and attachment on the electron-transport coefficients are considered using physical arguments. It is found that the difference between the two sets of transport coefficients, bulk and flux, resulting from the explicit effects of nonconservative collisions, can be controlled either by the variation in the magnetic field strengths or by the angles between the fields. In addition, it is shown that the phenomena of ionization cooling and/or attachment cooling/heating previously reported for dc electric fields carry over directly to the crossed electric and magnetic fields. The results of the Boltzmann equation analysis are compared with those obtained by a Monte Carlo simulation technique. The comparison confirms the theoretical basis and numerical integrity of the moment method for solving the Boltzmann equation and gives a set of well-established data that can be used to test future codes and plasma models.

Kinetic analysis of thermally relativistic flow with dissipation. II. Relativistic Boltzmann equation versus its kinetic models

NASA Astrophysics Data System (ADS)

Yano, Ryosuke; Matsumoto, Jun; Suzuki, Kojiro

2011-06-01

Thermally relativistic flow with dissipation was analyzed by solving the rarefied supersonic flow of thermally relativistic matter around a triangle prism by Yano and Suzuki [Phys. Rev. DPRVDAQ1550-7998 83, 023517 (2011)10.1103/PhysRevD.83.023517], where the Anderson-Witting (AW) model was used as a solver. In this paper, we solve the same problem, which was analyzed by Yano and Suzuki, using the relativistic Boltzmann equation (RBE). To solve the RBE, the conventional direct simulation Monte Carlo method for the nonrelativistic Boltzmann equation is extended to a new direct simulation Monte Carlo method for the RBE. Additionally, we solve the modified Marle (MM) model proposed by Yano-Suzuki-Kuroda for comparisons. The solution of the thermally relativistic shock layer around the triangle prism obtained using the relativistic Boltzmann equation is considered by focusing on profiles of macroscopic quantities, such as the density, velocity, temperature, heat flux and dynamic pressure along the stagnation streamline (SSL). Differences among profiles of the number density, velocity and temperature along the SSL obtained using the RBE, the AW and MM. models are described in the framework of the relativistic Navier-Stokes-Fourier law. Finally, distribution functions on the SSL obtained using the RBE are compared with those obtained using the AW and MM models. The distribution function inside the shock wave obtained using the RBE does not indicate a bimodal form, which is obtained using the AW and MM models, but a smooth deceleration of thermally relativistic matter inside a shock wave.

Electric Conductivity in a Beam, Plasma System.

DTIC Science & Technology

1977-09-15

Green ’s function solution to the Boltzmann equation and arrived at a stationary state. However Balescu has accounted for the potential energy of...R. Balescu , Statistical Mechanics of Charged Particles , (In terscience Publishers , New York , 1963) 21. P.M. Morse and H. Feshbach, Methods of

Rectification of thermal fluctuations in ideal gases.

PubMed

Meurs, P; Van den Broeck, C; Garcia, A

2004-11-01

We calculate the systematic average speed of the adiabatic piston and a thermal Brownian motor, introduced by C. Van den Broeck, R, Kawai and P. Meurs [Phys. Rev. Lett. 93, 090601 (2004)], by an expansion of the Boltzmann equation and compare with the exact numerical solution.

Interpolation methods and the accuracy of lattice-Boltzmann mesh refinement

DOE PAGES

Guzik, Stephen M.; Weisgraber, Todd H.; Colella, Phillip; ...

2013-12-10

A lattice-Boltzmann model to solve the equivalent of the Navier-Stokes equations on adap- tively refined grids is presented. A method for transferring information across interfaces between different grid resolutions was developed following established techniques for finite- volume representations. This new approach relies on a space-time interpolation and solving constrained least-squares problems to ensure conservation. The effectiveness of this method at maintaining the second order accuracy of lattice-Boltzmann is demonstrated through a series of benchmark simulations and detailed mesh refinement studies. These results exhibit smaller solution errors and improved convergence when compared with similar approaches relying only on spatial interpolation. Examplesmore » highlighting the mesh adaptivity of this method are also provided.« less

The electric double layer at a metal electrode in pure water

NASA Astrophysics Data System (ADS)

Brüesch, Peter; Christen, Thomas

2004-03-01

Pure water is a weak electrolyte that dissociates into hydronium ions and hydroxide ions. In contact with a charged electrode a double layer forms for which neither experimental nor theoretical studies exist, in contrast to electrolytes containing extrinsic ions like acids, bases, and solute salts. Starting from a self-consistent solution of the one-dimensional modified Poisson-Boltzmann equation, which takes into account activity coefficients of point-like ions, we explore the properties of the electric double layer by successive incorporation of various correction terms like finite ion size, polarization, image charge, and field dissociation. We also discuss the effect of the usual approximation of an average potential as required for the one-dimensional Poisson-Boltzmann equation, and conclude that the one-dimensional approximation underestimates the ion density. We calculate the electric potential, the ion distributions, the pH-values, the ion-size corrected activity coefficients, and the dissociation constants close to the electric double layer and compare the results for the various model corrections.

Navier-Stokes Dynamics by a Discrete Boltzmann Model

NASA Technical Reports Server (NTRS)

Rubinstein, Robet

2010-01-01

This work investigates the possibility of particle-based algorithms for the Navier-Stokes equations and higher order continuum approximations of the Boltzmann equation; such algorithms would generalize the well-known Pullin scheme for the Euler equations. One such method is proposed in the context of a discrete velocity model of the Boltzmann equation. Preliminary results on shock structure are consistent with the expectation that the shock should be much broader than the near discontinuity predicted by the Pullin scheme, yet narrower than the prediction of the Boltzmann equation. We discuss the extension of this essentially deterministic method to a stochastic particle method that, like DSMC, samples the distribution function rather than resolving it completely.

Dielectric Boundary Force in Molecular Solvation with the Poisson–Boltzmann Free Energy: A Shape Derivative Approach

PubMed Central

Li, Bo; Cheng, Xiaoliang; Zhang, Zhengfang

2013-01-01

In an implicit-solvent description of molecular solvation, the electrostatic free energy is given through the electrostatic potential. This potential solves a boundary-value problem of the Poisson–Boltzmann equation in which the dielectric coefficient changes across the solute-solvent interface—the dielectric boundary. The dielectric boundary force acting on such a boundary is the negative first variation of the electrostatic free energy with respect to the location change of the boundary. In this work, the concept of shape derivative is used to define such variations and formulas of the dielectric boundary force are derived. It is shown that such a force is always in the direction toward the charged solute molecules. PMID:24058212

Solitons of the Kadomtsev-Petviashvili equation based on lattice Boltzmann model

NASA Astrophysics Data System (ADS)

Wang, Huimin

2017-01-01

In this paper, a lattice Boltzmann model for the Kadomtsev-Petviashvili equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales are obtained. Due to the asymmetry in x direction and y direction of the equation, the moments of the equilibrium distribution function are selected are asymmetric. The numerical results demonstrate the lattice Boltzmann method is an effective method to simulate the solitons of the Kadomtsev-Petviashvili equation.

Trapping hydrogen atoms from a neon-gas matrix: a theoretical simulation.

PubMed

Bovino, S; Zhang, P; Kharchenko, V; Dalgarno, A

2009-08-07

Hydrogen is of critical importance in atomic and molecular physics and the development of a simple and efficient technique for trapping cold and ultracold hydrogen atoms would be a significant advance. In this study we simulate a recently proposed trap-loading mechanism for trapping hydrogen atoms released from a neon matrix. Accurate ab initio quantum calculations are reported of the neon-hydrogen interaction potential and the energy- and angular-dependent elastic scattering cross sections that control the energy transfer of initially cold atoms are obtained. They are then used to construct the Boltzmann kinetic equation, describing the energy relaxation process. Numerical solutions of the Boltzmann equation predict the time evolution of the hydrogen energy distribution function. Based on the simulations we discuss the prospects of the technique.

Boltzmann equations for a binary one-dimensional ideal gas.

PubMed

Boozer, A D

2011-09-01

We consider a time-reversal invariant dynamical model of a binary ideal gas of N molecules in one spatial dimension. By making time-asymmetric assumptions about the behavior of the gas, we derive Boltzmann and anti-Boltzmann equations that describe the evolution of the single-molecule velocity distribution functions for an ensemble of such systems. We show that for a special class of initial states of the ensemble one can obtain an exact expression for the N-molecule velocity distribution function, and we use this expression to rigorously prove that the time-asymmetric assumptions needed to derive the Boltzmann and anti-Boltzmann equations hold in the limit of large N. Our results clarify some subtle issues regarding the origin of the time asymmetry of Boltzmann's H theorem.

Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution.

PubMed

Lu, Benzhuo; Zhou, Y C; Huber, Gary A; Bond, Stephen D; Holst, Michael J; McCammon, J Andrew

2007-10-07

A computational framework is presented for the continuum modeling of cellular biomolecular diffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of (adaptive) finite element and boundary element methods is adopted to solve the Smoluchowski equation (SE), the Poisson equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equations solutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.

Kinetic solvers with adaptive mesh in phase space

NASA Astrophysics Data System (ADS)

Arslanbekov, Robert R.; Kolobov, Vladimir I.; Frolova, Anna A.

2013-12-01

An adaptive mesh in phase space (AMPS) methodology has been developed for solving multidimensional kinetic equations by the discrete velocity method. A Cartesian mesh for both configuration (r) and velocity (v) spaces is produced using a “tree of trees” (ToT) data structure. The r mesh is automatically generated around embedded boundaries, and is dynamically adapted to local solution properties. The v mesh is created on-the-fly in each r cell. Mappings between neighboring v-space trees is implemented for the advection operator in r space. We have developed algorithms for solving the full Boltzmann and linear Boltzmann equations with AMPS. Several recent innovations were used to calculate the discrete Boltzmann collision integral with dynamically adaptive v mesh: the importance sampling, multipoint projection, and variance reduction methods. We have developed an efficient algorithm for calculating the linear Boltzmann collision integral for elastic and inelastic collisions of hot light particles in a Lorentz gas. Our AMPS technique has been demonstrated for simulations of hypersonic rarefied gas flows, ion and electron kinetics in weakly ionized plasma, radiation and light-particle transport through thin films, and electron streaming in semiconductors. We have shown that AMPS allows minimizing the number of cells in phase space to reduce the computational cost and memory usage for solving challenging kinetic problems.

Kinetic solvers with adaptive mesh in phase space.

PubMed

Arslanbekov, Robert R; Kolobov, Vladimir I; Frolova, Anna A

2013-12-01

An adaptive mesh in phase space (AMPS) methodology has been developed for solving multidimensional kinetic equations by the discrete velocity method. A Cartesian mesh for both configuration (r) and velocity (v) spaces is produced using a "tree of trees" (ToT) data structure. The r mesh is automatically generated around embedded boundaries, and is dynamically adapted to local solution properties. The v mesh is created on-the-fly in each r cell. Mappings between neighboring v-space trees is implemented for the advection operator in r space. We have developed algorithms for solving the full Boltzmann and linear Boltzmann equations with AMPS. Several recent innovations were used to calculate the discrete Boltzmann collision integral with dynamically adaptive v mesh: the importance sampling, multipoint projection, and variance reduction methods. We have developed an efficient algorithm for calculating the linear Boltzmann collision integral for elastic and inelastic collisions of hot light particles in a Lorentz gas. Our AMPS technique has been demonstrated for simulations of hypersonic rarefied gas flows, ion and electron kinetics in weakly ionized plasma, radiation and light-particle transport through thin films, and electron streaming in semiconductors. We have shown that AMPS allows minimizing the number of cells in phase space to reduce the computational cost and memory usage for solving challenging kinetic problems.

From the Boltzmann to the Lattice-Boltzmann Equation:. Beyond BGK Collision Models

NASA Astrophysics Data System (ADS)

Philippi, Paulo Cesar; Hegele, Luiz Adolfo; Surmas, Rodrigo; Siebert, Diogo Nardelli; Dos Santos, Luís Orlando Emerich

In this work, we present a derivation for the lattice-Boltzmann equation directly from the linearized Boltzmann equation, combining the following main features: multiple relaxation times and thermodynamic consistency in the description of non isothermal compressible flows. The method presented here is based on the discretization of increasingly order kinetic models of the Boltzmann equation. Following a Gross-Jackson procedure, the linearized collision term is developed in Hermite polynomial tensors and the resulting infinite series is diagonalized after a chosen integer N, establishing the order of approximation of the collision term. The velocity space is discretized, in accordance with a quadrature method based on prescribed abscissas (Philippi et al., Phys. Rev E 73, 056702, 2006). The problem of describing the energy transfer is discussed, in relation with the order of approximation of a two relaxation-times lattice Boltzmann model. The velocity-step, temperature-step and the shock tube problems are investigated, adopting lattices with 37, 53 and 81 velocities.

Unified implicit kinetic scheme for steady multiscale heat transfer based on the phonon Boltzmann transport equation

NASA Astrophysics Data System (ADS)

Zhang, Chuang; Guo, Zhaoli; Chen, Songze

2017-12-01

An implicit kinetic scheme is proposed to solve the stationary phonon Boltzmann transport equation (BTE) for multiscale heat transfer problem. Compared to the conventional discrete ordinate method, the present method employs a macroscopic equation to accelerate the convergence in the diffusive regime. The macroscopic equation can be taken as a moment equation for phonon BTE. The heat flux in the macroscopic equation is evaluated from the nonequilibrium distribution function in the BTE, while the equilibrium state in BTE is determined by the macroscopic equation. These two processes exchange information from different scales, such that the method is applicable to the problems with a wide range of Knudsen numbers. Implicit discretization is implemented to solve both the macroscopic equation and the BTE. In addition, a memory reduction technique, which is originally developed for the stationary kinetic equation, is also extended to phonon BTE. Numerical comparisons show that the present scheme can predict reasonable results both in ballistic and diffusive regimes with high efficiency, while the memory requirement is on the same order as solving the Fourier law of heat conduction. The excellent agreement with benchmark and the rapid converging history prove that the proposed macro-micro coupling is a feasible solution to multiscale heat transfer problems.

Rarefied gas flow simulations using high-order gas-kinetic unified algorithms for Boltzmann model equations

NASA Astrophysics Data System (ADS)

Li, Zhi-Hui; Peng, Ao-Ping; Zhang, Han-Xin; Yang, Jaw-Yen

2015-04-01

This article reviews rarefied gas flow computations based on nonlinear model Boltzmann equations using deterministic high-order gas-kinetic unified algorithms (GKUA) in phase space. The nonlinear Boltzmann model equations considered include the BGK model, the Shakhov model, the Ellipsoidal Statistical model and the Morse model. Several high-order gas-kinetic unified algorithms, which combine the discrete velocity ordinate method in velocity space and the compact high-order finite-difference schemes in physical space, are developed. The parallel strategies implemented with the accompanying algorithms are of equal importance. Accurate computations of rarefied gas flow problems using various kinetic models over wide ranges of Mach numbers 1.2-20 and Knudsen numbers 0.0001-5 are reported. The effects of different high resolution schemes on the flow resolution under the same discrete velocity ordinate method are studied. A conservative discrete velocity ordinate method to ensure the kinetic compatibility condition is also implemented. The present algorithms are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the steady normal shock wave structures for different Mach numbers, the two-dimensional flows past a circular cylinder and a NACA 0012 airfoil to verify the present methodology and to simulate gas transport phenomena covering various flow regimes. Illustrations of large scale parallel computations of three-dimensional hypersonic rarefied flows over the reusable sphere-cone satellite and the re-entry spacecraft using almost the largest computer systems available in China are also reported. The present computed results are compared with the theoretical prediction from gas dynamics, related DSMC results, slip N-S solutions and experimental data, and good agreement can be found. The numerical experience indicates that although the direct model Boltzmann equation solver in phase space can be computationally expensive, nevertheless, the present GKUAs for kinetic model Boltzmann equations in conjunction with current available high-performance parallel computer power can provide a vital engineering tool for analyzing rarefied gas flows covering the whole range of flow regimes in aerospace engineering applications.

Properties of the Boltzmann equation in the classical approximation

DOE PAGES

Epelbaum, Thomas; Gelis, François; Tanji, Naoto; ...

2014-12-30

We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since onemore » has also access to the non-approximated result for comparison.« less

Second-order Boltzmann equation: gauge dependence and gauge invariance

NASA Astrophysics Data System (ADS)

Naruko, Atsushi; Pitrou, Cyril; Koyama, Kazuya; Sasaki, Misao

2013-08-01

In the context of cosmological perturbation theory, we derive the second-order Boltzmann equation describing the evolution of the distribution function of radiation without a specific gauge choice. The essential steps in deriving the Boltzmann equation are revisited and extended given this more general framework: (i) the polarization of light is incorporated in this formalism by using a tensor-valued distribution function; (ii) the importance of a choice of the tetrad field to define the local inertial frame in the description of the distribution function is emphasized; (iii) we perform a separation between temperature and spectral distortion, both for the intensity and polarization for the first time; (iv) the gauge dependence of all perturbed quantities that enter the Boltzmann equation is derived, and this enables us to check the correctness of the perturbed Boltzmann equation by explicitly showing its gauge-invariance for both intensity and polarization. We finally discuss several implications of the gauge dependence for the observed temperature.

Multiple-Relaxation-Time Lattice Boltzmann Models in 3D

NASA Technical Reports Server (NTRS)

dHumieres, Dominique; Ginzburg, Irina; Krafczyk, Manfred; Lallemand, Pierre; Luo, Li-Shi; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

This article provides a concise exposition of the multiple-relaxation-time lattice Boltzmann equation, with examples of fifteen-velocity and nineteen-velocity models in three dimensions. Simulation of a diagonally lid-driven cavity flow in three dimensions at Re=500 and 2000 is performed. The results clearly demonstrate the superior numerical stability of the multiple-relaxation-time lattice Boltzmann equation over the popular lattice Bhatnagar-Gross-Krook equation.

Electroosmotic flow and mixing in microchannels with the lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Tang, G. H.; Li, Zhuo; Wang, J. K.; He, Y. L.; Tao, W. Q.

2006-11-01

Understanding the electroosmotic flow in microchannels is of both fundamental and practical significance for the design and optimization of various microfluidic devices to control fluid motion. In this paper, a lattice Boltzmann equation, which recovers the nonlinear Poisson-Boltzmann equation, is used to solve the electric potential distribution in the electrolytes, and another lattice Boltzmann equation, which recovers the Navier-Stokes equation including the external force term, is used to solve the velocity fields. The method is validated by the electric potential distribution in the electrolytes and the pressure driven pulsating flow. Steady-state and pulsating electroosmotic flows in two-dimensional parallel uniform and nonuniform charged microchannels are studied with this lattice Boltzmann method. The simulation results show that the heterogeneous surface potential distribution and the electroosmotic pulsating flow can induce chaotic advection and thus enhance the mixing in microfluidic systems efficiently.

A Unified Theory of Non-Ideal Gas Lattice Boltzmann Models

NASA Technical Reports Server (NTRS)

Luo, Li-Shi

1998-01-01

A non-ideal gas lattice Boltzmann model is directly derived, in an a priori fashion, from the Enskog equation for dense gases. The model is rigorously obtained by a systematic procedure to discretize the Enskog equation (in the presence of an external force) in both phase space and time. The lattice Boltzmann model derived here is thermodynamically consistent and is free of the defects which exist in previous lattice Boltzmann models for non-ideal gases. The existing lattice Boltzmann models for non-ideal gases are analyzed and compared with the model derived here.

Surface-slip equations for multicomponent nonequilibrium air flow

NASA Technical Reports Server (NTRS)

Gupta, R. N.; Scott, C. D.; Moss, J. N.

1985-01-01

Equations are presented for the surface-slip (or jump) values of species concentration, pressure, velocity, and temperature in the low-Reynolds number, high-altitude flight regime of a space vehicle. The equations are obtained from closed form solutions of the mass, momentum, and energy flux equations using the Chapman-Enskog velocity distribution function. This function represents a solution of the Boltzmann equation in the Navier-Stokes approximation. The analysis, obtained for nonequilibrium multicomponent air flow, includes the finite-rate surface catalytic recombination and changes in the internal energy during reflection from the surface. Expressions for the various slip quantities were obtained in a form which can be employed in flowfield computations. A consistent set of equations is provided for multicomponent, binary, and single species mixtures. Expression is also provided for the finite-rate, species-concentration boundary condition for a multicomponent mixture in absence of slip.

A particle-particle hybrid method for kinetic and continuum equations

NASA Astrophysics Data System (ADS)

Tiwari, Sudarshan; Klar, Axel; Hardt, Steffen

2009-10-01

We present a coupling procedure for two different types of particle methods for the Boltzmann and the Navier-Stokes equations. A variant of the DSMC method is applied to simulate the Boltzmann equation, whereas a meshfree Lagrangian particle method, similar to the SPH method, is used for simulations of the Navier-Stokes equations. An automatic domain decomposition approach is used with the help of a continuum breakdown criterion. We apply adaptive spatial and time meshes. The classical Sod's 1D shock tube problem is solved for a large range of Knudsen numbers. Results from Boltzmann, Navier-Stokes and hybrid solvers are compared. The CPU time for the hybrid solver is 3-4 times faster than for the Boltzmann solver.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Noronha, Jorge; Denicol, Gabriel S.

In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime AdS 2 Ⓧ S 2. We further derive explicit analytic expressions for the momentum dependence of the single-particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density doesmore » not match the equilibrium form. The nonequilibrium contribution to the entropy density is shown to be due to higher-order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Furthermore, in this system the slowly moving hydrodynamic degrees of freedom can exhibit true perfect fluidity while being totally decoupled from the fast moving, nonhydrodynamical microscopic degrees of freedom that lead to entropy production.« less

Reconstruction of phonon relaxation times from systems featuring interfaces with unknown properties

NASA Astrophysics Data System (ADS)

Forghani, Mojtaba; Hadjiconstantinou, Nicolas G.

2018-05-01

We present a method for reconstructing the phonon relaxation-time function τω=τ (ω ) (including polarization) and associated phonon free-path distribution from thermal spectroscopy data for systems featuring interfaces with unknown properties. Our method does not rely on the effective thermal-conductivity approximation or a particular physical model of the interface behavior. The reconstruction is formulated as an optimization problem in which the relaxation times are determined as functions of frequency by minimizing the discrepancy between the experimentally measured temperature profiles and solutions of the Boltzmann transport equation for the same system. Interface properties such as transmissivities are included as unknowns in the optimization; however, because for the thermal spectroscopy problems considered here the reconstruction is not very sensitive to the interface properties, the transmissivities are only approximately reconstructed and can be considered as byproducts of the calculation whose primary objective is the accurate determination of the relaxation times. The proposed method is validated using synthetic experimental data obtained from Monte Carlo solutions of the Boltzmann transport equation. The method is shown to remain robust in the presence of uncertainty (noise) in the measurement.

Structural interactions in ionic liquids linked to higher-order Poisson-Boltzmann equations

NASA Astrophysics Data System (ADS)

Blossey, R.; Maggs, A. C.; Podgornik, R.

2017-06-01

We present a derivation of generalized Poisson-Boltzmann equations starting from classical theories of binary fluid mixtures, employing an approach based on the Legendre transform as recently applied to the case of local descriptions of the fluid free energy. Under specific symmetry assumptions, and in the linearized regime, the Poisson-Boltzmann equation reduces to a phenomenological equation introduced by Bazant et al. [Phys. Rev. Lett. 106, 046102 (2011)], 10.1103/PhysRevLett.106.046102, whereby the structuring near the surface is determined by bulk coefficients.

Distributional Monte Carlo Methods for the Boltzmann Equation

DTIC Science & Technology

2013-03-01

Presented to the Faculty Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command...Interim Dean, Graduate School of Engineering and Management 8 Mar 2013 Date AFIT-ENC-DS-13-M-06 Abstract Stochastic particle methods (SPMs) for the...applied to the well-studied Bobylev-Krook-Wu solution as a numerical test case. Accuracy and variance of the solutions are examined as functions of various

PB-AM: An open-source, fully analytical linear poisson-boltzmann solver

DOE Office of Scientific and Technical Information (OSTI.GOV)

Felberg, Lisa E.; Brookes, David H.; Yap, Eng-Hui

2016-11-02

We present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized Poisson Boltzmann equation. The PB-AM software package includes the generation of outputs files appropriate for visualization using VMD, a Brownian dynamics scheme that uses periodic boundary conditions to simulate dynamics, the ability to specify docking criteria, and offers two different kinetics schemes to evaluate biomolecular association rate constants. Given that PB-AM defines mutual polarization completely and accurately, it can be refactored as a many-body expansion to explore 2- and 3-body polarization. Additionally, the software has been integrated into the Adaptive Poisson-Boltzmannmore » Solver (APBS) software package to make it more accessible to a larger group of scientists, educators and students that are more familiar with the APBS framework.« less

Generalized time evolution of the homogeneous cooling state of a granular gas with positive and negative coefficient of normal restitution

NASA Astrophysics Data System (ADS)

Khalil, Nagi

2018-04-01

The homogeneous cooling state (HCS) of a granular gas described by the inelastic Boltzmann equation is reconsidered. As usual, particles are taken as inelastic hard disks or spheres, but now the coefficient of normal restitution α is allowed to take negative values , which is a simple way of modeling more complicated inelastic interactions. The distribution function of the HCS is studied at the long-time limit, as well as intermediate times. At the long-time limit, the relevant information of the HCS is given by a scaling distribution function , where the time dependence occurs through a dimensionless velocity c. For , remains close to the Gaussian distribution in the thermal region, its cumulants and exponential tails being well described by the first Sonine approximation. In contrast, for , the distribution function becomes multimodal, its maxima located at , and its observable tails algebraic. The latter is a consequence of an unbalanced relaxation–dissipation competition, and is analytically demonstrated for , thanks to a reduction of the Boltzmann equation to a Fokker–Plank-like equation. Finally, a generalized scaling solution to the Boltzmann equation is also found . Apart from the time dependence occurring through the dimensionless velocity, depends on time through a new parameter β measuring the departure of the HCS from its long-time limit. It is shown that describes the time evolution of the HCS for almost all times. The relevance of the new scaling is also discussed.

Collisionless stellar hydrodynamics as an efficient alternative to N-body methods

NASA Astrophysics Data System (ADS)

Mitchell, Nigel L.; Vorobyov, Eduard I.; Hensler, Gerhard

2013-01-01

The dominant constituents of the Universe's matter are believed to be collisionless in nature and thus their modelling in any self-consistent simulation is extremely important. For simulations that deal only with dark matter or stellar systems, the conventional N-body technique is fast, memory efficient and relatively simple to implement. However when extending simulations to include the effects of gas physics, mesh codes are at a distinct disadvantage compared to Smooth Particle Hydrodynamics (SPH) codes. Whereas implementing the N-body approach into SPH codes is fairly trivial, the particle-mesh technique used in mesh codes to couple collisionless stars and dark matter to the gas on the mesh has a series of significant scientific and technical limitations. These include spurious entropy generation resulting from discreteness effects, poor load balancing and increased communication overhead which spoil the excellent scaling in massively parallel grid codes. In this paper we propose the use of the collisionless Boltzmann moment equations as a means to model the collisionless material as a fluid on the mesh, implementing it into the massively parallel FLASH Adaptive Mesh Refinement (AMR) code. This approach which we term `collisionless stellar hydrodynamics' enables us to do away with the particle-mesh approach and since the parallelization scheme is identical to that used for the hydrodynamics, it preserves the excellent scaling of the FLASH code already demonstrated on peta-flop machines. We find that the classic hydrodynamic equations and the Boltzmann moment equations can be reconciled under specific conditions, allowing us to generate analytic solutions for collisionless systems using conventional test problems. We confirm the validity of our approach using a suite of demanding test problems, including the use of a modified Sod shock test. By deriving the relevant eigenvalues and eigenvectors of the Boltzmann moment equations, we are able to use high order accurate characteristic tracing methods with Riemann solvers to generate numerical solutions which show excellent agreement with our analytic solutions. We conclude by demonstrating the ability of our code to model complex phenomena by simulating the evolution of a two-armed spiral galaxy whose properties agree with those predicted by the swing amplification theory.

Quantum dynamics in continuum for proton transport II: Variational solvent-solute interface.

PubMed

Chen, Duan; Chen, Zhan; Wei, Guo-Wei

2012-01-01

Proton transport plays an important role in biological energy transduction and sensory systems. Therefore, it has attracted much attention in biological science and biomedical engineering in the past few decades. The present work proposes a multiscale/multiphysics model for the understanding of the molecular mechanism of proton transport in transmembrane proteins involving continuum, atomic, and quantum descriptions, assisted with the evolution, formation, and visualization of membrane channel surfaces. We describe proton dynamics quantum mechanically via a new density functional theory based on the Boltzmann statistics, while implicitly model numerous solvent molecules as a dielectric continuum to reduce the number of degrees of freedom. The density of all other ions in the solvent is assumed to obey the Boltzmann distribution in a dynamic manner. The impact of protein molecular structure and its charge polarization on the proton transport is considered explicitly at the atomic scale. A variational solute-solvent interface is designed to separate the explicit molecule and implicit solvent regions. We formulate a total free-energy functional to put proton kinetic and potential energies, the free energy of all other ions, and the polar and nonpolar energies of the whole system on an equal footing. The variational principle is employed to derive coupled governing equations for the proton transport system. Generalized Laplace-Beltrami equation, generalized Poisson-Boltzmann equation, and generalized Kohn-Sham equation are obtained from the present variational framework. The variational solvent-solute interface is generated and visualized to facilitate the multiscale discrete/continuum/quantum descriptions. Theoretical formulations for the proton density and conductance are constructed based on fundamental laws of physics. A number of mathematical algorithms, including the Dirichlet-to-Neumann mapping, matched interface and boundary method, Gummel iteration, and Krylov space techniques are utilized to implement the proposed model in a computationally efficient manner. The gramicidin A channel is used to validate the performance of the proposed proton transport model and demonstrate the efficiency of the proposed mathematical algorithms. The proton channel conductances are studied over a number of applied voltages and reference concentrations. A comparison with experimental data verifies the present model predictions and confirms the proposed model. Copyright © 2011 John Wiley & Sons, Ltd.

Nonlinear Boltzmann equation for the homogeneous isotropic case: Minimal deterministic Matlab program

NASA Astrophysics Data System (ADS)

Asinari, Pietro

2010-10-01

The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both these corrections make possible to derive very accurate reference solutions for this test case. Moreover this work aims to distribute an open-source program (called HOMISBOLTZ), which can be redistributed and/or modified for dealing with different applications, under the terms of the GNU General Public License. The program has been purposely designed in order to be minimal, not only with regards to the reduced number of lines (less than 1000), but also with regards to the coding style (as simple as possible). Program summaryProgram title: HOMISBOLTZ Catalogue identifier: AEGN_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 23 340 No. of bytes in distributed program, including test data, etc.: 7 635 236 Distribution format: tar.gz Programming language: Tested with Matlab version ⩽6.5. However, in principle, any recent version of Matlab or Octave should work Computer: All supporting Matlab or Octave Operating system: All supporting Matlab or Octave RAM: 300 MBytes Classification: 23 Nature of problem: The problem consists in integrating the homogeneous Boltzmann equation for a generic collisional kernel in case of isotropic symmetry, by a deterministic direct method. Difficulties arise from the multi-dimensionality of the collisional operator and from satisfying the conservation of particle number and energy (momentum is trivial for this test case) as accurately as possible, in order to preserve the late dynamics. Solution method: The solution is based on the method proposed by Aristov (2001) [1], but with two substantial improvements: (a) the original problem is reformulated in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium). Both these corrections make possible to derive very accurate reference solutions for this test case. Restrictions: The nonlinear Boltzmann equation is extremely challenging from the computational point of view, in particular for deterministic methods, despite the increased computational power of recent hardware. In this work, only the homogeneous isotropic case is considered, for making possible the development of a minimal program (by a simple scripting language) and allowing the user to check the advantages of the proposed improvements beyond Aristov's (2001) method [1]. The initial conditions are supposed parameterized according to a fixed analytical expression, but this can be easily modified. Running time: From minutes to hours (depending on the adopted discretization of the kinetic energy space). For example, on a 64 bit workstation with Intel CoreTM i7-820Q Quad Core CPU at 1.73 GHz and 8 MBytes of RAM, the provided test run (with the corresponding binary data file storing the pre-computed relaxation rates) requires 154 seconds. References:V.V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Kluwer Academic Publishers, 2001.

An advanced kinetic theory for morphing continuum with inner structures

NASA Astrophysics Data System (ADS)

Chen, James

2017-12-01

Advanced kinetic theory with the Boltzmann-Curtiss equation provides a promising tool for polyatomic gas flows, especially for fluid flows containing inner structures, such as turbulence, polyatomic gas flows and others. Although a Hamiltonian-based distribution function was proposed for diatomic gas flow, a general distribution function for the generalized Boltzmann-Curtiss equations and polyatomic gas flow is still out of reach. With assistance from Boltzmann's entropy principle, a generalized Boltzmann-Curtiss distribution for polyatomic gas flow is introduced. The corresponding governing equations at equilibrium state are derived and compared with Eringen's morphing (micropolar) continuum theory derived under the framework of rational continuum thermomechanics. Although rational continuum thermomechanics has the advantages of mathematical rigor and simplicity, the presented statistical kinetic theory approach provides a clear physical picture for what the governing equations represent.

Four-Wave-Mixing Oscillations in a simplified Boltzmannian semiconductor model with LO-phonons

NASA Astrophysics Data System (ADS)

Tamborenea, P. I.; Bányai, L.; Haug, H.

1996-03-01

The recently discovered(L. Bányai, D. B. Tran Thoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, M. Wegener, T. Marschner and W. Stolz, Phys. Rev. Lett. 75), 2188 (1995). oscillations of the integrated four-wave-mixing signal in semiconductors due to electron-LO-phonon scattering are studied within a simplified Boltzmann-type model. Although several aspects of the experimental results require a description within the framework of non-Markovian quantum-kinetic theory, our simplified Boltzmannian model is well suited to analyze the origin of the observed novel oscillations of frequency (1+m_e/m_h) hbarω_LO. To this end, we developed a third-order, analytic solution of the semiconductor Bloch equations (SBE) with Boltzmann-type, LO-phonon collision terms. Results of this theory along with numerical solutions of the SBE will be presented.

Surface-slip equations for multicomponent, nonequilibrium air flow

NASA Technical Reports Server (NTRS)

Gupta, Roop N.; Scott, Carl D.; Moss, James N.; Goglia, Gene

1985-01-01

Equations are presented for the surface slip (or jump) values of species concentration, pressure, velocity, and temperature in the low-Reynolds-number, high-altitude flight regime of a space vehicle. These are obtained from closed-form solutions of the mass, momentum, and energy flux equations using the Chapman-Enskog velocity distribution function. This function represents a solution of the Boltzmann equation in the Navier-Stokes approximation. The analysis, obtained for nonequilibrium multicomponent air flow, includes the finite-rate surface catalytic recombination and changes in the internal energy during reflection from the surface. Expressions for the various slip quantities have been obtained in a form which can readily be employed in flow-field computations. A consistent set of equations is provided for multicomponent, binary, and single species mixtures. Expression is also provided for the finite-rate species-concentration boundary condition for a multicomponent mixture in absence of slip.

Boundary condition at a two-phase interface in the lattice Boltzmann method for the convection-diffusion equation.

PubMed

Yoshida, Hiroaki; Kobayashi, Takayuki; Hayashi, Hidemitsu; Kinjo, Tomoyuki; Washizu, Hitoshi; Fukuzawa, Kenji

2014-07-01

A boundary scheme in the lattice Boltzmann method (LBM) for the convection-diffusion equation, which correctly realizes the internal boundary condition at the interface between two phases with different transport properties, is presented. The difficulty in satisfying the continuity of flux at the interface in a transient analysis, which is inherent in the conventional LBM, is overcome by modifying the collision operator and the streaming process of the LBM. An asymptotic analysis of the scheme is carried out in order to clarify the role played by the adjustable parameters involved in the scheme. As a result, the internal boundary condition is shown to be satisfied with second-order accuracy with respect to the lattice interval, if we assign appropriate values to the adjustable parameters. In addition, two specific problems are numerically analyzed, and comparison with the analytical solutions of the problems numerically validates the proposed scheme.

Lattice Boltzmann Equation On a 2D Rectangular Grid

NASA Technical Reports Server (NTRS)

Bouzidi, MHamed; DHumieres, Dominique; Lallemand, Pierre; Luo, Li-Shi; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

We construct a multi-relaxation lattice Boltzmann model on a two-dimensional rectangular grid. The model is partly inspired by a previous work of Koelman to construct a lattice BGK model on a two-dimensional rectangular grid. The linearized dispersion equation is analyzed to obtain the constraints on the isotropy of the transport coefficients and Galilean invariance for various wave propagations in the model. The linear stability of the model is also studied. The model is numerically tested for three cases: (a) a vortex moving with a constant velocity on a mesh periodic boundary conditions; (b) Poiseuille flow with an arbitrasy inclined angle with respect to the lattice orientation: and (c) a cylinder &symmetrically placed in a channel. The numerical results of these tests are compared with either analytic solutions or the results obtained by other methods. Satisfactory results are obtained for the numerical simulations.

Evaluation of the Lattice-Boltzmann Equation Solver PowerFLOW for Aerodynamic Applications

NASA Technical Reports Server (NTRS)

Lockard, David P.; Luo, Li-Shi; Singer, Bart A.; Bushnell, Dennis M. (Technical Monitor)

2000-01-01

A careful comparison of the performance of a commercially available Lattice-Boltzmann Equation solver (Power-FLOW) was made with a conventional, block-structured computational fluid-dynamics code (CFL3D) for the flow over a two-dimensional NACA-0012 airfoil. The results suggest that the version of PowerFLOW used in the investigation produced solutions with large errors in the computed flow field; these errors are attributed to inadequate resolution of the boundary layer for reasons related to grid resolution and primitive turbulence modeling. The requirement of square grid cells in the PowerFLOW calculations limited the number of points that could be used to span the boundary layer on the wing and still keep the computation size small enough to fit on the available computers. Although not discussed in detail, disappointing results were also obtained with PowerFLOW for a cavity flow and for the flow around a generic helicopter configuration.

Thermal transport at the nanoscale: A Fourier's law vs. phonon Boltzmann equation study

NASA Astrophysics Data System (ADS)

Kaiser, J.; Feng, T.; Maassen, J.; Wang, X.; Ruan, X.; Lundstrom, M.

2017-01-01

Steady-state thermal transport in nanostructures with dimensions comparable to the phonon mean-free-path is examined. Both the case of contacts at different temperatures with no internal heat generation and contacts at the same temperature with internal heat generation are considered. Fourier's law results are compared to finite volume method solutions of the phonon Boltzmann equation in the gray approximation. When the boundary conditions are properly specified, results obtained using Fourier's law without modifying the bulk thermal conductivity are in essentially exact quantitative agreement with the phonon Boltzmann equation in the ballistic and diffusive limits. The errors between these two limits are examined in this paper. For the four cases examined, the error in the apparent thermal conductivity as deduced from a correct application of Fourier's law is less than 6%. We also find that the Fourier's law results presented here are nearly identical to those obtained from a widely used ballistic-diffusive approach but analytically much simpler. Although limited to steady-state conditions with spatial variations in one dimension and to a gray model of phonon transport, the results show that Fourier's law can be used for linear transport from the diffusive to the ballistic limit. The results also contribute to an understanding of how heat transport at the nanoscale can be understood in terms of the conceptual framework that has been established for electron transport at the nanoscale.

Stern potential and Debye length measurements in dilute ionic solutions with electrostatic force microscopy.

PubMed

Kumar, Bharat; Crittenden, Scott R

2013-11-01

We demonstrate the ability to measure Stern potential and Debye length in dilute ionic solution with atomic force microscopy. We develop an analytic expression for the second harmonic force component of the capacitive force in an ionic solution from the linearized Poisson-Boltzmann equation. This allows us to calibrate the AFM tip potential and, further, obtain the Stern potential of sample surfaces. In addition, the measured capacitive force is independent of van der Waals and double layer forces, thus providing a more accurate measure of Debye length.

PB-AM: An open-source, fully analytical linear poisson-boltzmann solver.

PubMed

Felberg, Lisa E; Brookes, David H; Yap, Eng-Hui; Jurrus, Elizabeth; Baker, Nathan A; Head-Gordon, Teresa

2017-06-05

We present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized PB equation, for molecules represented as non-overlapping spherical cavities. The PB-AM software package includes the generation of outputs files appropriate for visualization using visual molecular dynamics, a Brownian dynamics scheme that uses periodic boundary conditions to simulate dynamics, the ability to specify docking criteria, and offers two different kinetics schemes to evaluate biomolecular association rate constants. Given that PB-AM defines mutual polarization completely and accurately, it can be refactored as a many-body expansion to explore 2- and 3-body polarization. Additionally, the software has been integrated into the Adaptive Poisson-Boltzmann Solver (APBS) software package to make it more accessible to a larger group of scientists, educators, and students that are more familiar with the APBS framework. © 2016 Wiley Periodicals, Inc. © 2016 Wiley Periodicals, Inc.

An effective lattice Boltzmann flux solver on arbitrarily unstructured meshes

NASA Astrophysics Data System (ADS)

Wu, Qi-Feng; Shu, Chang; Wang, Yan; Yang, Li-Ming

2018-05-01

The recently proposed lattice Boltzmann flux solver (LBFS) is a new approach for the simulation of incompressible flow problems. It applies the finite volume method (FVM) to discretize the governing equations, and the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers. In the previous application of the LBFS, the structured meshes have been commonly employed, which may cause inconvenience for problems with complex geometries. In this paper, the LBFS is extended to arbitrarily unstructured meshes for effective simulation of incompressible flows. Two test cases, the lid-driven flow in a triangular cavity and flow around a circular cylinder, are carried out for validation. The obtained results are compared with the data available in the literature. Good agreement has been achieved, which demonstrates the effectiveness and reliability of the LBFS in simulating flows on arbitrarily unstructured meshes.

A nonequilibrium model for a moderate pressure hydrogen microwave discharge plasma

NASA Technical Reports Server (NTRS)

Scott, Carl D.

1993-01-01

This document describes a simple nonequilibrium energy exchange and chemical reaction model to be used in a computational fluid dynamics calculation for a hydrogen plasma excited by microwaves. The model takes into account the exchange between the electrons and excited states of molecular and atomic hydrogen. Specifically, electron-translation, electron-vibration, translation-vibration, ionization, and dissociation are included. The model assumes three temperatures, translational/rotational, vibrational, and electron, each describing a Boltzmann distribution for its respective energy mode. The energy from the microwave source is coupled to the energy equation via a source term that depends on an effective electric field which must be calculated outside the present model. This electric field must be found by coupling the results of the fluid dynamics and kinetics solution with a solution to Maxwell's equations that includes the effects of the plasma permittivity. The solution to Maxwell's equations is not within the scope of this present paper.

Nonlinear Boltzmann equation for the homogeneous isotropic case: Some improvements to deterministic methods and applications to relaxation towards local equilibrium

NASA Astrophysics Data System (ADS)

Asinari, P.

2011-03-01

Boltzmann equation is one the most powerful paradigms for explaining transport phenomena in fluids. Since early fifties, it received a lot of attention due to aerodynamic requirements for high altitude vehicles, vacuum technology requirements and nowadays, micro-electro-mechanical systems (MEMs). Because of the intrinsic mathematical complexity of the problem, Boltzmann himself started his work by considering first the case when the distribution function does not depend on space (homogeneous case), but only on time and the magnitude of the molecular velocity (isotropic collisional integral). The interest with regards to the homogeneous isotropic Boltzmann equation goes beyond simple dilute gases. In the so-called econophysics, a Boltzmann type model is sometimes introduced for studying the distribution of wealth in a simple market. Another recent application of the homogeneous isotropic Boltzmann equation is given by opinion formation modeling in quantitative sociology, also called socio-dynamics or sociophysics. The present work [1] aims to improve the deterministic method for solving homogenous isotropic Boltzmann equation proposed by Aristov [2] by two ideas: (a) the homogeneous isotropic problem is reformulated first in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium).

MPBEC, a Matlab Program for Biomolecular Electrostatic Calculations

NASA Astrophysics Data System (ADS)

Vergara-Perez, Sandra; Marucho, Marcelo

2016-01-01

One of the most used and efficient approaches to compute electrostatic properties of biological systems is to numerically solve the Poisson-Boltzmann (PB) equation. There are several software packages available that solve the PB equation for molecules in aqueous electrolyte solutions. Most of these software packages are useful for scientists with specialized training and expertise in computational biophysics. However, the user is usually required to manually take several important choices, depending on the complexity of the biological system, to successfully obtain the numerical solution of the PB equation. This may become an obstacle for researchers, experimentalists, even students with no special training in computational methodologies. Aiming to overcome this limitation, in this article we present MPBEC, a free, cross-platform, open-source software that provides non-experts in the field an easy and efficient way to perform biomolecular electrostatic calculations on single processor computers. MPBEC is a Matlab script based on the Adaptative Poisson-Boltzmann Solver, one of the most popular approaches used to solve the PB equation. MPBEC does not require any user programming, text editing or extensive statistical skills, and comes with detailed user-guide documentation. As a unique feature, MPBEC includes a useful graphical user interface (GUI) application which helps and guides users to configure and setup the optimal parameters and approximations to successfully perform the required biomolecular electrostatic calculations. The GUI also incorporates visualization tools to facilitate users pre- and post-analysis of structural and electrical properties of biomolecules.

MPBEC, a Matlab Program for Biomolecular Electrostatic Calculations

PubMed Central

Vergara-Perez, Sandra; Marucho, Marcelo

2015-01-01

One of the most used and efficient approaches to compute electrostatic properties of biological systems is to numerically solve the Poisson-Boltzmann (PB) equation. There are several software packages available that solve the PB equation for molecules in aqueous electrolyte solutions. Most of these software packages are useful for scientists with specialized training and expertise in computational biophysics. However, the user is usually required to manually take several important choices, depending on the complexity of the biological system, to successfully obtain the numerical solution of the PB equation. This may become an obstacle for researchers, experimentalists, even students with no special training in computational methodologies. Aiming to overcome this limitation, in this article we present MPBEC, a free, cross-platform, open-source software that provides non-experts in the field an easy and efficient way to perform biomolecular electrostatic calculations on single processor computers. MPBEC is a Matlab script based on the Adaptative Poisson Boltzmann Solver, one of the most popular approaches used to solve the PB equation. MPBEC does not require any user programming, text editing or extensive statistical skills, and comes with detailed user-guide documentation. As a unique feature, MPBEC includes a useful graphical user interface (GUI) application which helps and guides users to configure and setup the optimal parameters and approximations to successfully perform the required biomolecular electrostatic calculations. The GUI also incorporates visualization tools to facilitate users pre- and post- analysis of structural and electrical properties of biomolecules. PMID:26924848

MPBEC, a Matlab Program for Biomolecular Electrostatic Calculations.

PubMed

Vergara-Perez, Sandra; Marucho, Marcelo

2016-01-01

One of the most used and efficient approaches to compute electrostatic properties of biological systems is to numerically solve the Poisson-Boltzmann (PB) equation. There are several software packages available that solve the PB equation for molecules in aqueous electrolyte solutions. Most of these software packages are useful for scientists with specialized training and expertise in computational biophysics. However, the user is usually required to manually take several important choices, depending on the complexity of the biological system, to successfully obtain the numerical solution of the PB equation. This may become an obstacle for researchers, experimentalists, even students with no special training in computational methodologies. Aiming to overcome this limitation, in this article we present MPBEC, a free, cross-platform, open-source software that provides non-experts in the field an easy and efficient way to perform biomolecular electrostatic calculations on single processor computers. MPBEC is a Matlab script based on the Adaptative Poisson Boltzmann Solver, one of the most popular approaches used to solve the PB equation. MPBEC does not require any user programming, text editing or extensive statistical skills, and comes with detailed user-guide documentation. As a unique feature, MPBEC includes a useful graphical user interface (GUI) application which helps and guides users to configure and setup the optimal parameters and approximations to successfully perform the required biomolecular electrostatic calculations. The GUI also incorporates visualization tools to facilitate users pre- and post- analysis of structural and electrical properties of biomolecules.

Differential geometry based solvation model. III. Quantum formulation

PubMed Central

Chen, Zhan; Wei, Guo-Wei

2011-01-01

Solvation is of fundamental importance to biomolecular systems. Implicit solvent models, particularly those based on the Poisson-Boltzmann equation for electrostatic analysis, are established approaches for solvation analysis. However, ad hoc solvent-solute interfaces are commonly used in the implicit solvent theory. Recently, we have introduced differential geometry based solvation models which allow the solvent-solute interface to be determined by the variation of a total free energy functional. Atomic fixed partial charges (point charges) are used in our earlier models, which depends on existing molecular mechanical force field software packages for partial charge assignments. As most force field models are parameterized for a certain class of molecules or materials, the use of partial charges limits the accuracy and applicability of our earlier models. Moreover, fixed partial charges do not account for the charge rearrangement during the solvation process. The present work proposes a differential geometry based multiscale solvation model which makes use of the electron density computed directly from the quantum mechanical principle. To this end, we construct a new multiscale total energy functional which consists of not only polar and nonpolar solvation contributions, but also the electronic kinetic and potential energies. By using the Euler-Lagrange variation, we derive a system of three coupled governing equations, i.e., the generalized Poisson-Boltzmann equation for the electrostatic potential, the generalized Laplace-Beltrami equation for the solvent-solute boundary, and the Kohn-Sham equations for the electronic structure. We develop an iterative procedure to solve three coupled equations and to minimize the solvation free energy. The present multiscale model is numerically validated for its stability, consistency and accuracy, and is applied to a few sets of molecules, including a case which is difficult for existing solvation models. Comparison is made to many other classic and quantum models. By using experimental data, we show that the present quantum formulation of our differential geometry based multiscale solvation model improves the prediction of our earlier models, and outperforms some explicit solvation model. PMID:22112067

Lattice Boltzmann model for high-order nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Lattice Boltzmann model for high-order nonlinear partial differential equations.

PubMed

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

The halo Boltzmann equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Biagetti, Matteo; Desjacques, Vincent; Kehagias, Alex

2016-04-01

Dark matter halos are the building blocks of the universe as they host galaxies and clusters. The knowledge of the clustering properties of halos is therefore essential for the understanding of the galaxy statistical properties. We derive an effective halo Boltzmann equation which can be used to describe the halo clustering statistics. In particular, we show how the halo Boltzmann equation encodes a statistically biased gravitational force which generates a bias in the peculiar velocities of virialized halos with respect to the underlying dark matter, as recently observed in N-body simulations.

Direct simulation Monte Carlo method for the Uehling-Uhlenbeck-Boltzmann equation.

PubMed

Garcia, Alejandro L; Wagner, Wolfgang

2003-11-01

In this paper we describe a direct simulation Monte Carlo algorithm for the Uehling-Uhlenbeck-Boltzmann equation in terms of Markov processes. This provides a unifying framework for both the classical Boltzmann case as well as the Fermi-Dirac and Bose-Einstein cases. We establish the foundation of the algorithm by demonstrating its link to the kinetic equation. By numerical experiments we study its sensitivity to the number of simulation particles and to the discretization of the velocity space, when approximating the steady-state distribution.

Direct modeling for computational fluid dynamics

NASA Astrophysics Data System (ADS)

Xu, Kun

2015-06-01

All fluid dynamic equations are valid under their modeling scales, such as the particle mean free path and mean collision time scale of the Boltzmann equation and the hydrodynamic scale of the Navier-Stokes (NS) equations. The current computational fluid dynamics (CFD) focuses on the numerical solution of partial differential equations (PDEs), and its aim is to get the accurate solution of these governing equations. Under such a CFD practice, it is hard to develop a unified scheme that covers flow physics from kinetic to hydrodynamic scales continuously because there is no such governing equation which could make a smooth transition from the Boltzmann to the NS modeling. The study of fluid dynamics needs to go beyond the traditional numerical partial differential equations. The emerging engineering applications, such as air-vehicle design for near-space flight and flow and heat transfer in micro-devices, do require further expansion of the concept of gas dynamics to a larger domain of physical reality, rather than the traditional distinguishable governing equations. At the current stage, the non-equilibrium flow physics has not yet been well explored or clearly understood due to the lack of appropriate tools. Unfortunately, under the current numerical PDE approach, it is hard to develop such a meaningful tool due to the absence of valid PDEs. In order to construct multiscale and multiphysics simulation methods similar to the modeling process of constructing the Boltzmann or the NS governing equations, the development of a numerical algorithm should be based on the first principle of physical modeling. In this paper, instead of following the traditional numerical PDE path, we introduce direct modeling as a principle for CFD algorithm development. Since all computations are conducted in a discretized space with limited cell resolution, the flow physics to be modeled has to be done in the mesh size and time step scales. Here, the CFD is more or less a direct construction of discrete numerical evolution equations, where the mesh size and time step will play dynamic roles in the modeling process. With the variation of the ratio between mesh size and local particle mean free path, the scheme will capture flow physics from the kinetic particle transport and collision to the hydrodynamic wave propagation. Based on the direct modeling, a continuous dynamics of flow motion will be captured in the unified gas-kinetic scheme. This scheme can be faithfully used to study the unexplored non-equilibrium flow physics in the transition regime.

Fast Maximum Entropy Moment Closure Approach to Solving the Boltzmann Equation

NASA Astrophysics Data System (ADS)

Summy, Dustin; Pullin, Dale

2015-11-01

We describe a method for a moment-based solution of the Boltzmann Equation (BE). This is applicable to an arbitrary set of velocity moments whose transport is governed by partial-differential equations (PDEs) derived from the BE. The equations are unclosed, containing both higher-order moments and molecular-collision terms. These are evaluated using a maximum-entropy reconstruction of the velocity distribution function f (c , x , t) , from the known moments, within a finite-box domain of single-particle velocity (c) space. Use of a finite-domain alleviates known problems (Junk and Unterreiter, Continuum Mech. Thermodyn., 2002) concerning existence and uniqueness of the reconstruction. Unclosed moments are evaluated with quadrature while collision terms are calculated using any desired method. This allows integration of the moment PDEs in time. The high computational cost of the general method is greatly reduced by careful choice of the velocity moments, allowing the necessary integrals to be reduced from three- to one-dimensional in the case of strictly 1D flows. A method to extend this enhancement to fully 3D flows is discussed. Comparison with relaxation and shock-wave problems using the DSMC method will be presented. Partially supported by NSF grant DMS-1418903.

Kinetic and fluid descriptions of charged particle swarms in gases and nonpolar fluids: Theory and applications

NASA Astrophysics Data System (ADS)

Dujko, Sasa

2016-09-01

In this work we review the progress achieved over the last few decades in the fundamental kinetic theory of charged particle swarms with the focus on numerical techniques for the solution of Boltzmann's equation for electrons, as well as on the development of fluid models. We present a time-dependent multi term solution of Boltzmann's equation valid for electrons and positrons in varying configurations of electric and magnetic fields. The capacity of a theory and associated computer code will be illustrated by considering the heating mechanisms for electrons in radio-frequency electric and magnetic fields in a collision-dominated regime under conditions when electron transport is greatly affected by non-conservative collisions. The kinetic theory for solving the Boltzmann equation will be followed by a fluid equation description of charged particle swarms in both the hydrodynamic and non-hydrodynamic regimes, highlighting (i) the utility of momentum transfer theory for evaluating collisional terms in the balance equations and (ii) closure assumptions and approximations. The applications of this theory are split into three sections. First, we will present our 1.5D model of Resistive Plate Chambers (RPCs) which are used for timing and triggering purposes in many high energy physics experiments. The model is employed to study the avalanche to streamer transition in RPCs under the influence of space charge effects and photoionization. Second, we will discuss our high-order fluid model for streamer discharges. Particular emphases will be placed on the correct implementation of transport data in streamer models as well as on the evaluation of the mean-energy-dependent collision rates for electrons required as an input in the high-order fluid model. In the last segment of this work, we will present our model to study the avalanche to streamer transition in non-polar fluids. Using a Monte Carlo simulation technique we have calculated transport coefficients for electrons in liquid argon and liquid xenon. We employ the two model processes in which only momentum and only energy are exchanged to account for structure dependent coherent elastic scattering at low energies. The specific treatment of inelastic collisions in our model will be also discussed using physical arguments.

A free software for pore-scale modelling: solving Stokes equation for velocity fields and permeability values in 3D pore geometries

NASA Astrophysics Data System (ADS)

Gerke, Kirill; Vasilyev, Roman; Khirevich, Siarhei; Karsanina, Marina; Collins, Daniel; Korost, Dmitry; Mallants, Dirk

2015-04-01

In this contribution we introduce a novel free software which solves the Stokes equation to obtain velocity fields for low Reynolds-number flows within externally generated 3D pore geometries. Provided with velocity fields, one can calculate permeability for known pressure gradient boundary conditions via Darcy's equation. Finite-difference schemes of 2nd and 4th order of accuracy are used together with an artificial compressibility method to iteratively converge to a steady-state solution of Stokes' equation. This numerical approach is much faster and less computationally demanding than the majority of open-source or commercial softwares employing other algorithms (finite elements/volumes, lattice Boltzmann, etc.) The software consists of two parts: 1) a pre and post-processing graphical interface, and 2) a solver. The latter is efficiently parallelized to use any number of available cores (the speedup on 16 threads was up to 10-12 depending on hardware). Due to parallelization and memory optimization our software can be used to obtain solutions for 300x300x300 voxels geometries on modern desktop PCs. The software was successfully verified by testing it against lattice Boltzmann simulations and analytical solutions. To illustrate the software's applicability for numerous problems in Earth Sciences, a number of case studies have been developed: 1) identifying the representative elementary volume for permeability determination within a sandstone sample, 2) derivation of permeability/hydraulic conductivity values for rock and soil samples and comparing those with experimentally obtained values, 3) revealing the influence of the amount of fine-textured material such as clay on filtration properties of sandy soil. This work was partially supported by RSF grant 14-17-00658 (pore-scale modelling) and RFBR grants 13-04-00409-a and 13-05-01176-a.

Strong and weak adsorptions of polyelectrolyte chains onto oppositely charged spheres

NASA Astrophysics Data System (ADS)

Cherstvy, A. G.; Winkler, R. G.

2006-08-01

We investigate the complexation of long thin polyelectrolyte (PE) chains with oppositely charged spheres. In the limit of strong adsorption, when strongly charged PE chains adapt a definite wrapped conformation on the sphere surface, we analytically solve the linear Poisson-Boltzmann equation and calculate the electrostatic potential and the energy of the complex. We discuss some biological applications of the obtained results. For weak adsorption, when a flexible weakly charged PE chain is localized next to the sphere in solution, we solve the Edwards equation for PE conformations in the Hulthén potential, which is used as an approximation for the screened Debye-Hückel potential of the sphere. We predict the critical conditions for PE adsorption. We find that the critical sphere charge density exhibits a distinctively different dependence on the Debye screening length than for PE adsorption onto a flat surface. We compare our findings with experimental measurements on complexation of various PEs with oppositely charged colloidal particles. We also present some numerical results of the coupled Poisson-Boltzmann and self-consistent field equation for PE adsorption in an assembly of oppositely charged spheres.

Asymptotic analysis of the Boltzmann equation for dark matter relics in the presence of a running dilaton and space-time defects

NASA Astrophysics Data System (ADS)

Bender, Carl M.; Mavromatos, Nick E.; Sarkar, Sarben

2013-03-01

The interplay of dilatonic effects in dilaton cosmology and stochastic quantum space-time defects within the framework of string/brane cosmologies is examined. The Boltzmann equation describes the physics of thermal dark-matter-relic abundances in the presence of rolling dilatons. These dilatons affect the coupling of stringy matter to D-particle defects, which are generic in string theory. This coupling leads to an additional source term in the Boltzmann equation. The techniques of asymptotic matching and boundary-layer theory, which were recently applied by two of the authors (Bender and Sarkar) to a Boltzmann equation, are used here to find the detailed asymptotic relic abundances for all ranges of the expectation value of the dilaton field. The phenomenological implications for the search for supersymmetric dark matter in current colliders, such as the LHC, are discussed.

Fast Laplace solver approach to pore-scale permeability

NASA Astrophysics Data System (ADS)

Arns, C. H.; Adler, P. M.

2018-02-01

We introduce a powerful and easily implemented method to calculate the permeability of porous media at the pore scale using an approximation based on the Poiseulle equation to calculate permeability to fluid flow with a Laplace solver. The method consists of calculating the Euclidean distance map of the fluid phase to assign local conductivities and lends itself naturally to the treatment of multiscale problems. We compare with analytical solutions as well as experimental measurements and lattice Boltzmann calculations of permeability for Fontainebleau sandstone. The solver is significantly more stable than the lattice Boltzmann approach, uses less memory, and is significantly faster. Permeabilities are in excellent agreement over a wide range of porosities.

AN EFFICIENT HIGHER-ORDER FAST MULTIPOLE BOUNDARY ELEMENT SOLUTION FOR POISSON-BOLTZMANN BASED MOLECULAR ELECTROSTATICS

PubMed Central

Bajaj, Chandrajit; Chen, Shun-Chuan; Rand, Alexander

2011-01-01

In order to compute polarization energy of biomolecules, we describe a boundary element approach to solving the linearized Poisson-Boltzmann equation. Our approach combines several important features including the derivative boundary formulation of the problem and a smooth approximation of the molecular surface based on the algebraic spline molecular surface. State of the art software for numerical linear algebra and the kernel independent fast multipole method is used for both simplicity and efficiency of our implementation. We perform a variety of computational experiments, testing our method on a number of actual proteins involved in molecular docking and demonstrating the effectiveness of our solver for computing molecular polarization energy. PMID:21660123

A differential equation for the Generalized Born radii.

PubMed

Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro

2013-06-28

The Generalized Born (GB) model offers a convenient way of representing electrostatics in complex macromolecules like proteins or nucleic acids. The computation of atomic GB radii is currently performed by different non-local approaches involving volume or surface integrals. Here we obtain a non-linear second-order partial differential equation for the Generalized Born radius, which may be solved using local iterative algorithms. The equation is derived under the assumption that the usual GB approximation to the reaction field obeys Laplace's equation. The equation admits as particular solutions the correct GB radii for the sphere and the plane. The tests performed on a set of 55 different proteins show an overall agreement with other reference GB models and "perfect" Poisson-Boltzmann based values.

Ultrasonic waves in classical gases

NASA Astrophysics Data System (ADS)

Magner, A. G.; Gorenstein, M. I.; Grygoriev, U. V.

2017-12-01

The velocity and absorption coefficient for the plane sound waves in a classical gas are obtained by solving the Boltzmann kinetic equation, which describes the reaction of the single-particle distribution function to a periodic external field. Within the linear response theory, the nonperturbative dispersion equation valid for all sound frequencies is derived and solved numerically. The results are in agreement with the approximate analytical solutions found for both the frequent- and rare-collision regimes. These results are also in qualitative agreement with the experimental data for ultrasonic waves in dilute gases.

Evolution of statistical averages: An interdisciplinary proposal using the Chapman-Enskog method

NASA Astrophysics Data System (ADS)

Mariscal-Sanchez, A.; Sandoval-Villalbazo, A.

2017-08-01

This work examines the idea of applying the Chapman-Enskog (CE) method for approximating the solution of the Boltzmann equation beyond the realm of physics, using an information theory approach. Equations describing the evolution of averages and their fluctuations in a generalized phase space are established up to first-order in the Knudsen parameter which is defined as the ratio of the time between interactions (mean free time) and a characteristic macroscopic time. Although the general equations here obtained may be applied in a wide range of disciplines, in this paper, only a particular case related to the evolution of averages in speculative markets is examined.

AFMPB: An adaptive fast multipole Poisson-Boltzmann solver for calculating electrostatics in biomolecular systems

NASA Astrophysics Data System (ADS)

Lu, Benzhuo; Cheng, Xiaolin; Huang, Jingfang; McCammon, J. Andrew

2010-06-01

A Fortran program package is introduced for rapid evaluation of the electrostatic potentials and forces in biomolecular systems modeled by the linearized Poisson-Boltzmann equation. The numerical solver utilizes a well-conditioned boundary integral equation (BIE) formulation, a node-patch discretization scheme, a Krylov subspace iterative solver package with reverse communication protocols, and an adaptive new version of fast multipole method in which the exponential expansions are used to diagonalize the multipole-to-local translations. The program and its full description, as well as several closely related libraries and utility tools are available at http://lsec.cc.ac.cn/~lubz/afmpb.html and a mirror site at http://mccammon.ucsd.edu/. This paper is a brief summary of the program: the algorithms, the implementation and the usage. Program summaryProgram title: AFMPB: Adaptive fast multipole Poisson-Boltzmann solver Catalogue identifier: AEGB_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGB_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GPL 2.0 No. of lines in distributed program, including test data, etc.: 453 649 No. of bytes in distributed program, including test data, etc.: 8 764 754 Distribution format: tar.gz Programming language: Fortran Computer: Any Operating system: Any RAM: Depends on the size of the discretized biomolecular system Classification: 3 External routines: Pre- and post-processing tools are required for generating the boundary elements and for visualization. Users can use MSMS ( http://www.scripps.edu/~sanner/html/msms_home.html) for pre-processing, and VMD ( http://www.ks.uiuc.edu/Research/vmd/) for visualization. Sub-programs included: An iterative Krylov subspace solvers package from SPARSKIT by Yousef Saad ( http://www-users.cs.umn.edu/~saad/software/SPARSKIT/sparskit.html), and the fast multipole methods subroutines from FMMSuite ( http://www.fastmultipole.org/). Nature of problem: Numerical solution of the linearized Poisson-Boltzmann equation that describes electrostatic interactions of molecular systems in ionic solutions. Solution method: A novel node-patch scheme is used to discretize the well-conditioned boundary integral equation formulation of the linearized Poisson-Boltzmann equation. Various Krylov subspace solvers can be subsequently applied to solve the resulting linear system, with a bounded number of iterations independent of the number of discretized unknowns. The matrix-vector multiplication at each iteration is accelerated by the adaptive new versions of fast multipole methods. The AFMPB solver requires other stand-alone pre-processing tools for boundary mesh generation, post-processing tools for data analysis and visualization, and can be conveniently coupled with different time stepping methods for dynamics simulation. Restrictions: Only three or six significant digits options are provided in this version. Unusual features: Most of the codes are in Fortran77 style. Memory allocation functions from Fortran90 and above are used in a few subroutines. Additional comments: The current version of the codes is designed and written for single core/processor desktop machines. Check http://lsec.cc.ac.cn/~lubz/afmpb.html and http://mccammon.ucsd.edu/ for updates and changes. Running time: The running time varies with the number of discretized elements ( N) in the system and their distributions. In most cases, it scales linearly as a function of N.

Variational multiscale models for charge transport.

PubMed

Wei, Guo-Wei; Zheng, Qiong; Chen, Zhan; Xia, Kelin

2012-01-01

This work presents a few variational multiscale models for charge transport in complex physical, chemical and biological systems and engineering devices, such as fuel cells, solar cells, battery cells, nanofluidics, transistors and ion channels. An essential ingredient of the present models, introduced in an earlier paper (Bulletin of Mathematical Biology, 72, 1562-1622, 2010), is the use of differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain from the microscopic domain, meanwhile, dynamically couple discrete and continuum descriptions. Our main strategy is to construct the total energy functional of a charge transport system to encompass the polar and nonpolar free energies of solvation, and chemical potential related energy. By using the Euler-Lagrange variation, coupled Laplace-Beltrami and Poisson-Nernst-Planck (LB-PNP) equations are derived. The solution of the LB-PNP equations leads to the minimization of the total free energy, and explicit profiles of electrostatic potential and densities of charge species. To further reduce the computational complexity, the Boltzmann distribution obtained from the Poisson-Boltzmann (PB) equation is utilized to represent the densities of certain charge species so as to avoid the computationally expensive solution of some Nernst-Planck (NP) equations. Consequently, the coupled Laplace-Beltrami and Poisson-Boltzmann-Nernst-Planck (LB-PBNP) equations are proposed for charge transport in heterogeneous systems. A major emphasis of the present formulation is the consistency between equilibrium LB-PB theory and non-equilibrium LB-PNP theory at equilibrium. Another major emphasis is the capability of the reduced LB-PBNP model to fully recover the prediction of the LB-PNP model at non-equilibrium settings. To account for the fluid impact on the charge transport, we derive coupled Laplace-Beltrami, Poisson-Nernst-Planck and Navier-Stokes equations from the variational principle for chemo-electro-fluid systems. A number of computational algorithms is developed to implement the proposed new variational multiscale models in an efficient manner. A set of ten protein molecules and a realistic ion channel, Gramicidin A, are employed to confirm the consistency and verify the capability. Extensive numerical experiment is designed to validate the proposed variational multiscale models. A good quantitative agreement between our model prediction and the experimental measurement of current-voltage curves is observed for the Gramicidin A channel transport. This paper also provides a brief review of the field.

Variational multiscale models for charge transport

PubMed Central

Wei, Guo-Wei; Zheng, Qiong; Chen, Zhan; Xia, Kelin

2012-01-01

This work presents a few variational multiscale models for charge transport in complex physical, chemical and biological systems and engineering devices, such as fuel cells, solar cells, battery cells, nanofluidics, transistors and ion channels. An essential ingredient of the present models, introduced in an earlier paper (Bulletin of Mathematical Biology, 72, 1562-1622, 2010), is the use of differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain from the microscopic domain, meanwhile, dynamically couple discrete and continuum descriptions. Our main strategy is to construct the total energy functional of a charge transport system to encompass the polar and nonpolar free energies of solvation, and chemical potential related energy. By using the Euler-Lagrange variation, coupled Laplace-Beltrami and Poisson-Nernst-Planck (LB-PNP) equations are derived. The solution of the LB-PNP equations leads to the minimization of the total free energy, and explicit profiles of electrostatic potential and densities of charge species. To further reduce the computational complexity, the Boltzmann distribution obtained from the Poisson-Boltzmann (PB) equation is utilized to represent the densities of certain charge species so as to avoid the computationally expensive solution of some Nernst-Planck (NP) equations. Consequently, the coupled Laplace-Beltrami and Poisson-Boltzmann-Nernst-Planck (LB-PBNP) equations are proposed for charge transport in heterogeneous systems. A major emphasis of the present formulation is the consistency between equilibrium LB-PB theory and non-equilibrium LB-PNP theory at equilibrium. Another major emphasis is the capability of the reduced LB-PBNP model to fully recover the prediction of the LB-PNP model at non-equilibrium settings. To account for the fluid impact on the charge transport, we derive coupled Laplace-Beltrami, Poisson-Nernst-Planck and Navier-Stokes equations from the variational principle for chemo-electro-fluid systems. A number of computational algorithms is developed to implement the proposed new variational multiscale models in an efficient manner. A set of ten protein molecules and a realistic ion channel, Gramicidin A, are employed to confirm the consistency and verify the capability. Extensive numerical experiment is designed to validate the proposed variational multiscale models. A good quantitative agreement between our model prediction and the experimental measurement of current-voltage curves is observed for the Gramicidin A channel transport. This paper also provides a brief review of the field. PMID:23172978

Shear viscosity for a heated granular binary mixture at low density.

PubMed

Montanero, José María; Garzó, Vicente

2003-02-01

The shear viscosity for a heated granular binary mixture of smooth hard spheres at low density is analyzed. The mixture is heated by the action of an external driving force (Gaussian thermostat) that exactly compensates for cooling effects associated with the dissipation of collisions. The study is made from the Boltzmann kinetic theory, which is solved by using two complementary approaches. First, a normal solution of the Boltzmann equation via the Chapman-Enskog method is obtained up to first order in the spatial gradients. The mass, heat, and momentum fluxes are determined and the corresponding transport coefficients identified. As in the free cooling case [V. Garzó and J. W. Dufty, Phys. Fluids 14, 1476 (2002)], practical evaluation requires a Sonine polynomial approximation, and here it is mainly illustrated in the case of the shear viscosity. Second, to check the accuracy of the Chapman-Enskog results, the Boltzmann equation is numerically solved by means of the direct simulation Monte Carlo method. The simulation is performed for a system under uniform shear flow, using the Gaussian thermostat to control inelastic cooling. The comparison shows an excellent agreement between theory and simulation over a wide range of values of the restitution coefficients and the parameters of the mixture (masses, concentrations, and sizes).

High order spectral difference lattice Boltzmann method for incompressible hydrodynamics

NASA Astrophysics Data System (ADS)

Li, Weidong

2017-09-01

This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order (≥3) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners-Burggraf flow; and (c) the unsteady Taylor-Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified.

Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders.

PubMed

Kanazawa, Kiyoshi; Sueshige, Takumi; Takayasu, Hideki; Takayasu, Misako

2018-03-30

A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs. We report first on a microscopic empirical law of traders' trend-following behavior by tracking the trajectories of all individuals, which quantifies the collective motion of HFTs but has not been captured in conventional order-book models. We next introduce the corresponding microscopic model of HFTs and present its theoretical solution paralleling molecular kinetic theory: Boltzmann-like and Langevin-like equations are derived from the microscopic dynamics via the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Our model is the first microscopic model that has been directly validated through data analysis of the microscopic dynamics, exhibiting quantitative agreements with mesoscopic and macroscopic empirical results.

Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders

NASA Astrophysics Data System (ADS)

Kanazawa, Kiyoshi; Sueshige, Takumi; Takayasu, Hideki; Takayasu, Misako

2018-03-01

A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs. We report first on a microscopic empirical law of traders' trend-following behavior by tracking the trajectories of all individuals, which quantifies the collective motion of HFTs but has not been captured in conventional order-book models. We next introduce the corresponding microscopic model of HFTs and present its theoretical solution paralleling molecular kinetic theory: Boltzmann-like and Langevin-like equations are derived from the microscopic dynamics via the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Our model is the first microscopic model that has been directly validated through data analysis of the microscopic dynamics, exhibiting quantitative agreements with mesoscopic and macroscopic empirical results.

A high-order gas-kinetic Navier-Stokes flow solver

DOE Office of Scientific and Technical Information (OSTI.GOV)

Li Qibing, E-mail: lqb@tsinghua.edu.c; Xu Kun, E-mail: makxu@ust.h; Fu Song, E-mail: fs-dem@tsinghua.edu.c

2010-09-20

The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge-Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge-Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to itsmore » spatial and temporal decoupling. Many recently developed high-order methods require a Navier-Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier-Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to construct a dissipative wave structure starting from an initially discontinuous flow condition on a time scale being larger than the particle collision time. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier-Stokes equations (BGK-NS). The novelty for the easy extension from a second-order to a higher order is due to the simple particle transport and collision mechanism on the microscopic level. This paper will present a hierarchy to construct such a high-order method. The necessity to couple spatial and temporal evolution nonlinearly in the flux evaluation can be clearly observed through the numerical performance of the scheme for the viscous flow computations.« less

A Revision on Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials

NASA Astrophysics Data System (ADS)

Alonso, Ricardo J.; Gamba, Irene M.

2011-05-01

This short note complements the recent paper of the authors (Alonso, Gamba in J. Stat. Phys. 137(5-6):1147-1165, 2009). We revisit the results on propagation of regularity and stability using L p estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator. We show here the correct range of exponents. We require a Lebesgue's exponent α>1 in the angular part of the collision kernel in order to obtain finiteness in some constants involved in the regularity and stability estimates. As a consequence the L p regularity associated to the Cauchy problem of the space inhomogeneous Boltzmann equation holds for a finite range of p≥1 explicitly determined.

Late-time behaviour of the Einstein–Boltzmann system with a positive cosmological constant

NASA Astrophysics Data System (ADS)

Lee, Ho; Nungesser, Ernesto

2018-01-01

In this paper we study the Einstein–Boltzmann system for Israel particles with a positive cosmological constant. We consider spatially homogeneous solutions of all Bianchi types except type IX and obtain future global existence and the asymptotic behaviour of solutions to the Einstein–Boltzmann system. The result shows that the solutions converge to the de Sitter solution at late times.

1D kinetic simulations of a short glow discharge in helium

NASA Astrophysics Data System (ADS)

Yuan, Chengxun; Bogdanov, E. A.; Eliseev, S. I.; Kudryavtsev, A. A.

2017-07-01

This paper presents a 1D model of a direct current glow discharge based on the solution of the kinetic Boltzmann equation in the two-term approximation. The model takes into account electron-electron coulomb collisions, the corresponding collision integral is written in both detailed and simplified forms. The Boltzmann equation for electrons is coupled with continuity equations for ions and metastable atoms and the Poisson equation for electric potential. Simulations are carried out self-consistently for the whole length of discharge in helium (from cathode to anode) for cases p = 1 Torr, L = 3.6 cm and p = 20 Torr, L = 1.8 mm, so that pL = 3.6 cm.Torr in both cases. It is shown that simulations based on the kinetic approach give lower values of electron temperature in plasma than fluid simulations. Peaks in spatial differential flux corresponding to the electrons originating from superelastic collisions and Penning ionization were observed in simulations. Different approaches of taking coulomb collisions into account give significantly different values of electron density and electron temperature in plasma. Analysis showed that using a simplified approach gives a non-zero contribution to the electron energy balance, which is comparable to energy losses on elastic and inelastic collisions and leads to significant errors and thus is not recommended.

Nano-particle drag prediction at low Reynolds number using a direct Boltzmann-BGK solution approach

NASA Astrophysics Data System (ADS)

Evans, B.

2018-01-01

This paper outlines a novel approach for solution of the Boltzmann-BGK equation describing molecular gas dynamics applied to the challenging problem of drag prediction of a 2D circular nano-particle at transitional Knudsen number (0.0214) and low Reynolds number (0.25-2.0). The numerical scheme utilises a discontinuous-Galerkin finite element discretisation for the physical space representing the problem particle geometry and a high order discretisation for molecular velocity space describing the molecular distribution function. The paper shows that this method produces drag predictions that are aligned well with the range of drag predictions for this problem generated from the alternative numerical approaches of molecular dynamics codes and a modified continuum scheme. It also demonstrates the sensitivity of flow-field solutions and therefore drag predictions to the wall absorption parameter used to construct the solid wall boundary condition used in the solver algorithm. The results from this work has applications in fields ranging from diagnostics and therapeutics in medicine to the fields of semiconductors and xerographics.

The Boltzmann equation in the difference formulation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Szoke, Abraham; Brooks III, Eugene D.

2015-05-06

First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.

Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows

NASA Astrophysics Data System (ADS)

Liang, Hong; Xu, Jiangrong; Chen, Jiangxing; Wang, Huili; Chai, Zhenhua; Shi, Baochang

2018-03-01

In this paper, we present a simple and accurate lattice Boltzmann (LB) model for immiscible two-phase flows, which is able to deal with large density contrasts. This model utilizes two LB equations, one of which is used to solve the conservative Allen-Cahn equation, and the other is adopted to solve the incompressible Navier-Stokes equations. A forcing distribution function is elaborately designed in the LB equation for the Navier-Stokes equations, which make it much simpler than the existing LB models. In addition, the proposed model can achieve superior numerical accuracy compared with previous Allen-Cahn type of LB models. Several benchmark two-phase problems, including static droplet, layered Poiseuille flow, and spinodal decomposition are simulated to validate the present LB model. It is found that the present model can achieve relatively small spurious velocity in the LB community, and the obtained numerical results also show good agreement with the analytical solutions or some available results. Lastly, we use the present model to investigate the droplet impact on a thin liquid film with a large density ratio of 1000 and the Reynolds number ranging from 20 to 500. The fascinating phenomena of droplet splashing is successfully reproduced by the present model and the numerically predicted spreading radius exhibits to obey the power law reported in the literature.

Physical scales in the Wigner–Boltzmann equation

PubMed Central

Nedjalkov, M.; Selberherr, S.; Ferry, D.K.; Vasileska, D.; Dollfus, P.; Querlioz, D.; Dimov, I.; Schwaha, P.

2013-01-01

The Wigner–Boltzmann equation provides the Wigner single particle theory with interactions with bosonic degrees of freedom associated with harmonic oscillators, such as phonons in solids. Quantum evolution is an interplay of two transport modes, corresponding to the common coherent particle-potential processes, or to the decoherence causing scattering due to the oscillators. Which evolution mode will dominate depends on the scales of the involved physical quantities. A dimensionless formulation of the Wigner–Boltzmann equation is obtained, where these scales appear as dimensionless strength parameters. A notion called scaling theorem is derived, linking the strength parameters to the coupling with the oscillators. It is shown that an increase of this coupling is equivalent to a reduction of both the strength of the electric potential, and the coherence length. Secondly, the existence of classes of physically different, but mathematically equivalent setups of the Wigner–Boltzmann evolution is demonstrated. PMID:23504194

Hierarchical Boltzmann simulations and model error estimation

NASA Astrophysics Data System (ADS)

Torrilhon, Manuel; Sarna, Neeraj

2017-08-01

A hierarchical simulation approach for Boltzmann's equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics, but a subsequent refinement allows to successively improve the result to the complete Boltzmann result. We use Hermite discretization, or moment equations, for the steady linearized Boltzmann equation for a proof-of-concept of such a framework. All representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous Galerkin formulation. We demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains. The hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations. We present various model errors for a flow through a curved channel with obstacles.

Mathematical modeling of influence of ion size effects in an electrolyte in a nanoslit with overlapped EDL

NASA Astrophysics Data System (ADS)

Rajni, Kumar, Prashant

2017-10-01

Many nanofluidic systems are being used in a wide range of applications due to advances in nanotechnology. Due to nanoscale size of the system, the physics involved in the electric double layer and consequently the different phenomena related to it are different than those at microscale. The Poisson-Boltzmann equation governing the electric double layer in the system has many shortcomings such as point sized ions. The inclusion of finite size of ions give rise to various electrokinetic phenomena. Electrocapillarity is one such phenomena where the size effect plays an important role. Theeffect of asymmetric finite ion sizes in nano-confinement in the view of osmotic pressure and electrocapillarity is analyzed. As the confinement width of the system becomes comparable with the Debye length, the overlapped electric double layer (EDL) is influenced and significantly deformed by the steric effects. The osmotic pressure from the modified Poisson-Boltzmann equation in nanoslit is obtained. Due to nonlinear nature of the modified PB equation, the solution is obtained through numerical method. Afterwards, the electrocapillarity due to the steric effect is analyzed under constant surface potential condition at the walls of the nanoslit along with the flat interface assumption.

Corner-transport-upwind lattice Boltzmann model for bubble cavitation

NASA Astrophysics Data System (ADS)

Sofonea, V.; Biciuşcǎ, T.; Busuioc, S.; Ambruş, Victor E.; Gonnella, G.; Lamura, A.

2018-02-01

Aiming to study the bubble cavitation problem in quiescent and sheared liquids, a third-order isothermal lattice Boltzmann model that describes a two-dimensional (2D) fluid obeying the van der Waals equation of state, is introduced. The evolution equations for the distribution functions in this off-lattice model with 16 velocities are solved using the corner-transport-upwind (CTU) numerical scheme on large square lattices (up to 6144 ×6144 nodes). The numerical viscosity and the regularization of the model are discussed for first- and second-order CTU schemes finding that the latter choice allows to obtain a very accurate phase diagram of a nonideal fluid. In a quiescent liquid, the present model allows us to recover the solution of the 2D Rayleigh-Plesset equation for a growing vapor bubble. In a sheared liquid, we investigated the evolution of the total bubble area, the bubble deformation, and the bubble tilt angle, for various values of the shear rate. A linear relation between the dimensionless deformation coefficient D and the capillary number Ca is found at small Ca but with a different factor than in equilibrium liquids. A nonlinear regime is observed for Ca≳0.2 .

Monte Carlo solution of Boltzmann equation for a simple model of highly nonequilibrium diatomic gases: Translational rotational energy relaxation

NASA Technical Reports Server (NTRS)

Yoshikawa, K. K.

1978-01-01

The semiclassical transition probability was incorporated in the simulation for energy exchange between rotational and translational energy. The results provide details on the fundamental mechanisms of gas kinetics where analytical methods were impractical. The validity of the local Maxwellian assumption and relaxation time, rotational-translational energy transition, and a velocity analysis of the inelastic collision were discussed in detail.

Efficient Asymptotic Preserving Deterministic methods for the Boltzmann Equation

DTIC Science & Technology

2011-04-01

history tracing back to Hilbert , Chapmann and Enskog (Cercignani, 1988) at the beginning of the last century. The mathematical difficulties related to the...accurate determin- istic computations of the stationary solutions, which may be treated by schemes aimed to capture the stationary state ( Greenberg and...Stokes model, can be considered using the Chapmann-Enskog and the Hilbert expansions. We refer to Levermore (1996) for a mathematical setting of the

A kinetic equation with kinetic entropy functions for scalar conservation laws

NASA Technical Reports Server (NTRS)

Perthame, Benoit; Tadmor, Eitan

1990-01-01

A nonlinear kinetic equation is constructed and proved to be well-adapted to describe general multidimensional scalar conservation laws. In particular, it is proved to be well-posed uniformly in epsilon - the microscopic scale. It is also shown that the proposed kinetic equation is equipped with a family of kinetic entropy functions - analogous to Boltzmann's microscopic H-function, such that they recover Krushkov-type entropy inequality on the macroscopic scale. Finally, it is proved by both - BV compactness arguments in the one-dimensional case, that the local density of kinetic particles admits a continuum limit, as it converges strongly with epsilon below 0 to the unique entropy solution of the corresponding conservation law.

A second-order accurate kinetic-theory-based method for inviscid compressible flows

NASA Technical Reports Server (NTRS)

Deshpande, Suresh M.

1986-01-01

An upwind method for the numerical solution of the Euler equations is presented. This method, called the kinetic numerical method (KNM), is based on the fact that the Euler equations are moments of the Boltzmann equation of the kinetic theory of gases when the distribution function is Maxwellian. The KNM consists of two phases, the convection phase and the collision phase. The method is unconditionally stable and explicit. It is highly vectorizable and can be easily made total variation diminishing for the distribution function by a suitable choice of the interpolation strategy. The method is applied to a one-dimensional shock-propagation problem and to a two-dimensional shock-reflection problem.

Nonlinear transport for a dilute gas in steady Couette flow

NASA Astrophysics Data System (ADS)

Garzó, V.; López de Haro, M.

1997-03-01

Transport properties of a dilute gas subjected to arbitrarily large velocity and temperature gradients (steady planar Couette flow) are determined. The results are obtained from the so-called ellipsoidal statistical (ES) kinetic model, which is an extension of the well-known BGK kinetic model to account for the correct Prandtl number. At a hydrodynamic level, the solution is characterized by constant pressure, and linear velocity and parabolic temperature profiles with respect to a scaled variable. The transport coefficients are explicitly evaluated as nonlinear functions of the shear rate. A comparison with previous results derived from a perturbative solution of the Boltzmann equation as well as from other kinetic models is carried out. Such a comparison shows that the ES predictions are in better agreement with the Boltzmann results than those of the other approximations. In addition, the velocity distribution function is also computed. Although the shear rates required for observing non-Newtonian effects are experimentally unrealizable, the conclusions obtained here may be relevant for analyzing computer results.

Essentially Entropic Lattice Boltzmann Model

NASA Astrophysics Data System (ADS)

Atif, Mohammad; Kolluru, Praveen Kumar; Thantanapally, Chakradhar; Ansumali, Santosh

2017-12-01

The entropic lattice Boltzmann model (ELBM), a discrete space-time kinetic theory for hydrodynamics, ensures nonlinear stability via the discrete time version of the second law of thermodynamics (the H theorem). Compliance with the H theorem is numerically enforced in this methodology and involves a search for the maximal discrete path length corresponding to the zero dissipation state by iteratively solving a nonlinear equation. We demonstrate that an exact solution for the path length can be obtained by assuming a natural criterion of negative entropy change, thereby reducing the problem to solving an inequality. This inequality is solved by creating a new framework for construction of Padé approximants via quadrature on appropriate convex function. This exact solution also resolves the issue of indeterminacy in case of nonexistence of the entropic involution step. Since our formulation is devoid of complex mathematical library functions, the computational cost is drastically reduced. To illustrate this, we have simulated a model setup of flow over the NACA-0012 airfoil at a Reynolds number of 2.88 ×106.

Wetting boundary condition for the color-gradient lattice Boltzmann method: Validation with analytical and experimental data

NASA Astrophysics Data System (ADS)

Akai, Takashi; Bijeljic, Branko; Blunt, Martin J.

2018-06-01

In the color gradient lattice Boltzmann model (CG-LBM), a fictitious-density wetting boundary condition has been widely used because of its ease of implementation. However, as we show, this may lead to inaccurate results in some cases. In this paper, a new scheme for the wetting boundary condition is proposed which can handle complicated 3D geometries. The validity of our method for static problems is demonstrated by comparing the simulated results to analytical solutions in 2D and 3D geometries with curved boundaries. Then, capillary rise simulations are performed to study dynamic problems where the three-phase contact line moves. The results are compared to experimental results in the literature (Heshmati and Piri, 2014). If a constant contact angle is assumed, the simulations agree with the analytical solution based on the Lucas-Washburn equation. However, to match the experiments, we need to implement a dynamic contact angle that varies with the flow rate.

Electronegative nonlinear oscillating modes in plasmas

NASA Astrophysics Data System (ADS)

Panguetna, Chérif Souleman; Tabi, Conrad Bertrand; Kofané, Timoléon Crépin

2018-02-01

The emergence of nonlinear modulated waves is addressed in an unmagnetized electronegative plasma made of Boltzmann electrons, Boltzmann negative ions and cold mobile positive ions. The reductive perturbation method is used to reduce the dynamics of the whole system to a cubic nonlinear Schrödinger equation, whose the nonlinear and dispersion coefficients, P and Q, are function of the negative ion parameters, namely the negative ion concentration ratio (α) and the electron-to-negative ion temperature ratio (σn). It is observed that these parameters importantly affect the formation of modulated ion-acoustic waves, either as exact solutions or via the activation of modulational instability. Especially, the theory of modulational instability is used to show the correlation between the parametric analysis and the formation of modulated solitons, obtained here as bright envelopes and kink-wave solitons.

Transient finite element analysis of electric double layer using Nernst-Planck-Poisson equations with a modified Stern layer.

PubMed

Lim, Jongil; Whitcomb, John; Boyd, James; Varghese, Julian

2007-01-01

A finite element implementation of the transient nonlinear Nernst-Planck-Poisson (NPP) and Nernst-Planck-Poisson-modified Stern (NPPMS) models is presented. The NPPMS model uses multipoint constraints to account for finite ion size, resulting in realistic ion concentrations even at high surface potential. The Poisson-Boltzmann equation is used to provide a limited check of the transient models for low surface potential and dilute bulk solutions. The effects of the surface potential and bulk molarity on the electric potential and ion concentrations as functions of space and time are studied. The ability of the models to predict realistic energy storage capacity is investigated. The predicted energy is much more sensitive to surface potential than to bulk solution molarity.

On the Boltzmann-Grad Limit for Smooth Hard-Sphere Systems

NASA Astrophysics Data System (ADS)

Tessarotto, Massimo; Cremaschini, Claudio; Mond, Michael; Asci, Claudio; Soranzo, Alessandro; Tironi, Gino

2018-03-01

The problem is posed of the prescription of the so-called Boltzmann-Grad limit operator (L_{BG}) for the N-body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. It is proved, that, despite the non-commutative property of the operator L_{BG}, the Boltzmann equation can nevertheless be uniquely determined. In particular, consistent with the claim of Uffink and Valente (Found Phys 45:404, 2015) that there is "no time-asymmetric ingredient" in its derivation, the Boltzmann equation is shown to be time-reversal symmetric. The proof is couched on the "ab initio" axiomatic approach to the classical statistical mechanics recently developed (Tessarotto et al. in Eur Phys J Plus 128:32, 2013). Implications relevant for the physical interpretation of the Boltzmann H-theorem and the phenomenon of decay to kinetic equilibrium are pointed out.

BRYNTRN: A baryon transport computer code, computation procedures and data base

NASA Technical Reports Server (NTRS)

Wilson, John W.; Townsend, Lawrence W.; Chun, Sang Y.; Buck, Warren W.; Khan, Ferdous; Cucinotta, Frank

1988-01-01

The development is described of an interaction data base and a numerical solution to the transport of baryons through the arbitrary shield material based on a straight ahead approximation of the Boltzmann equation. The code is most accurate for continuous energy boundary values but gives reasonable results for discrete spectra at the boundary with even a relatively coarse energy grid (30 points) and large spatial increments (1 cm in H2O).

A mass-conserving multiphase lattice Boltzmann model for simulation of multiphase flows

NASA Astrophysics Data System (ADS)

Niu, Xiao-Dong; Li, You; Ma, Yi-Ren; Chen, Mu-Feng; Li, Xiang; Li, Qiao-Zhong

2018-01-01

In this study, a mass-conserving multiphase lattice Boltzmann (LB) model is proposed for simulating the multiphase flows. The proposed model developed in the present study is to improve the model of Shao et al. ["Free-energy-based lattice Boltzmann model for simulation of multiphase flows with density contrast," Phys. Rev. E 89, 033309 (2014)] by introducing a mass correction term in the lattice Boltzmann model for the interface. The model of Shao et al. [(the improved Zheng-Shu-Chew (Z-S-C model)] correctly considers the effect of the local density variation in momentum equation and has an obvious improvement over the Zheng-Shu-Chew (Z-S-C) model ["A lattice Boltzmann model for multiphase flows with large density ratio," J. Comput. Phys. 218(1), 353-371 (2006)] in terms of solution accuracy. However, due to the physical diffusion and numerical dissipation, the total mass of each fluid phase cannot be conserved correctly. To solve this problem, a mass correction term, which is similar to the one proposed by Wang et al. ["A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio," J. Comput. Phys. 290, 336-351 (2015)], is introduced into the lattice Boltzmann equation for the interface to compensate the mass losses or offset the mass increase. Meanwhile, to implement the wetting boundary condition and the contact angle, a geometric formulation and a local force are incorporated into the present mass-conserving LB model. The proposed model is validated by verifying the Laplace law, simulating both one and two aligned droplets splashing onto a liquid film, droplets standing on an ideal wall, droplets with different wettability splashing onto smooth wax, and bubbles rising under buoyancy. Numerical results show that the proposed model can correctly simulate multiphase flows. It was found that the mass is well-conserved in all cases considered by the model developed in the present study. The developed model has been found to perform better than the improved Z-S-C model in this aspect.

Modeling self-consistent multi-class dynamic traffic flow

NASA Astrophysics Data System (ADS)

Cho, Hsun-Jung; Lo, Shih-Ching

2002-09-01

In this study, we present a systematic self-consistent multiclass multilane traffic model derived from the vehicular Boltzmann equation and the traffic dispersion model. The multilane domain is considered as a two-dimensional space and the interaction among vehicles in the domain is described by a dispersion model. The reason we consider a multilane domain as a two-dimensional space is that the driving behavior of road users may not be restricted by lanes, especially motorcyclists. The dispersion model, which is a nonlinear Poisson equation, is derived from the car-following theory and the equilibrium assumption. Under the concept that all kinds of users share the finite section, the density is distributed on a road by the dispersion model. In addition, the dynamic evolution of the traffic flow is determined by the systematic gas-kinetic model derived from the Boltzmann equation. Multiplying Boltzmann equation by the zeroth, first- and second-order moment functions, integrating both side of the equation and using chain rules, we can derive continuity, motion and variance equation, respectively. However, the second-order moment function, which is the square of the individual velocity, is employed by previous researches does not have physical meaning in traffic flow. Although the second-order expansion results in the velocity variance equation, additional terms may be generated. The velocity variance equation we propose is derived from multiplying Boltzmann equation by the individual velocity variance. It modifies the previous model and presents a new gas-kinetic traffic flow model. By coupling the gas-kinetic model and the dispersion model, a self-consistent system is presented.

Generalized Boltzmann-Type Equations for Aggregation in Gases

NASA Astrophysics Data System (ADS)

Adzhiev, S. Z.; Vedenyapin, V. V.; Volkov, Yu. A.; Melikhov, I. V.

2017-12-01

The coalescence and fragmentation of particles in a dispersion system are investigated by applying kinetic theory methods, namely, by generalizing the Boltzmann kinetic equation to coalescence and fragmentation processes. Dynamic equations for the particle concentrations in the system are derived using the kinetic equations of motion. For particle coalescence and fragmentation, equations for the particle momentum, coordinate, and mass distribution functions are obtained and the coalescence and fragmentation coefficients are calculated. The equilibrium mass and velocity distribution functions of the particles in the dispersion system are found in the approximation of an active terminal group (Becker-Döring-type equation). The transition to a continuum description is performed.

Modeling electrokinetic flows by consistent implicit incompressible smoothed particle hydrodynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pan, Wenxiao; Kim, Kyungjoo; Perego, Mauro

2017-04-01

We present an efficient implicit incompressible smoothed particle hydrodynamics (I2SPH) discretization of Navier-Stokes, Poisson-Boltzmann, and advection-diffusion equations subject to Dirichlet or Robin boundary conditions. It is applied to model various two and three dimensional electrokinetic flows in simple or complex geometries. The I2SPH's accuracy and convergence are examined via comparison with analytical solutions, grid-based numerical solutions, or empirical models. The new method provides a framework to explore broader applications of SPH in microfluidics and complex fluids with charged objects, such as colloids and biomolecules, in arbitrary complex geometries.

A useful observable for estimating keff in fast subcritical systems

NASA Astrophysics Data System (ADS)

Saracco, Paolo; Borreani, Walter; Chersola, Davide; Lomonaco, Guglielmo; Ricco, Gianni; Ripani, Marco

2017-09-01

The neutron multiplication factor keff is a key quantity to characterize subcritical neutron multiplying devices and for understanting their physical behaviour, being related to the fundamental eigenvalue of Boltzmann transport equation. Both the maximum available power - and all quantities related to it, like, e.g. the effectiveness in burning nuclear wastes - as well as reactor kinetics and dynamics depend on keff. Nevertheless, keff is not directly measurable and its determination results from the solution of an inverse problem: minimizing model dependence of the solution for keff then becomes a critical issue, relevant both for practical and theoretical reasons.

Thermal conductivity and phonon transport properties of silicon using perturbation theory and the environment-dependent interatomic potential

NASA Astrophysics Data System (ADS)

Pascual-Gutiérrez, José A.; Murthy, Jayathi Y.; Viskanta, Raymond

2009-09-01

Silicon thermal conductivities are obtained from the solution of the linearized phonon Boltzmann transport equation without the use of any parameter-fitting. Perturbation theory is used to compute the strength of three-phonon and isotope scattering mechanisms. Matrix elements based on Fermi's golden rule are computed exactly without assuming either average or mode-dependent Grüeisen parameters, and with no underlying assumptions of crystal isotropy. The environment-dependent interatomic potential is employed to describe the interatomic force constants and the perturbing Hamiltonians. A detailed methodology to accurately find three-phonon processes satisfying energy- and momentum-conservation rules is also described. Bulk silicon thermal conductivity values are computed across a range of temperatures and shown to match experimental data very well. It is found that about two-thirds of the heat transport in bulk silicon may be attributed to transverse acoustic modes. Effective relaxation times and mean free paths are computed in order to provide a more complete picture of the detailed transport mechanisms and for use with carrier transport models based on the Boltzmann transport equation.

Relaxation of energetic S(1D) atoms in Xe gas: comparison of ab initio calculations with experimental data.

PubMed

Bovino, S; Zhang, P; Kharchenko, V; Dalgarno, A

2011-07-14

In this paper, we report our investigation of the translational energy relaxation of fast S((1)D) atoms in a Xe thermal bath. The interaction potential of Xe-S was constructed using ab initio methods. Total and differential cross sections were then calculated. The latter have been incorporated into the construction of the kernel of the Boltzmann equation describing the energy relaxation process. The solution of the Boltzmann equation was obtained and results were compared with those reported in experiments [G. Nan, and P. L. Houston, J. Chem. Phys. 97, 7865 (1992)]. Good agreement with the measured time-dependent relative velocity of fast S((1)D) atoms was obtained except at long relaxation times. The discrepancy may be due to the error accumulation caused by the use of hard sphere approximation and the Monte Carlo analysis of the experimental data. Our accurate description of the energy relaxation process led to an increase in the number of collisions required to achieve equilibrium by an order of magnitude compared to the number given by the hard-sphere approximation.

Numerical solution of Boltzmann tranport equation for TEA CO 2 laser having nitrogen-lean gas mixtures to predict laser characteristics and gas lifetime

NASA Astrophysics Data System (ADS)

Kumar, Manoj; Khare, Jai; Nath, A. K.

2007-02-01

Selective laser isotope separation by TEA CO 2 laser often needs short tail-free pulses. Using laser mixtures having very little nitrogen almost tail free laser pulses can be generated. The laser pulse characteristics and its gas lifetime is an important issue for long-term laser operation. Boltzmann transport equation is therefore solved numerically for TEA CO 2 laser gas mixtures having very little nitrogen to predict electron energy distribution function (EEDF). The distribution function is used to calculate various excitation and dissociation rate of CO 2 to predict laser pulse characteristics and laser gas lifetime, respectively. Laser rate equations have been solved with the calculated excitation rates for numerically evaluated discharge current and voltage profiles to calculate laser pulse shape. The calculated laser pulse shape and duration are in good agreement with the measured laser characteristics. The gas lifetime is estimated by integrating the equation governing the dissociation of CO 2. An experimental study of gas lifetime was carried out using quadrapole mass analyzer for such mixtures to estimate the O 2 being produced due to dissociation of CO 2 in the pulse discharge. The theoretically calculated O 2 concentration in the laser gas mixture matches with experimentally observed value. In the present TEA CO 2 laser system, for stable discharge the O 2 concentration should be below 0.2%.

Comment on ``Numerics of the lattice Boltzmann method: Effects of collision models on the lattice Boltzmann simulations''

NASA Astrophysics Data System (ADS)

Karlin, I. V.; Succi, S.; Chikatamarla, S. S.

2011-12-01

Critical comments on the entropic lattice Boltzmann equation (ELBE), by Li-Shi Luo, Wei Liao, Xingwang Chen, Yan Peng, and Wei Zhang in Ref. , are based on simulations, which make use of a model that, despite being referred to as the ELBE by the authors, is in fact equivalent to the standard lattice Bhatnagar-Gross-Krook equation for low Mach number simulations. In this Comment, a concise review of the ELBE is provided and illustrated by means of a three-dimensional turbulent flow simulation, which highlights the subgrid features of the ELBE.

Numerical Solution of the Gyrokinetic Poisson Equation in TEMPEST

NASA Astrophysics Data System (ADS)

Dorr, Milo; Cohen, Bruce; Cohen, Ronald; Dimits, Andris; Hittinger, Jeffrey; Kerbel, Gary; Nevins, William; Rognlien, Thomas; Umansky, Maxim; Xiong, Andrew; Xu, Xueqiao

2006-10-01

The gyrokinetic Poisson (GKP) model in the TEMPEST continuum gyrokinetic edge plasma code yields the electrostatic potential due to the charge density of electrons and an arbitrary number of ion species including the effects of gyroaveraging in the limit kρ1. The TEMPEST equations are integrated as a differential algebraic system involving a nonlinear system solve via Newton-Krylov iteration. The GKP preconditioner block is inverted using a multigrid preconditioned conjugate gradient (CG) algorithm. Electrons are treated as kinetic or adiabatic. The Boltzmann relation in the adiabatic option employs flux surface averaging to maintain neutrality within field lines and is solved self-consistently with the GKP equation. A decomposition procedure circumvents the near singularity of the GKP Jacobian block that otherwise degrades CG convergence.

A Lattice Boltzmann Method for Turbomachinery Simulations

NASA Technical Reports Server (NTRS)

Hsu, A. T.; Lopez, I.

2003-01-01

Lattice Boltzmann (LB) Method is a relatively new method for flow simulations. The start point of LB method is statistic mechanics and Boltzmann equation. The LB method tries to set up its model at molecular scale and simulate the flow at macroscopic scale. LBM has been applied to mostly incompressible flows and simple geometry.

Re-ionization and decaying dark matter

NASA Technical Reports Server (NTRS)

Dodelson, Scott; Jubas, Jay M.

1991-01-01

Gunn-Peterson tests suggest that the Universe was reionized after the standard recombination epoch. A systematic treatment is presented of the ionization process by deriving the Boltzmann equations appropriate to this regime. A compact solution for the photon spectrum is found in terms of the ionization ratio. These equations are then solved numerically for the Decaying Dark Matter scenario, wherein neutrinos with mass of order 30 eV radiatively decay producing photons which ionize the intergalactic medium. It was found that the neutrino mass and lifetime are severely constrained by Gunn-Peterson tests, observations of the diffuse photon spectrum in the ultraviolet regime, and the Hubble parameter.

Implicit versus explicit momentum relaxation time solution for semiconductor nanowires

DOE Office of Scientific and Technical Information (OSTI.GOV)

Marin, E. G., E-mail: egmarin@ugr.es; Ruiz, F. G., E-mail: franruiz@ugr.es; Godoy, A., E-mail: agodoy@ugr.es

2015-07-14

We discuss the necessity of the exact implicit Momentum Relaxation Time (MRT) solution of the Boltzmann transport equation in order to achieve reliable carrier mobility results in semiconductor nanowires. Firstly, the implicit solution for a 1D electron gas with a isotropic bandstructure is presented resulting in the formulation of a simple matrix system. Using this solution as a reference, the explicit approach is demonstrated to be inaccurate for the calculation of inelastic anisotropic mechanisms such as polar optical phonons, characteristic of III-V materials. Its validity for elastic and isotropic mechanisms is also evaluated. Finally, the implications of the MRT explicitmore » approach inaccuracies on the total mobility of Si and III-V NWs are studied.« less

Effects of nanoparticles on melting process with phase-change using the lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Ibrahem, Ahmed M.; El-Amin, Mohamed F.; Sun, Shuyu

In this work, the problem of nanoparticles dispersion effects on coupled heat transfer and solid-liquid phase change has been studied. The lattice Boltzmann method (LBM) enthalpy-based is employed. The collision model of lattice Bhatnagar-Gross-Krook (LBGK) is used to solve the problem of 1D melting by conduction. On the other hand, we use the model of multi-distribution functions (MDF) to calculate the density, the velocity and the temperature for the problem of 2D melting by free convection, associated with different boundary conditions. In these simulations, the volume fractions of copper nanoparticles (0-2%) added to water-base fluid and Rayleigh numbers of 103-105. We use the Chapman-Enskog expansion to derive the governing macroscopic quantities from the mesoscopic lattice Boltzmann equation. The results obtained by these models have been compared to an analytical solution or other numerical methods. The effects of nanoparticles on conduction and natural convection during the melting process have been investigated. Moreover, the influences of nanoparticles on moving of the phase change front, the thermal conductivity and the latent heat of fusion are also studied.

A Fast and Robust Poisson-Boltzmann Solver Based on Adaptive Cartesian Grids

PubMed Central

Boschitsch, Alexander H.; Fenley, Marcia O.

2011-01-01

An adaptive Cartesian grid (ACG) concept is presented for the fast and robust numerical solution of the 3D Poisson-Boltzmann Equation (PBE) governing the electrostatic interactions of large-scale biomolecules and highly charged multi-biomolecular assemblies such as ribosomes and viruses. The ACG offers numerous advantages over competing grid topologies such as regular 3D lattices and unstructured grids. For very large biological molecules and multi-biomolecule assemblies, the total number of grid-points is several orders of magnitude less than that required in a conventional lattice grid used in the current PBE solvers thus allowing the end user to obtain accurate and stable nonlinear PBE solutions on a desktop computer. Compared to tetrahedral-based unstructured grids, ACG offers a simpler hierarchical grid structure, which is naturally suited to multigrid, relieves indirect addressing requirements and uses fewer neighboring nodes in the finite difference stencils. Construction of the ACG and determination of the dielectric/ionic maps are straightforward, fast and require minimal user intervention. Charge singularities are eliminated by reformulating the problem to produce the reaction field potential in the molecular interior and the total electrostatic potential in the exterior ionic solvent region. This approach minimizes grid-dependency and alleviates the need for fine grid spacing near atomic charge sites. The technical portion of this paper contains three parts. First, the ACG and its construction for general biomolecular geometries are described. Next, a discrete approximation to the PBE upon this mesh is derived. Finally, the overall solution procedure and multigrid implementation are summarized. Results obtained with the ACG-based PBE solver are presented for: (i) a low dielectric spherical cavity, containing interior point charges, embedded in a high dielectric ionic solvent – analytical solutions are available for this case, thus allowing rigorous assessment of the solution accuracy; (ii) a pair of low dielectric charged spheres embedded in a ionic solvent to compute electrostatic interaction free energies as a function of the distance between sphere centers; (iii) surface potentials of proteins, nucleic acids and their larger-scale assemblies such as ribosomes; and (iv) electrostatic solvation free energies and their salt sensitivities – obtained with both linear and nonlinear Poisson-Boltzmann equation – for a large set of proteins. These latter results along with timings can serve as benchmarks for comparing the performance of different PBE solvers. PMID:21984876

Involving the Navier-Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method.

PubMed

Verschaeve, Joris C G

2011-06-13

By means of the continuity equation of the incompressible Navier-Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.

Heating-frequency-dependent thermal conductivity: An analytical solution from diffusive to ballistic regime and its relevance to phonon scattering measurements

NASA Astrophysics Data System (ADS)

Yang, Fan; Dames, Chris

2015-04-01

The heating-frequency dependence of the apparent thermal conductivity in a semi-infinite body with periodic planar surface heating is explained by an analytical solution to the Boltzmann transport equation. This solution is obtained using a two-flux model and gray mean free time approximation and verified numerically with a lattice Boltzmann method and numerical results from the literature. Extending the gray solution to the nongray regime leads to an integral transform and accumulation-function representation of the phonon scattering spectrum, where the natural variable is mean free time rather than mean free path, as often used in previous work. The derivation leads to an approximate cutoff conduction similar in spirit to that of Koh and Cahill [Phys. Rev. B 76, 075207 (2007), 10.1103/PhysRevB.76.075207] except that the most appropriate criterion involves the heater frequency rather than thermal diffusion length. The nongray calculations are consistent with Koh and Cahill's experimental observation that the apparent thermal conductivity shows a stronger heater-frequency dependence in a SiGe alloy than in natural Si. Finally these results are demonstrated using a virtual experiment, which fits the phase lag between surface temperature and heat flux to obtain the apparent thermal conductivity and accumulation function.

Free-energy-based lattice Boltzmann model for the simulation of multiphase flows with density contrast.

PubMed

Shao, J Y; Shu, C; Huang, H B; Chew, Y T

2014-03-01

A free-energy-based phase-field lattice Boltzmann method is proposed in this work to simulate multiphase flows with density contrast. The present method is to improve the Zheng-Shu-Chew (ZSC) model [Zheng, Shu, and Chew, J. Comput. Phys. 218, 353 (2006)] for correct consideration of density contrast in the momentum equation. The original ZSC model uses the particle distribution function in the lattice Boltzmann equation (LBE) for the mean density and momentum, which cannot properly consider the effect of local density variation in the momentum equation. To correctly consider it, the particle distribution function in the LBE must be for the local density and momentum. However, when the LBE of such distribution function is solved, it will encounter a severe numerical instability. To overcome this difficulty, a transformation, which is similar to the one used in the Lee-Lin (LL) model [Lee and Lin, J. Comput. Phys. 206, 16 (2005)] is introduced in this work to change the particle distribution function for the local density and momentum into that for the mean density and momentum. As a result, the present model still uses the particle distribution function for the mean density and momentum, and in the meantime, considers the effect of local density variation in the LBE as a forcing term. Numerical examples demonstrate that both the present model and the LL model can correctly simulate multiphase flows with density contrast, and the present model has an obvious improvement over the ZSC model in terms of solution accuracy. In terms of computational time, the present model is less efficient than the ZSC model, but is much more efficient than the LL model.

Quantum weak turbulence with applications to semiconductor lasers

NASA Astrophysics Data System (ADS)

Lvov, Yuri Victorovich

Based on a model Hamiltonian appropriate for the description of fermionic systems such as semiconductor lasers, we describe a natural asymptotic closure of the BBGKY hierarchy in complete analogy with that derived for classical weak turbulence. The main features of the interaction Hamiltonian are the inclusion of full Fermi statistics containing Pauli blocking and a simple, phenomenological, uniformly weak two particle interaction potential equivalent to the static screening approximation. The resulting asymytotic closure and quantum kinetic Boltzmann equation are derived in a self consistent manner without resorting to a priori statistical hypotheses or cumulant discard assumptions. We find a new class of solutions to the quantum kinetic equation which are analogous to the Kolmogorov spectra of hydrodynamics and classical weak turbulence. They involve finite fluxes of particles and energy across momentum space and are particularly relevant for describing the behavior of systems containing sources and sinks. We explore these solutions by using differential approximation to collision integral. We make a prima facie case that these finite flux solutions can be important in the context of semiconductor lasers. We show that semiconductor laser output efficiency can be improved by exciting these finite flux solutions. Numerical simulations of the semiconductor Maxwell Bloch equations support the claim.

Protein-ion binding process on finite macromolecular concentration. A Poisson-Boltzmann and Monte Carlo study.

PubMed

de Carvalho, Sidney Jurado; Fenley, Márcia O; da Silva, Fernando Luís Barroso

2008-12-25

Electrostatic interactions are one of the key driving forces for protein-ligands complexation. Different levels for the theoretical modeling of such processes are available on the literature. Most of the studies on the Molecular Biology field are performed within numerical solutions of the Poisson-Boltzmann Equation and the dielectric continuum models framework. In such dielectric continuum models, there are two pivotal questions: (a) how the protein dielectric medium should be modeled, and (b) what protocol should be used when solving this effective Hamiltonian. By means of Monte Carlo (MC) and Poisson-Boltzmann (PB) calculations, we define the applicability of the PB approach with linear and nonlinear responses for macromolecular electrostatic interactions in electrolyte solution, revealing some physical mechanisms and limitations behind it especially due the raise of both macromolecular charge and concentration out of the strong coupling regime. A discrepancy between PB and MC for binding constant shifts is shown and explained in terms of the manner PB approximates the excess chemical potentials of the ligand, and not as a consequence of the nonlinear thermal treatment and/or explicit ion-ion interactions as it could be argued. Our findings also show that the nonlinear PB predictions with a low dielectric response well reproduce the pK shifts calculations carried out with an uniform dielectric model. This confirms and completes previous results obtained by both MC and linear PB calculations.

Wigner distribution functions for complex dynamical systems: the emergence of the Wigner-Boltzmann equation.

PubMed

Sels, Dries; Brosens, Fons

2013-10-01

The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.

Transport tensors in perfectly aligned low-density fluids: Self-diffusion and thermal conductivity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Singh, G. S.; Kumar, B.

2001-06-01

The modified Taxman equation for the kinetic theory of low-density fluids composed of rigid aspherical molecules possessing internal degrees of freedom is generalized to obtain the transport tensors in a fluid of aligned molecules. The theory takes care of the shape of the particles exactly but the solution has been obtained only for the case of perfectly aligned hard spheroids within the framework of the first Sonine polynomial approximation. The expressions for the thermal-conductivity components have been obtained for the first time whereas the self-diffusion components obtained here turn out to be exactly the same as those derived by Kumarmore » and Masters [Mol. Phys. >81, 491 (1994)] through the solution of the Lorentz-Boltzmann equation. All our expressions yield correct results in the hard-sphere limit.« less

Boltzmann's "H"-Theorem and the Assumption of Molecular Chaos

ERIC Educational Resources Information Center

Boozer, A. D.

2011-01-01

We describe a simple dynamical model of a one-dimensional ideal gas and use computer simulations of the model to illustrate two fundamental results of kinetic theory: the Boltzmann transport equation and the Boltzmann "H"-theorem. Although the model is time-reversal invariant, both results predict that the behaviour of the gas is time-asymmetric.…

Boltzmann equation and hydrodynamics beyond Navier-Stokes.

PubMed

Bobylev, A V

2018-04-28

We consider in this paper the problem of derivation and regularization of higher (in Knudsen number) equations of hydrodynamics. The author's approach based on successive changes of hydrodynamic variables is presented in more detail for the Burnett level. The complete theory is briefly discussed for the linearized Boltzmann equation. It is shown that the best results in this case can be obtained by using the 'diagonal' equations of hydrodynamics. Rigorous estimates of accuracy of the Navier-Stokes and Burnett approximations are also presented.This article is part of the theme issue 'Hilbert's sixth problem'. © 2018 The Author(s).

Poisson-Boltzmann-Nernst-Planck model

NASA Astrophysics Data System (ADS)

Zheng, Qiong; Wei, Guo-Wei

2011-05-01

The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external voltages. Extensive numerical experiments show that there is an excellent consistency between the results predicted from the present PBNP model and those obtained from the PNP model in terms of the electrostatic potentials, ion concentration profiles, and current-voltage (I-V) curves. The present PBNP model is further validated by a comparison with experimental measurements of I-V curves under various ion bulk concentrations. Numerical experiments indicate that the proposed PBNP model is more efficient than the original PNP model in terms of simulation time.

Poisson-Boltzmann-Nernst-Planck model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zheng Qiong; Wei Guowei; Department of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824

2011-05-21

The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species inmore » the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external voltages. Extensive numerical experiments show that there is an excellent consistency between the results predicted from the present PBNP model and those obtained from the PNP model in terms of the electrostatic potentials, ion concentration profiles, and current-voltage (I-V) curves. The present PBNP model is further validated by a comparison with experimental measurements of I-V curves under various ion bulk concentrations. Numerical experiments indicate that the proposed PBNP model is more efficient than the original PNP model in terms of simulation time.« less

Poisson-Boltzmann-Nernst-Planck model.

PubMed

Zheng, Qiong; Wei, Guo-Wei

2011-05-21

The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external voltages. Extensive numerical experiments show that there is an excellent consistency between the results predicted from the present PBNP model and those obtained from the PNP model in terms of the electrostatic potentials, ion concentration profiles, and current-voltage (I-V) curves. The present PBNP model is further validated by a comparison with experimental measurements of I-V curves under various ion bulk concentrations. Numerical experiments indicate that the proposed PBNP model is more efficient than the original PNP model in terms of simulation time. © 2011 American Institute of Physics.

Differential geometry-based solvation and electrolyte transport models for biomolecular modeling: a review

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wei, Guowei; Baker, Nathan A.

2016-11-11

This chapter reviews the differential geometry-based solvation and electrolyte transport for biomolecular solvation that have been developed over the past decade. A key component of these methods is the differential geometry of surfaces theory, as applied to the solvent-solute boundary. In these approaches, the solvent-solute boundary is determined by a variational principle that determines the major physical observables of interest, for example, biomolecular surface area, enclosed volume, electrostatic potential, ion density, electron density, etc. Recently, differential geometry theory has been used to define the surfaces that separate the microscopic (solute) domains for biomolecules from the macroscopic (solvent) domains. In thesemore » approaches, the microscopic domains are modeled with atomistic or quantum mechanical descriptions, while continuum mechanics models (including fluid mechanics, elastic mechanics, and continuum electrostatics) are applied to the macroscopic domains. This multiphysics description is integrated through an energy functional formalism and the resulting Euler-Lagrange equation is employed to derive a variety of governing partial differential equations for different solvation and transport processes; e.g., the Laplace-Beltrami equation for the solvent-solute interface, Poisson or Poisson-Boltzmann equations for electrostatic potentials, the Nernst-Planck equation for ion densities, and the Kohn-Sham equation for solute electron density. Extensive validation of these models has been carried out over hundreds of molecules, including proteins and ion channels, and the experimental data have been compared in terms of solvation energies, voltage-current curves, and density distributions. We also propose a new quantum model for electrolyte transport.« less

Fundamental equations of a mixture of gas and small spherical solid particles from simple kinetic theory.

NASA Technical Reports Server (NTRS)

Pai, S. I.

1973-01-01

The fundamental equations of a mixture of a gas and pseudofluid of small spherical solid particles are derived from the Boltzmann equation of two-fluid theory. The distribution function of the gas molecules is defined in the same manner as in the ordinary kinetic theory of gases, but the distribution function for the solid particles is different from that of the gas molecules, because it is necessary to take into account the different size and physical properties of solid particles. In the proposed simple kinetic theory, two additional parameters are introduced: one is the radius of the spheres and the other is the instantaneous temperature of the solid particles in the distribution of the solid particles. The Boltzmann equation for each species of the mixture is formally written, and the transfer equations of these Boltzmann equations are derived and compared to the well-known fundamental equations of the mixture of a gas and small solid particles from continuum theory. The equations obtained reveal some insight into various terms in the fundamental equations. For instance, the partial pressure of the pseudofluid of solid particles is not negligible if the volume fraction of solid particles is not negligible as in the case of lunar ash flow.

Computational attributes of the integral form of the equation of transfer

NASA Technical Reports Server (NTRS)

Frankel, J. I.

1991-01-01

Difficulties can arise in radiative and neutron transport calculations when a highly anisotropic scattering phase function is present. In the presence of anisotropy, currently used numerical solutions are based on the integro-differential form of the linearized Boltzmann transport equation. This paper, departs from classical thought and presents an alternative numerical approach based on application of the integral form of the transport equation. Use of the integral formalism facilitates the following steps: a reduction in dimensionality of the system prior to discretization, the use of symbolic manipulation to augment the computational procedure, and the direct determination of key physical quantities which are derivable through the various Legendre moments of the intensity. The approach is developed in the context of radiative heat transfer in a plane-parallel geometry, and results are presented and compared with existing benchmark solutions. Encouraging results are presented to illustrate the potential of the integral formalism for computation. The integral formalism appears to possess several computational attributes which are well-suited to radiative and neutron transport calculations.

SU-G-TeP1-15: Toward a Novel GPU Accelerated Deterministic Solution to the Linear Boltzmann Transport Equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yang, R; Fallone, B; Cross Cancer Institute, Edmonton, AB

Purpose: To develop a Graphic Processor Unit (GPU) accelerated deterministic solution to the Linear Boltzmann Transport Equation (LBTE) for accurate dose calculations in radiotherapy (RT). A deterministic solution yields the potential for major speed improvements due to the sparse matrix-vector and vector-vector multiplications and would thus be of benefit to RT. Methods: In order to leverage the massively parallel architecture of GPUs, the first order LBTE was reformulated as a second order self-adjoint equation using the Least Squares Finite Element Method (LSFEM). This produces a symmetric positive-definite matrix which is efficiently solved using a parallelized conjugate gradient (CG) solver. Themore » LSFEM formalism is applied in space, discrete ordinates is applied in angle, and the Multigroup method is applied in energy. The final linear system of equations produced is tightly coupled in space and angle. Our code written in CUDA-C was benchmarked on an Nvidia GeForce TITAN-X GPU against an Intel i7-6700K CPU. A spatial mesh of 30,950 tetrahedral elements was used with an S4 angular approximation. Results: To avoid repeating a full computationally intensive finite element matrix assembly at each Multigroup energy, a novel mapping algorithm was developed which minimized the operations required at each energy. Additionally, a parallelized memory mapping for the kronecker product between the sparse spatial and angular matrices, including Dirichlet boundary conditions, was created. Atomicity is preserved by graph-coloring overlapping nodes into separate kernel launches. The one-time mapping calculations for matrix assembly, kronecker product, and boundary condition application took 452±1ms on GPU. Matrix assembly for 16 energy groups took 556±3s on CPU, and 358±2ms on GPU using the mappings developed. The CG solver took 93±1s on CPU, and 468±2ms on GPU. Conclusion: Three computationally intensive subroutines in deterministically solving the LBTE have been formulated on GPU, resulting in two orders of magnitude speedup. Funding support from Natural Sciences and Engineering Research Council and Alberta Innovates Health Solutions. Dr. Fallone is a co-founder and CEO of MagnetTx Oncology Solutions (under discussions to license Alberta bi-planar linac MR for commercialization).« less

Poisson-Boltzmann versus Size-Modified Poisson-Boltzmann Electrostatics Applied to Lipid Bilayers.

PubMed

Wang, Nuo; Zhou, Shenggao; Kekenes-Huskey, Peter M; Li, Bo; McCammon, J Andrew

2014-12-26

Mean-field methods, such as the Poisson-Boltzmann equation (PBE), are often used to calculate the electrostatic properties of molecular systems. In the past two decades, an enhancement of the PBE, the size-modified Poisson-Boltzmann equation (SMPBE), has been reported. Here, the PBE and the SMPBE are reevaluated for realistic molecular systems, namely, lipid bilayers, under eight different sets of input parameters. The SMPBE appears to reproduce the molecular dynamics simulation results better than the PBE only under specific parameter sets, but in general, it performs no better than the Stern layer correction of the PBE. These results emphasize the need for careful discussions of the accuracy of mean-field calculations on realistic systems with respect to the choice of parameters and call for reconsideration of the cost-efficiency and the significance of the current SMPBE formulation.

Core-Collapse Supernovae Explored by Multi-D Boltzmann Hydrodynamic Simulations

NASA Astrophysics Data System (ADS)

Sumiyoshi, Kohsuke; Nagakura, Hiroki; Iwakami, Wakana; Furusawa, Shun; Matsufuru, Hideo; Imakura, Akira; Yamada, Shoichi

We report the latest results of numerical simulations of core-collapse supernovae by solving multi-D neutrino-radiation hydrodynamics with Boltzmann equations. One of the longstanding issues of the explosion mechanism of supernovae has been uncertainty in the approximations of the neutrino transfer in multi-D such as the diffusion approximation and ray-by-ray method. The neutrino transfer is essential, together with 2D/3D hydrodynamical instabilities, to evaluate the neutrino heating behind the shock wave for successful explosions and to predict the neutrino burst signals. We tackled this difficult problem by utilizing our solver of the 6D Boltzmann equation for neutrinos in 3D space and 3D neutrino momentum space coupled with multi-D hydrodynamics adding special and general relativistic extensions. We have performed a set of 2D core-collapse simulations from 11M ⊙ and 15M ⊙ stars on K-computer in Japan by following long-term evolution over 400 ms after bounce to reveal the outcome from the full Boltzmann hydrodynamic simulations with a sophisticated equation of state with multi-nuclear species and updated rates for electron captures on nuclei.

Multi-Group Maximum Entropy Model for Translational Non-Equilibrium

NASA Technical Reports Server (NTRS)

Jayaraman, Vegnesh; Liu, Yen; Panesi, Marco

2017-01-01

The aim of the current work is to describe a new model for flows in translational non- equilibrium. Starting from the statistical description of a gas proposed by Boltzmann, the model relies on a domain decomposition technique in velocity space. Using the maximum entropy principle, the logarithm of the distribution function in each velocity sub-domain (group) is expressed with a power series in molecular velocity. New governing equations are obtained using the method of weighted residuals by taking the velocity moments of the Boltzmann equation. The model is applied to a spatially homogeneous Boltzmann equation with a Bhatnagar-Gross-Krook1(BGK) model collision operator and the relaxation of an initial non-equilibrium distribution to a Maxwellian is studied using the model. In addition, numerical results obtained using the model for a 1D shock tube problem are also reported.

Modeling electrokinetic flows by consistent implicit incompressible smoothed particle hydrodynamics

DOE PAGES

Pan, Wenxiao; Kim, Kyungjoo; Perego, Mauro; ...

2017-01-03

In this paper, we present a consistent implicit incompressible smoothed particle hydrodynamics (I 2SPH) discretization of Navier–Stokes, Poisson–Boltzmann, and advection–diffusion equations subject to Dirichlet or Robin boundary conditions. It is applied to model various two and three dimensional electrokinetic flows in simple or complex geometries. The accuracy and convergence of the consistent I 2SPH are examined via comparison with analytical solutions, grid-based numerical solutions, or empirical models. Lastly, the new method provides a framework to explore broader applications of SPH in microfluidics and complex fluids with charged objects, such as colloids and biomolecules, in arbitrary complex geometries.

Computation of Hypersonic Shock Wave Flows of Multi-Component Reactive Gas Mixtures Using the Generalized Boltzmann Equation

DTIC Science & Technology

2009-03-27

ones like the Lennard - Jones potential with established parameters for each gas (e.g. N2 and 02), and for inelastic collisions DSMC method employs...solution of the collision integral. Lennard - Jones potential with two free parameters is used to obtain the elastic cross-section of the gas molecules...and the so called "combinatory relations" are used to obtain parameters of Lennard - Jones potential for an interaction of molecule A with molecule B

Energy and technology review: Engineering modeling

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cabayan, H.S.; Goudreau, G.L.; Ziolkowski, R.W.

1986-10-01

This report presents information concerning: Modeling Canonical Problems in Electromagnetic Coupling Through Apertures; Finite-Element Codes for Computing Electrostatic Fields; Finite-Element Modeling of Electromagnetic Phenomena; Modeling Microwave-Pulse Compression in a Resonant Cavity; Lagrangian Finite-Element Analysis of Penetration Mechanics; Crashworthiness Engineering; Computer Modeling of Metal-Forming Processes; Thermal-Mechanical Modeling of Tungsten Arc Welding; Modeling Air Breakdown Induced by Electromagnetic Fields; Iterative Techniques for Solving Boltzmann's Equations for p-Type Semiconductors; Semiconductor Modeling; and Improved Numerical-Solution Techniques in Large-Scale Stress Analysis.

Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks

NASA Astrophysics Data System (ADS)

Pogan, Alin; Zumbrun, Kevin

2018-06-01

We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman-Enskog picture of near-equilibrium flow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dynamical systems point of view, with a number of interesting modifications to accommodate ill-posedness of the underlying evolution equation.

Recurrent procedure for constructing nonisotropic matrix elements of the collision integral of the nonlinear Boltzmann equation

NASA Astrophysics Data System (ADS)

Ender, I. A.; Bakaleinikov, L. A.; Flegontova, E. Yu.; Gerasimenko, A. B.

2017-08-01

We have proposed an algorithm for the sequential construction of nonisotropic matrix elements of the collision integral, which are required to solve the nonlinear Boltzmann equation using the moments method. The starting elements of the matrix are isotropic and assumed to be known. The algorithm can be used for an arbitrary law of interactions for any ratio of the masses of colliding particles.

Nonlinear stability of the 1D Boltzmann equation in a periodic box

NASA Astrophysics Data System (ADS)

Wu, Kung-Chien

2018-05-01

We study the nonlinear stability of the Boltzmann equation in the 1D periodic box with size , where is the Knudsen number. The convergence rate is for small time region and exponential for large time region. Moreover, the exponential rate depends on the size of the domain (Knudsen number). This problem is highly nonlinear and hence we need more careful analysis to control the nonlinear term.

Statistical mechanics of unsupervised feature learning in a restricted Boltzmann machine with binary synapses

NASA Astrophysics Data System (ADS)

Huang, Haiping

2017-05-01

Revealing hidden features in unlabeled data is called unsupervised feature learning, which plays an important role in pretraining a deep neural network. Here we provide a statistical mechanics analysis of the unsupervised learning in a restricted Boltzmann machine with binary synapses. A message passing equation to infer the hidden feature is derived, and furthermore, variants of this equation are analyzed. A statistical analysis by replica theory describes the thermodynamic properties of the model. Our analysis confirms an entropy crisis preceding the non-convergence of the message passing equation, suggesting a discontinuous phase transition as a key characteristic of the restricted Boltzmann machine. Continuous phase transition is also confirmed depending on the embedded feature strength in the data. The mean-field result under the replica symmetric assumption agrees with that obtained by running message passing algorithms on single instances of finite sizes. Interestingly, in an approximate Hopfield model, the entropy crisis is absent, and a continuous phase transition is observed instead. We also develop an iterative equation to infer the hyper-parameter (temperature) hidden in the data, which in physics corresponds to iteratively imposing Nishimori condition. Our study provides insights towards understanding the thermodynamic properties of the restricted Boltzmann machine learning, and moreover important theoretical basis to build simplified deep networks.

On the Debye-Hückel effect of electric screening

NASA Astrophysics Data System (ADS)

Campos, L. M. B. C.; Lau, F. J. P.

2014-07-01

The paper considers non-linear self-consistent electric potential equation (Sec. I), due to a cloud made of a single species of electric charges, satisfying a Boltzmann distribution law (Sec. II). Exact solutions are obtained in a simple logarithmic form, in three cases: (Sec. III) spherical radial symmetry; (Sec. IV) plane parallel symmetry; (Sec. V) a special case of azimuthal-cylindrical symmetry. All these solutions, and their transformations (Sec. VI), involve the Debye-Hückel radius; the latter was originally defined from a solution of the linearized self-consistent potential equation. Using an exact solution of the self-consistent potential equation, the distance at which the potential vanishes differs from the Debye-Hückel radius by a factor of √2 . The preceding (Secs. II-VI) simple logarithmic exact solutions of the self-consistent potential equations involve no arbitrary constants, and thus are special or singular integrals not the general integral. The general solution of the self-consistent potential equation is obtained in the plane parallel case (Sec. VII), and it involves two arbitrary constants that can be reduced to one via a translation (Sec. VIII). The plots of dimensionless potential (Figure 1), electric field (Figure 2), charge density (Figure 3), and total charge between ζ and infinity (Figure 4), versus distance normalized to Debye-Hückel radius ζ ≡ z/a, show that (Sec. IX) there is a continuum of solutions, ranging from a charge distribution concentrated inside the Debye-Hückel radius to one spread-out beyond it. The latter case leads to the limiting case of logarithmic potential, and stronger electric field; the former case, of very concentrated charge distribution, leads to a fratricide effect and weaker electric field.

On the Debye–Hückel effect of electric screening

DOE Office of Scientific and Technical Information (OSTI.GOV)

Campos, L. M. B. C.; Lau, F. J. P.

2014-07-15

The paper considers non-linear self-consistent electric potential equation (Sec. I), due to a cloud made of a single species of electric charges, satisfying a Boltzmann distribution law (Sec. II). Exact solutions are obtained in a simple logarithmic form, in three cases: (Sec. III) spherical radial symmetry; (Sec. IV) plane parallel symmetry; (Sec. V) a special case of azimuthal-cylindrical symmetry. All these solutions, and their transformations (Sec. VI), involve the Debye-Hückel radius; the latter was originally defined from a solution of the linearized self-consistent potential equation. Using an exact solution of the self-consistent potential equation, the distance at which the potentialmore » vanishes differs from the Debye-Hückel radius by a factor of √(2). The preceding (Secs. II–VI) simple logarithmic exact solutions of the self-consistent potential equations involve no arbitrary constants, and thus are special or singular integrals not the general integral. The general solution of the self-consistent potential equation is obtained in the plane parallel case (Sec. VII), and it involves two arbitrary constants that can be reduced to one via a translation (Sec. VIII). The plots of dimensionless potential (Figure 1), electric field (Figure 2), charge density (Figure 3), and total charge between ζ and infinity (Figure 4), versus distance normalized to Debye-Hückel radius ζ ≡ z/a, show that (Sec. IX) there is a continuum of solutions, ranging from a charge distribution concentrated inside the Debye-Hückel radius to one spread-out beyond it. The latter case leads to the limiting case of logarithmic potential, and stronger electric field; the former case, of very concentrated charge distribution, leads to a fratricide effect and weaker electric field.« less

Modeling Electrokinetic Flows by the Smoothed Profile Method

PubMed Central

Luo, Xian; Beskok, Ali; Karniadakis, George Em

2010-01-01

We propose an efficient modeling method for electrokinetic flows based on the Smoothed Profile Method (SPM) [1–4] and spectral element discretizations. The new method allows for arbitrary differences in the electrical conductivities between the charged surfaces and the the surrounding electrolyte solution. The electrokinetic forces are included into the flow equations so that the Poisson-Boltzmann and electric charge continuity equations are cast into forms suitable for SPM. The method is validated by benchmark problems of electroosmotic flow in straight channels and electrophoresis of charged cylinders. We also present simulation results of electrophoresis of charged microtubules, and show that the simulated electrophoretic mobility and anisotropy agree with the experimental values. PMID:20352076

Unsteady non-Newtonian hydrodynamics in granular gases.

PubMed

Astillero, Antonio; Santos, Andrés

2012-02-01

The temporal evolution of a dilute granular gas, both in a compressible flow (uniform longitudinal flow) and in an incompressible flow (uniform shear flow), is investigated by means of the direct simulation Monte Carlo method to solve the Boltzmann equation. Emphasis is laid on the identification of a first "kinetic" stage (where the physical properties are strongly dependent on the initial state) subsequently followed by an unsteady "hydrodynamic" stage (where the momentum fluxes are well-defined non-Newtonian functions of the rate of strain). The simulation data are seen to support this two-stage scenario. Furthermore, the rheological functions obtained from simulation are well described by an approximate analytical solution of a model kinetic equation. © 2012 American Physical Society

Electron Transport Coefficients and Effective Ionization Coefficients in SF6-O2 and SF6-Air Mixtures Using Boltzmann Analysis

NASA Astrophysics Data System (ADS)

Wei, Linsheng; Xu, Min; Yuan, Dingkun; Zhang, Yafang; Hu, Zhaoji; Tan, Zhihong

2014-10-01

The electron drift velocity, electron energy distribution function (EEDF), density-normalized effective ionization coefficient and density-normalized longitudinal diffusion velocity are calculated in SF6-O2 and SF6-Air mixtures. The experimental results from a pulsed Townsend discharge are plotted for comparison with the numerical results. The reduced field strength varies from 40 Td to 500 Td (1 Townsend=10-17 V·cm2) and the SF6 concentration ranges from 10% to 100%. A Boltzmann equation associated with the two-term spherical harmonic expansion approximation is utilized to gain the swarm parameters in steady-state Townsend. Results show that the accuracy of the Boltzmann solution with a two-term expansion in calculating the electron drift velocity, electron energy distribution function, and density-normalized effective ionization coefficient is acceptable. The effective ionization coefficient presents a distinct relationship with the SF6 content in the mixtures. Moreover, the E/Ncr values in SF6-Air mixtures are higher than those in SF6-O2 mixtures and the calculated value E/Ncr in SF6-O2 and SF6-Air mixtures is lower than the measured value in SF6-N2. Parametric studies conducted on these parameters using the Boltzmann analysis offer substantial insight into the plasma physics, as well as a basis to explore the ozone generation process.

Lattice Boltzmann and Navier-Stokes Cartesian CFD Approaches for Airframe Noise Predictions

NASA Technical Reports Server (NTRS)

Barad, Michael F.; Kocheemoolayil, Joseph G.; Kiris, Cetin C.

2017-01-01

Lattice Boltzmann (LB) and compressible Navier-Stokes (NS) equations based computational fluid dynamics (CFD) approaches are compared for simulating airframe noise. Both LB and NS CFD approaches are implemented within the Launch Ascent and Vehicle Aerodynamics (LAVA) framework. Both schemes utilize the same underlying Cartesian structured mesh paradigm with provision for local adaptive grid refinement and sub-cycling in time. We choose a prototypical massively separated, wake-dominated flow ideally suited for Cartesian-grid based approaches in this study - The partially-dressed, cavity-closed nose landing gear (PDCC-NLG) noise problem from AIAA's Benchmark problems for Airframe Noise Computations (BANC) series of workshops. The relative accuracy and computational efficiency of the two approaches are systematically compared. Detailed comments are made on the potential held by LB to significantly reduce time-to-solution for a desired level of accuracy within the context of modeling airframes noise from first principles.

A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows

NASA Astrophysics Data System (ADS)

Güçlü, Y.; Hitchon, W. N. G.

2012-04-01

The term 'Convected Scheme' (CS) refers to a family of algorithms, most usually applied to the solution of Boltzmann's equation, which uses a method of characteristics in an integral form to project an initial cell forward to a group of final cells. As such the CS is a 'forward-trajectory' semi-Lagrangian scheme. For multi-dimensional simulations of neutral gas flows, the cell-centered version of this semi-Lagrangian (CCSL) scheme has advantages over other options due to its implementation simplicity, low memory requirements, and easier treatment of boundary conditions. The main drawback of the CCSL-CS to date has been its high numerical diffusion in physical space, because of the 2nd order remapping that takes place at the end of each time step. By means of a modified equation analysis, it is shown that a high order estimate of the remapping error can be obtained a priori, and a small correction to the final position of the cells can be applied upon remapping, in order to achieve full compensation of this error. The resulting scheme is 4th order accurate in space while retaining the desirable properties of the CS: it is conservative and positivity-preserving, and the overall algorithm complexity is not appreciably increased. Two monotone (i.e. non-oscillating) versions of the fourth order CCSL-CS are also presented: one uses a common flux-limiter approach; the other uses a non-polynomial reconstruction to evaluate the derivatives of the density function. The method is illustrated in simple one- and two-dimensional examples, and a fully 3D solution of the Boltzmann equation describing expansion of a gas into vacuum through a cylindrical tube.

Foundations of modelling of nonequilibrium low-temperature plasmas

NASA Astrophysics Data System (ADS)

Alves, L. L.; Bogaerts, A.; Guerra, V.; Turner, M. M.

2018-02-01

This work explains the need for plasma models, introduces arguments for choosing the type of model that better fits the purpose of each study, and presents the basics of the most common nonequilibrium low-temperature plasma models and the information available from each one, along with an extensive list of references for complementary in-depth reading. The paper presents the following models, organised according to the level of multi-dimensional description of the plasma: kinetic models, based on either a statistical particle-in-cell/Monte-Carlo approach or the solution to the Boltzmann equation (in the latter case, special focus is given to the description of the electron kinetics); multi-fluid models, based on the solution to the hydrodynamic equations; global (spatially-average) models, based on the solution to the particle and energy rate-balance equations for the main plasma species, usually including a very complete reaction chemistry; mesoscopic models for plasma-surface interaction, adopting either a deterministic approach or a stochastic dynamical Monte-Carlo approach. For each plasma model, the paper puts forward the physics context, introduces the fundamental equations, presents advantages and limitations, also from a numerical perspective, and illustrates its application with some examples. Whenever pertinent, the interconnection between models is also discussed, in view of multi-scale hybrid approaches.

Nonlinear waves in electron-positron-ion plasmas including charge separation

NASA Astrophysics Data System (ADS)

Mugemana, A.; Moolla, S.; Lazarus, I. J.

2017-02-01

Nonlinear low-frequency electrostatic waves in a magnetized, three-component plasma consisting of hot electrons, hot positrons and warm ions have been investigated. The electrons and positrons are assumed to have Boltzmann density distributions while the motion of the ions are governed by fluid equations. The system is closed with the Poisson equation. This set of equations is numerically solved for the electric field. The effects of the driving electric field, ion temperature, positron density, ion drift, Mach number and propagation angle are investigated. It is shown that depending on the driving electric field, ion temperature, positron density, ion drift, Mach number and propagation angle, the numerical solutions exhibit waveforms that are sinusoidal, sawtooth and spiky. The introduction of the Poisson equation increased the Mach number required to generate the waveforms but the driving electric field E 0 was reduced. The results are compared with satellite observations.

Development of a Prototype Lattice Boltzmann Code for CFD of Fusion Systems.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pattison, Martin J; Premnath, Kannan N; Banerjee, Sanjoy

2007-02-26

Designs of proposed fusion reactors, such as the ITER project, typically involve the use of liquid metals as coolants in components such as heat exchangers, which are generally subjected to strong magnetic fields. These fields induce electric currents in the fluids, resulting in magnetohydrodynamic (MHD) forces which have important effects on the flow. The objective of this SBIR project was to develop computational techniques based on recently developed lattice Boltzmann techniques for the simulation of these MHD flows and implement them in a computational fluid dynamics (CFD) code for the study of fluid flow systems encountered in fusion engineering. Themore » code developed during this project, solves the lattice Boltzmann equation, which is a kinetic equation whose behaviour represents fluid motion. This is in contrast to most CFD codes which are based on finite difference/finite volume based solvers. The lattice Boltzmann method (LBM) is a relatively new approach which has a number of advantages compared with more conventional methods such as the SIMPLE or projection method algorithms that involve direct solution of the Navier-Stokes equations. These are that the LBM is very well suited to parallel processing, with almost linear scaling even for very large numbers of processors. Unlike other methods, the LBM does not require solution of a Poisson pressure equation leading to a relatively fast execution time. A particularly attractive property of the LBM is that it can handle flows in complex geometries very easily. It can use simple rectangular grids throughout the computational domain -- generation of a body-fitted grid is not required. A recent advance in the LBM is the introduction of the multiple relaxation time (MRT) model; the implementation of this model greatly enhanced the numerical stability when used in lieu of the single relaxation time model, with only a small increase in computer time. Parallel processing was implemented using MPI and demonstrated the ability of the LBM to scale almost linearly. The equation for magnetic induction was also solved using a lattice Boltzmann method. This approach has the advantage that it fits in well to the framework used for the hydrodynamic equations, but more importantly that it preserves the ability of the code to run efficiently on parallel architectures. Since the LBM is a relatively recent model, a number of new developments were needed to solve the magnetic induction equation for practical problems. Existing methods were only suitable for cases where the fluid viscosity and the magnetic resistivity are of the same order, and a preconditioning method was used to allow the simulation of liquid metals, where these properties differ by several orders of magnitude. An extension of this method to the hydrodynamic equations allowed faster convergence to steady state. A new method of imposing boundary conditions using an extrapolation technique was derived, enabling the magnetic field at a boundary to be specified. Also, a technique by which the grid can be stretched was formulated to resolve thin layers at high imposed magnetic fields, allowing flows with Hartmann numbers of several thousand to be quickly and efficiently simulated. In addition, a module has been developed to calculate the temperature field and heat transfer. This uses a total variation diminishing scheme to solve the equations and is again very amenable to parallelisation. Although, the module was developed with thermal modelling in mind, it can also be applied to passive scalar transport. The code is fully three dimensional and has been applied to a wide variety of cases, including both laminar and turbulent flows. Validations against a series of canonical problems involving both MHD effects and turbulence have clearly demonstrated the ability of the LBM to properly model these types of flow. As well as applications to fusion engineering, the resulting code is flexible enough to be applied to a wide range of other flows, in particular those requiring parallel computations with many processors. For example, at present it is being used for studies in aerodynamics and acoustics involving flows at high Reynolds numbers. It is anticipated that it will be used for multiphase flow applications in the near future.« less

A new splitting scheme to the discrete Boltzmann equation for non-ideal gases on non-uniform meshes

NASA Astrophysics Data System (ADS)

Patel, Saumil; Lee, Taehun

2016-12-01

We present a novel numerical procedure for solving the discrete Boltzmann equations (DBE) on non-uniform meshes. Our scheme is based on the Strang splitting method where we seek to investigate two-phase flow applications. In this note, we investigate the onset of parasitic currents which arise in many computational two-phase algorithms. To the best of our knowledge, the results presented in this work show, for the first time, a spectral element discontinuous Galerkin (SEDG) discretization of a discrete Boltzmann equation which successfully eliminates parasitic currents on non-uniform meshes. With the hope that this technique can be used for applications in complex geometries, calculations are performed on non-uniform mesh distributions by using high-order (spectral), body-fitting quadrilateral elements. Validation and verification of our work is carried out by comparing results against the classical 2D Young-Laplace law problem for a static drop.

Heat dissipation in the quasiballistic regime studied using the Boltzmann equation in the spatial frequency domain

DOE PAGES

Hua, Chengyun; Minnich, Austin J.

2018-01-10

Quasiballistic heat conduction, in which some phonons propagate ballistically over a thermal gradient, has recently become of intense interest. Most works report that the thermal resistance associated with nanoscale heat sources is far larger than predicted by Fourier's law; however, recent experiments show that in certain cases the difference is negligible despite the heaters being far smaller than phonon mean-free paths. In this work, we examine how thermal resistance depends on the heater geometry using analytical solutions of the Boltzmann equation. We show that the spatial frequencies of the heater pattern play the key role in setting the thermal resistancemore » rather than any single geometric parameter, and that for many geometries the thermal resistance in the quasiballistic regime is no different than the Fourier prediction. We further demonstrate that the spectral distribution of the heat source also plays a major role in the resulting transport, unlike in the diffusion regime. Our work provides an intuitive link between the heater geometry, spectral heating distribution, and the effective thermal resistance in the quasiballistic regime, a finding that could impact strategies for thermal management in electronics and other applications.« less

The Physical Mechanism for Retinal Discrete Dark Noise: Thermal Activation or Cellular Ultraweak Photon Emission?

PubMed

Salari, Vahid; Scholkmann, Felix; Bokkon, Istvan; Shahbazi, Farhad; Tuszynski, Jack

2016-01-01

For several decades the physical mechanism underlying discrete dark noise of photoreceptors in the eye has remained highly controversial and poorly understood. It is known that the Arrhenius equation, which is based on the Boltzmann distribution for thermal activation, can model only a part (e.g. half of the activation energy) of the retinal dark noise experimentally observed for vertebrate rod and cone pigments. Using the Hinshelwood distribution instead of the Boltzmann distribution in the Arrhenius equation has been proposed as a solution to the problem. Here, we show that the using the Hinshelwood distribution does not solve the problem completely. As the discrete components of noise are indistinguishable in shape and duration from those produced by real photon induced photo-isomerization, the retinal discrete dark noise is most likely due to 'internal photons' inside cells and not due to thermal activation of visual pigments. Indeed, all living cells exhibit spontaneous ultraweak photon emission (UPE), mainly in the optical wavelength range, i.e., 350-700 nm. We show here that the retinal discrete dark noise has a similar rate as UPE and therefore dark noise is most likely due to spontaneous cellular UPE and not due to thermal activation.

GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method

NASA Astrophysics Data System (ADS)

Gong, Chunye; Liu, Jie; Chi, Lihua; Huang, Haowei; Fang, Jingyue; Gong, Zhenghu

2011-07-01

Graphics Processing Unit (GPU), originally developed for real-time, high-definition 3D graphics in computer games, now provides great faculty in solving scientific applications. The basis of particle transport simulation is the time-dependent, multi-group, inhomogeneous Boltzmann transport equation. The numerical solution to the Boltzmann equation involves the discrete ordinates ( Sn) method and the procedure of source iteration. In this paper, we present a GPU accelerated simulation of one energy group time-independent deterministic discrete ordinates particle transport in 3D Cartesian geometry (Sweep3D). The performance of the GPU simulations are reported with the simulations of vacuum boundary condition. The discussion of the relative advantages and disadvantages of the GPU implementation, the simulation on multi GPUs, the programming effort and code portability are also reported. The results show that the overall performance speedup of one NVIDIA Tesla M2050 GPU ranges from 2.56 compared with one Intel Xeon X5670 chip to 8.14 compared with one Intel Core Q6600 chip for no flux fixup. The simulation with flux fixup on one M2050 is 1.23 times faster than on one X5670.

Heat dissipation in the quasiballistic regime studied using the Boltzmann equation in the spatial frequency domain

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hua, Chengyun; Minnich, Austin J.

Quasiballistic heat conduction, in which some phonons propagate ballistically over a thermal gradient, has recently become of intense interest. Most works report that the thermal resistance associated with nanoscale heat sources is far larger than predicted by Fourier's law; however, recent experiments show that in certain cases the difference is negligible despite the heaters being far smaller than phonon mean-free paths. In this work, we examine how thermal resistance depends on the heater geometry using analytical solutions of the Boltzmann equation. We show that the spatial frequencies of the heater pattern play the key role in setting the thermal resistancemore » rather than any single geometric parameter, and that for many geometries the thermal resistance in the quasiballistic regime is no different than the Fourier prediction. We further demonstrate that the spectral distribution of the heat source also plays a major role in the resulting transport, unlike in the diffusion regime. Our work provides an intuitive link between the heater geometry, spectral heating distribution, and the effective thermal resistance in the quasiballistic regime, a finding that could impact strategies for thermal management in electronics and other applications.« less

Ca/Na selectivity coefficients from the Poisson-Boltzmann theory

NASA Astrophysics Data System (ADS)

Hedström, Magnus; Karnland, Ola

As a model for ion equilibrium in montmorillonite, the Poisson-Boltzmann (PB) equation was solved for two parallel charged surfaces in contact with an external NaCl/CaCl 2 mixed solution. The ion concentration profiles in the montmorillonite interlayer were obtained from the PB equation and integration of those gave the occupancy of Na + and Ca 2+ in the clay. That information together with the composition of the external electrolyte were then used for the calculation of the Gaines-Thomas selectivity coefficient K GT. The predictions from the model were compared to experimental data from batch as well as compacted conditions, and the agreement was generally good. With a surface layer-charge density of one unit charge per 145 Å 2, which is close to the value for Wyoming-type montmorillonite, the calculated selectivity coefficients were found to vary from about 4 in batch to 8 in compacted montmorillonite with dry density ∼1700 kg/m 3. From the point of view of assessing the evolution, with regard to sodium-calcium ion exchange, of the bentonite buffer in a repository for spent nuclear fuel, these results justify the use of data obtained in batch experiments.

Ionization of Interstellar Hydrogen

NASA Astrophysics Data System (ADS)

Whang, Y. C.

1996-09-01

Interstellar hydrogen can penetrate through the heliopause, enter the heliosphere, and may become ionized by photoionization and by charge exchange with solar wind protons. A fluid model is introduced to study the flow of interstellar hydrogen in the heliosphere. The flow is governed by moment equations obtained from integration of the Boltzmann equation over the velocity space. Under the assumption that the flow is steady axisymmetric and the pressure is isotropic, we develop a method of solution for this fluid model. This model and the method of solution can be used to study the flow of neutral hydrogen with various forms of ionization rate β and boundary conditions for the flow on the upwind side. We study the solution of a special case in which the ionization rate β is inversely proportional to R2 and the interstellar hydrogen flow is uniform at infinity on the upwind side. We solve the moment equations directly for the normalized density NH/NN∞, bulk velocity VH/VN∞, and temperature TH/TN∞ of interstellar hydrogen as functions of r/λ and z/λ, where λ is the ionization scale length. The solution is compared with the kinetic theory solution of Lallement et al. The fluid solution is much less time-consuming than the kinetic theory solutions. Since the ionization rate for production of pickup protons is directly proportional to the local density of neutral hydrogen, the high-resolution solution of interstellar neutral hydrogen obtained here will be used to study the global distribution of pickup protons.

Continuum description of ionic and dielectric shielding for molecular-dynamics simulations of proteins in solution

NASA Astrophysics Data System (ADS)

Egwolf, Bernhard; Tavan, Paul

2004-01-01

We extend our continuum description of solvent dielectrics in molecular-dynamics (MD) simulations [B. Egwolf and P. Tavan, J. Chem. Phys. 118, 2039 (2003)], which has provided an efficient and accurate solution of the Poisson equation, to ionic solvents as described by the linearized Poisson-Boltzmann (LPB) equation. We start with the formulation of a general theory for the electrostatics of an arbitrarily shaped molecular system, which consists of partially charged atoms and is embedded in a LPB continuum. This theory represents the reaction field induced by the continuum in terms of charge and dipole densities localized within the molecular system. Because these densities cannot be calculated analytically for systems of arbitrary shape, we introduce an atom-based discretization and a set of carefully designed approximations. This allows us to represent the densities by charges and dipoles located at the atoms. Coupled systems of linear equations determine these multipoles and can be rapidly solved by iteration during a MD simulation. The multipoles yield the reaction field forces and energies. Finally, we scrutinize the quality of our approach by comparisons with an analytical solution restricted to perfectly spherical systems and with results of a finite difference method.

Physical scales in the Wigner-Boltzmann equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nedjalkov, M., E-mail: mixi@iue.tuwien.ac.at; Selberherr, S.; Ferry, D.K.

2013-01-15

The Wigner-Boltzmann equation provides the Wigner single particle theory with interactions with bosonic degrees of freedom associated with harmonic oscillators, such as phonons in solids. Quantum evolution is an interplay of two transport modes, corresponding to the common coherent particle-potential processes, or to the decoherence causing scattering due to the oscillators. Which evolution mode will dominate depends on the scales of the involved physical quantities. A dimensionless formulation of the Wigner-Boltzmann equation is obtained, where these scales appear as dimensionless strength parameters. A notion called scaling theorem is derived, linking the strength parameters to the coupling with the oscillators. Itmore » is shown that an increase of this coupling is equivalent to a reduction of both the strength of the electric potential, and the coherence length. Secondly, the existence of classes of physically different, but mathematically equivalent setups of the Wigner-Boltzmann evolution is demonstrated. - Highlights: Black-Right-Pointing-Pointer Dimensionless parameters determine the ratio of quantum or classical WB evolution. Black-Right-Pointing-Pointer The scaling theorem evaluates the decoherence effect due to scattering. Black-Right-Pointing-Pointer Evolution processes are grouped into classes of equivalence.« less

SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson-Boltzmann equation.

PubMed

Xie, Yang; Ying, Jinyong; Xie, Dexuan

2017-03-30

SMPBS (Size Modified Poisson-Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson-Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson-Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile-friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware-accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.

Mean-field message-passing equations in the Hopfield model and its generalizations

NASA Astrophysics Data System (ADS)

Mézard, Marc

2017-02-01

Motivated by recent progress in using restricted Boltzmann machines as preprocessing algorithms for deep neural network, we revisit the mean-field equations [belief-propagation and Thouless-Anderson Palmer (TAP) equations] in the best understood of such machines, namely the Hopfield model of neural networks, and we explicit how they can be used as iterative message-passing algorithms, providing a fast method to compute the local polarizations of neurons. In the "retrieval phase", where neurons polarize in the direction of one memorized pattern, we point out a major difference between the belief propagation and TAP equations: The set of belief propagation equations depends on the pattern which is retrieved, while one can use a unique set of TAP equations. This makes the latter method much better suited for applications in the learning process of restricted Boltzmann machines. In the case where the patterns memorized in the Hopfield model are not independent, but are correlated through a combinatorial structure, we show that the TAP equations have to be modified. This modification can be seen either as an alteration of the reaction term in TAP equations or, more interestingly, as the consequence of message passing on a graphical model with several hidden layers, where the number of hidden layers depends on the depth of the correlations in the memorized patterns. This layered structure is actually necessary when one deals with more general restricted Boltzmann machines.

Theoretical and numerical study of axisymmetric lattice Boltzmann models

NASA Astrophysics Data System (ADS)

Huang, Haibo; Lu, Xi-Yun

2009-07-01

The forcing term in the lattice Boltzmann equation (LBE) is usually used to mimic Navier-Stokes equations with a body force. To derive axisymmetric model, forcing terms are incorporated into the two-dimensional (2D) LBE to mimic the additional axisymmetric contributions in 2D Navier-Stokes equations in cylindrical coordinates. Many axisymmetric lattice Boltzmann D2Q9 models were obtained through the Chapman-Enskog expansion to recover the 2D Navier-Stokes equations in cylindrical coordinates [I. Halliday , Phys. Rev. E 64, 011208 (2001); K. N. Premnath and J. Abraham, Phys. Rev. E 71, 056706 (2005); T. S. Lee, H. Huang, and C. Shu, Int. J. Mod. Phys. C 17, 645 (2006); T. Reis and T. N. Phillips, Phys. Rev. E 75, 056703 (2007); J. G. Zhou, Phys. Rev. E 78, 036701 (2008)]. The theoretical differences between them are discussed in detail. Numerical studies were also carried out by simulating two different flows to make a comparison on these models’ accuracy and τ sensitivity. It is found all these models are able to obtain accurate results and have the second-order spatial accuracy. However, the model C [J. G. Zhou, Phys. Rev. E 78, 036701 (2008)] is the most stable one in terms of τ sensitivity. It is also found that if density of fluid is defined in its usual way and not directly relevant to source terms, the lattice Boltzmann model seems more stable.

Development of an Innovative Algorithm for Aerodynamics-Structure Interaction Using Lattice Boltzmann Method

NASA Technical Reports Server (NTRS)

Mei, Ren-Wei; Shyy, Wei; Yu, Da-Zhi; Luo, Li-Shi; Rudy, David (Technical Monitor)

2001-01-01

The lattice Boltzmann equation (LBE) is a kinetic formulation which offers an alternative computational method capable of solving fluid dynamics for various systems. Major advantages of the method are owing to the fact that the solution for the particle distribution functions is explicit, easy to implement, and the algorithm is natural to parallelize. In this final report, we summarize the works accomplished in the past three years. Since most works have been published, the technical details can be found in the literature. Brief summary will be provided in this report. In this project, a second-order accurate treatment of boundary condition in the LBE method is developed for a curved boundary and tested successfully in various 2-D and 3-D configurations. To evaluate the aerodynamic force on a body in the context of LBE method, several force evaluation schemes have been investigated. A simple momentum exchange method is shown to give reliable and accurate values for the force on a body in both 2-D and 3-D cases. Various 3-D LBE models have been assessed in terms of efficiency, accuracy, and robustness. In general, accurate 3-D results can be obtained using LBE methods. The 3-D 19-bit model is found to be the best one among the 15-bit, 19-bit, and 27-bit LBE models. To achieve desired grid resolution and to accommodate the far field boundary conditions in aerodynamics computations, a multi-block LBE method is developed by dividing the flow field into various blocks each having constant lattice spacing. Substantial contribution to the LBE method is also made through the development of a new, generalized lattice Boltzmann equation constructed in the moment space in order to improve the computational stability, detailed theoretical analysis on the stability, dispersion, and dissipation characteristics of the LBE method, and computational studies of high Reynolds number flows with singular gradients. Finally, a finite difference-based lattice Boltzmann method is developed for inviscid compressible flows.

Laser short-pulse heating of an aluminum thin film: Energy transfer in electron and lattice sub-systems

NASA Astrophysics Data System (ADS)

Bin Mansoor, Saad; Sami Yilbas, Bekir

2015-08-01

Laser short-pulse heating of an aluminum thin film is considered and energy transfer in the film is formulated using the Boltzmann equation. Since the heating duration is short and the film thickness is considerably small, thermal separation of electron and lattice sub-systems is incorporated in the analysis. The electron-phonon coupling is used to formulate thermal communication of both sub-systems during the heating period. Equivalent equilibrium temperature is introduced to account for the average energy of all phonons around a local point when they redistribute adiabatically to an equilibrium state. Temperature predictions of the Boltzmann equation are compared with those obtained from the two-equation model. It is found that temperature predictions from the Boltzmann equation differ slightly from the two-equation model results. Temporal variation of equivalent equilibrium temperature does not follow the laser pulse intensity in the electron sub-system. The time occurrence of the peak equivalent equilibrium temperature differs for electron and lattice sub-systems, which is attributed to phonon scattering in the irradiated field in the lattice sub-system. In this case, time shift is observed for occurrence of the peak temperature in the lattice sub-system.

Simulation on Thermocapillary-Driven Drop Coalescence by Hybrid Lattice Boltzmann Method

NASA Astrophysics Data System (ADS)

Xie, Haiqiong; Zeng, Zhong; Zhang, Liangqi; Yokota, Yuui; Kawazoe, Yoshiyuki; Yoshikawa, Akira

2016-04-01

A hybrid two-phase model, incorporating lattice Boltzmann method (LBM) and finite difference method (FDM), was developed to investigate the coalescence of two drops during their thermocapillary migration. The lattice Boltzmann method with a multi-relaxation-time (MRT) collision model was applied to solve the flow field for incompressible binary fluids, and the method was implemented in an axisymmetric form. The deformation of the drop interface was captured with the phase-field theory, and the continuum surface force model (CSF) was adopted to introduce the surface tension, which depends on the temperature. Both phase-field equation and the energy equation were solved with the finite difference method. The effects of Marangoni number and Capillary numbers on the drop's motion and coalescence were investigated.

Evaluation of the Performance of the Hybrid Lattice Boltzmann Based Numerical Flux

NASA Astrophysics Data System (ADS)

Zheng, H. W.; Shu, C.

2016-06-01

It is well known that the numerical scheme is a key factor to the stability and accuracy of a Navier-Stokes solver. Recently, a new hybrid lattice Boltzmann numerical flux (HLBFS) is developed by Shu's group. It combines two different LBFS schemes by a switch function. It solves the Boltzmann equation instead of the Euler equation. In this article, the main object is to evaluate the ability of this HLBFS scheme by our in-house cell centered hybrid mesh based Navier-Stokes code. Its performance is examined by several widely-used bench-mark test cases. The comparisons on results between calculation and experiment are conducted. They show that the scheme can capture the shock wave as well as the resolving of boundary layer.

Multiscale Multiphysics and Multidomain Models I: Basic Theory

PubMed Central

Wei, Guo-Wei

2013-01-01

This work extends our earlier two-domain formulation of a differential geometry based multiscale paradigm into a multidomain theory, which endows us the ability to simultaneously accommodate multiphysical descriptions of aqueous chemical, physical and biological systems, such as fuel cells, solar cells, nanofluidics, ion channels, viruses, RNA polymerases, molecular motors and large macromolecular complexes. The essential idea is to make use of the differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain of solvent from the microscopic domain of solute, and dynamically couple continuum and discrete descriptions. Our main strategy is to construct energy functionals to put on an equal footing of multiphysics, including polar (i.e., electrostatic) solvation, nonpolar solvation, chemical potential, quantum mechanics, fluid mechanics, molecular mechanics, coarse grained dynamics and elastic dynamics. The variational principle is applied to the energy functionals to derive desirable governing equations, such as multidomain Laplace-Beltrami (LB) equations for macromolecular morphologies, multidomain Poisson-Boltzmann (PB) equation or Poisson equation for electrostatic potential, generalized Nernst-Planck (NP) equations for the dynamics of charged solvent species, generalized Navier-Stokes (NS) equation for fluid dynamics, generalized Newton's equations for molecular dynamics (MD) or coarse-grained dynamics and equation of motion for elastic dynamics. Unlike the classical PB equation, our PB equation is an integral-differential equation due to solvent-solute interactions. To illustrate the proposed formalism, we have explicitly constructed three models, a multidomain solvation model, a multidomain charge transport model and a multidomain chemo-electro-fluid-MD-elastic model. Each solute domain is equipped with distinct surface tension, pressure, dielectric function, and charge density distribution. In addition to long-range Coulombic interactions, various non-electrostatic solvent-solute interactions are considered in the present modeling. We demonstrate the consistency between the non-equilibrium charge transport model and the equilibrium solvation model by showing the systematical reduction of the former to the latter at equilibrium. This paper also offers a brief review of the field. PMID:25382892

Multiscale Multiphysics and Multidomain Models I: Basic Theory.

PubMed

Wei, Guo-Wei

2013-12-01

This work extends our earlier two-domain formulation of a differential geometry based multiscale paradigm into a multidomain theory, which endows us the ability to simultaneously accommodate multiphysical descriptions of aqueous chemical, physical and biological systems, such as fuel cells, solar cells, nanofluidics, ion channels, viruses, RNA polymerases, molecular motors and large macromolecular complexes. The essential idea is to make use of the differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain of solvent from the microscopic domain of solute, and dynamically couple continuum and discrete descriptions. Our main strategy is to construct energy functionals to put on an equal footing of multiphysics, including polar (i.e., electrostatic) solvation, nonpolar solvation, chemical potential, quantum mechanics, fluid mechanics, molecular mechanics, coarse grained dynamics and elastic dynamics. The variational principle is applied to the energy functionals to derive desirable governing equations, such as multidomain Laplace-Beltrami (LB) equations for macromolecular morphologies, multidomain Poisson-Boltzmann (PB) equation or Poisson equation for electrostatic potential, generalized Nernst-Planck (NP) equations for the dynamics of charged solvent species, generalized Navier-Stokes (NS) equation for fluid dynamics, generalized Newton's equations for molecular dynamics (MD) or coarse-grained dynamics and equation of motion for elastic dynamics. Unlike the classical PB equation, our PB equation is an integral-differential equation due to solvent-solute interactions. To illustrate the proposed formalism, we have explicitly constructed three models, a multidomain solvation model, a multidomain charge transport model and a multidomain chemo-electro-fluid-MD-elastic model. Each solute domain is equipped with distinct surface tension, pressure, dielectric function, and charge density distribution. In addition to long-range Coulombic interactions, various non-electrostatic solvent-solute interactions are considered in the present modeling. We demonstrate the consistency between the non-equilibrium charge transport model and the equilibrium solvation model by showing the systematical reduction of the former to the latter at equilibrium. This paper also offers a brief review of the field.

Electrostatic Interactions Between Glycosaminoglycan Molecules

NASA Astrophysics Data System (ADS)

Song, Fan; Moyne, Christian; Bai, Yi-Long

2005-02-01

The electrostatic interactions between nearest-neighbouring chondroitin sulfate glycosaminoglycan (CS-GAG) molecular chains are obtained on the bottle brush conformation of proteoglycan aggrecan based on an asymptotic solution of the Poisson-Boltzmann equation the CS-GAGs satisfy under the physiological conditions of articular cartilage. The present results show that the interactions are associated intimately with the minimum separation distance and mutual angle between the molecular chains themselves. Further analysis indicates that the electrostatic interactions are not only expressed to be purely exponential in separation distance and decrease with the increasing mutual angle but also dependent sensitively on the saline concentration in the electrolyte solution within the tissue, which is in agreement with the existed relevant conclusions.

Anisotropic nonequilibrium hydrodynamic attractor

NASA Astrophysics Data System (ADS)

Strickland, Michael; Noronha, Jorge; Denicol, Gabriel S.

2018-02-01

We determine the dynamical attractors associated with anisotropic hydrodynamics (aHydro) and the DNMR equations for a 0 +1 d conformal system using kinetic theory in the relaxation time approximation. We compare our results to the nonequilibrium attractor obtained from the exact solution of the 0 +1 d conformal Boltzmann equation, the Navier-Stokes theory, and the second-order Mueller-Israel-Stewart theory. We demonstrate that the aHydro attractor equation resums an infinite number of terms in the inverse Reynolds number. The resulting resummed aHydro attractor possesses a positive longitudinal-to-transverse pressure ratio and is virtually indistinguishable from the exact attractor. This suggests that an optimized hydrodynamic treatment of kinetic theory involves a resummation not only in gradients (Knudsen number) but also in the inverse Reynolds number. We also demonstrate that the DNMR result provides a better approximation of the exact kinetic theory attractor than the Mueller-Israel-Stewart theory. Finally, we introduce a new method for obtaining approximate aHydro equations which relies solely on an expansion in the inverse Reynolds number. We then carry this expansion out to the third order, and compare these third-order results to the exact kinetic theory solution.

Gas-kinetic theory and Boltzmann equation of share price within an equilibrium market hypothesis and ad hoc strategy

NASA Astrophysics Data System (ADS)

Ausloos, M.

2000-09-01

Recent observations have indicated that the traditional equilibrium market hypothesis (EMH; also known as Efficient Market Hypothesis) is unrealistic. It is shown here that it is the analog of a Boltzmann equation in physics, thus having some bad properties of mean-field approximations like a Gaussian distribution of price fluctuations. A kinetic theory for prices can be simply derived, considering in a first approach that market actors have all identical relaxation times, and solved within a Chapman-Enskog like formalism. In closing the set of equations, (i) an equation of state with a pressure and (ii) the equilibrium (isothermal) equation for the price (taken as the order parameter) of a stock as a function of the volume of money available are obtained.

Nonlinear excitations in electron-positron-ion plasmas in accretion disks of active galactic nuclei

DOE Office of Scientific and Technical Information (OSTI.GOV)

Moslem, W. M.; Kourakis, I.; Shukla, P. K.

2007-10-15

The propagation of acoustic nonlinear excitations in an electron-positron-ion (e-p-i) plasma composed of warm electrons and positrons, as well as hot ions, has been investigated by adopting a two-dimensional cylindrical geometry. The electrons and positrons are modeled by hydrodynamic fluid equations, while the ions are assumed to follow a temperature-parametrized Boltzmann distribution (the fixed ion model is recovered in the appropriate limit). This situation applies in the accretion disk near a black hole in active galactic nuclei, where the ion temperature may be as high as 3 to 300 times that of the electrons. Using a reductive perturbation technique, amore » cylindrical Kadomtsev-Petviashvili equation is derived and its exact soliton solutions are presented. Furthermore, real situations in which the strength of the nonlinearity may be weak are considered, so that higher-order nonlinearity plays an important role. Accordingly, an extended cylindrical Kadomtsev-Petviashvili equation is derived, which admits both soliton and double-layer solutions. The characteristics of the nonlinear excitations obtained are investigated in detail.« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dustin Popp; Zander Mausolff; Sedat Goluoglu

We are proposing to use the code, TDKENO, to model TREAT. TDKENO solves the time dependent, three dimensional Boltzmann transport equation with explicit representation of delayed neutrons. Instead of directly integrating this equation, the neutron flux is factored into two components – a rapidly varying amplitude equation and a slowly varying shape equation and each is solved separately on different time scales. The shape equation is solved using the 3D Monte Carlo transport code KENO, from Oak Ridge National Laboratory’s SCALE code package. Using the Monte Carlo method to solve the shape equation is still computationally intensive, but the operationmore » is only performed when needed. The amplitude equation is solved deterministically and frequently, so the solution gives an accurate time-dependent solution without having to repeatedly We have modified TDKENO to incorporate KENO-VI so that we may accurately represent the geometries within TREAT. This paper explains the motivation behind using generalized geometry, and provides the results of our modifications. TDKENO uses the Improved Quasi-Static method to accomplish this. In this method, the neutron flux is factored into two components. One component is a purely time-dependent and rapidly varying amplitude function, which is solved deterministically and very frequently (small time steps). The other is a slowly varying flux shape function that weakly depends on time and is only solved when needed (significantly larger time steps).« less

Electrostatic similarity of proteins: Application of three dimensional spherical harmonic decomposition

PubMed Central

Długosz, Maciej; Trylska, Joanna

2008-01-01

We present a method for describing and comparing global electrostatic properties of biomolecules based on the spherical harmonic decomposition of electrostatic potential data. Unlike other approaches our method does not require any prior three dimensional structural alignment. The electrostatic potential, given as a volumetric data set from a numerical solution of the Poisson or Poisson–Boltzmann equation, is represented with descriptors that are rotation invariant. The method can be applied to large and structurally diverse sets of biomolecules enabling to cluster them according to their electrostatic features. PMID:18624502

Finite difference numerical method for the superlattice Boltzmann transport equation and case comparison of CPU(C) and GPU(CUDA) implementations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Priimak, Dmitri

2014-12-01

We present a finite difference numerical algorithm for solving two dimensional spatially homogeneous Boltzmann transport equation which describes electron transport in a semiconductor superlattice subject to crossed time dependent electric and constant magnetic fields. The algorithm is implemented both in C language targeted to CPU and in CUDA C language targeted to commodity NVidia GPU. We compare performances and merits of one implementation versus another and discuss various software optimisation techniques.

3D Navier-Stokes Flow Analysis for Shared and Distributed Memory MIMD Computers

DTIC Science & Technology

1992-09-15

arithmetical averaged density or Stefan -Boltzmann constant (= 5.67032 x 10-8 ) Oai+1/2 intermediate term for Harten-Yee fluxes - k, O’ constants for k...system of algebraic equations. These equations I are solved using point Gauss- Seidel relaxation. This relaxation scheme is modified to be a lower-upper...interaction of the radiation with the gas. The radiative heat flux per unit area is then I = -(T [EwT - awTdb] (19) Here a is the Stefan Boltzmann

Kinetics of Electrons from Plasma Discharge in a Latent Track Region Induced by Swift Heavy ION Irradiation

NASA Astrophysics Data System (ADS)

Minárik, Stanislav

2015-08-01

While passing swift heavy ion through a material structure, it produces a region of radiation affected material which is known as a "latent track". Scattering motions of electrons interacting with a swift heavy ion are dominant in the latent track region. These phenomena include the electron impurity and phonon scattering processes modified by the interaction with the ion projectile as well as the Coulomb scattering between two electrons. In this paper, we provide detailed derivation of a 3D Boltzmann scattering equation for the description of the relative scattering motion of such electrons. Phase-space distribution function for this non-equilibrioum system of scattering electrons can be found by the solution of mentioned equation.

Modelling of electric characteristics of 150-watt peak solar panel using Boltzmann sigmoid function under various temperature and irradiance

NASA Astrophysics Data System (ADS)

Sapteka, A. A. N. G.; Narottama, A. A. N. M.; Winarta, A.; Amerta Yasa, K.; Priambodo, P. S.; Putra, N.

2018-01-01

Solar energy utilized with solar panel is a renewable energy that needs to be studied further. The site nearest to the equator, it is not surprising, receives the highest solar energy. In this paper, a modelling of electrical characteristics of 150-Watt peak solar panels using Boltzmann sigmoid function under various temperature and irradiance is reported. Current, voltage, temperature and irradiance data in Denpasar, a city located at just south of equator, was collected. Solar power meter is used to measure irradiance level, meanwhile digital thermometer is used to measure temperature of front and back panels. Short circuit current and open circuit voltage data was also collected at different temperature and irradiance level. Statistically, the electrical characteristics of 150-Watt peak solar panel can be modelled using Boltzmann sigmoid function with good fit. Therefore, it can be concluded that Boltzmann sigmoid function might be used to determine current and voltage characteristics of 150-Watt peak solar panel under various temperature and irradiance.

Entropic lattice Boltzmann representations required to recover Navier-Stokes flows.

PubMed

Keating, Brian; Vahala, George; Yepez, Jeffrey; Soe, Min; Vahala, Linda

2007-03-01

There are two disparate formulations of the entropic lattice Boltzmann scheme: one of these theories revolves around the analog of the discrete Boltzmann H function of standard extensive statistical mechanics, while the other revolves around the nonextensive Tsallis entropy. It is shown here that it is the nonenforcement of the pressure tensor moment constraints that lead to extremizations of entropy resulting in Tsallis-like forms. However, with the imposition of the pressure tensor moment constraint, as is fundamentally necessary for the recovery of the Navier-Stokes equations, it is proved that the entropy function must be of the discrete Boltzmann form. Three-dimensional simulations are performed which illustrate some of the differences between standard lattice Boltzmann and entropic lattice Boltzmann schemes, as well as the role played by the number of phase-space velocities used in the discretization.

A resonance shift prediction based on the Boltzmann-Ehrenfest principle for cylindrical cavities with a rigid sphere.

PubMed

Santillan, Arturo O; Cutanda-Henríquez, Vicente

2008-11-01

An investigation on the resonance frequency shift for a plane-wave mode in a cylindrical cavity produced by a rigid sphere is reported in this paper. This change of the resonance frequency has been previously considered as a cause of oscillational instabilities in single-mode acoustic levitation devices. It is shown that the use of the Boltzmann-Ehrenfest principle of adiabatic invariance allows the derivation of an expression for the resonance frequency shift in a simpler and more direct way than a method based on a Green's function reported in literature. The position of the sphere can be any point along the axis of the cavity. Obtained predictions of the resonance frequency shift with the deduced equation agree quite well with numerical simulations based on the boundary element method. The results are also confirmed by experiments. The equation derived from the Boltzmann-Ehrenfest principle appears to be more general, and for large spheres, it gives a better approximation than the equation previously reported.

Lattice Boltzmann method for bosons and fermions and the fourth-order Hermite polynomial expansion.

PubMed

Coelho, Rodrigo C V; Ilha, Anderson; Doria, Mauro M; Pereira, R M; Aibe, Valter Yoshihiko

2014-04-01

The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution functions. We show that the expansion of the microscopic velocity in terms of Hermite polynomials must be carried to the fourth order to correctly describe the energy equation. The viscosity and thermal coefficients, previously obtained by Yang et al. [Shi and Yang, J. Comput. Phys. 227, 9389 (2008); Yang and Hung, Phys. Rev. E 79, 056708 (2009)] through the Uehling-Uhlenbeck approach, are also derived here. Thus the construction of a lattice Boltzmann method for the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac equilibrium distribution functions are expanded to fourth order in the Hermite polynomials.

Extension of the Helmholtz-Smoluchowski velocity to the hydrophobic microchannels with velocity slip.

PubMed

Park, H M; Kim, T W

2009-01-21

Electrokinetic flows through hydrophobic microchannels experience velocity slip at the microchannel wall, which affects volumetric flow rate and solute retention time. The usual method of predicting the volumetric flow rate and velocity profile for hydrophobic microchannels is to solve the Navier-Stokes equation and the Poisson-Boltzmann equation for the electric potential with the boundary condition of velocity slip expressed by the Navier slip coefficient, which is computationally demanding and defies analytic solutions. In the present investigation, we have devised a simple method of predicting the velocity profiles and volumetric flow rates of electrokinetic flows by extending the concept of the Helmholtz-Smoluchowski velocity to microchannels with Navier slip. The extended Helmholtz-Smoluchowski velocity is simple to use and yields accurate results as compared to the exact solutions. Employing the extended Helmholtz-Smoluchowski velocity, the analytical expressions for volumetric flow rate and velocity profile for electrokinetic flows through rectangular microchannels with Navier slip have been obtained at high values of zeta potential. The range of validity of the extended Helmholtz-Smoluchowski velocity is also investigated.

Consistent transport coefficients in astrophysics

NASA Technical Reports Server (NTRS)

Fontenla, Juan M.; Rovira, M.; Ferrofontan, C.

1986-01-01

A consistent theory for dealing with transport phenomena in stellar atmospheres starting with the kinetic equations and introducing three cases (LTE, partial LTE, and non-LTE) was developed. The consistent hydrodynamical equations were presented for partial-LTE, the transport coefficients defined, and a method shown to calculate them. The method is based on the numerical solution of kinetic equations considering Landau, Boltzmann, and Focker-Planck collision terms. Finally a set of results for the transport coefficients derived for a partially ionized hydrogen gas with radiation was shown, considering ionization and recombination as well as elastic collisions. The results obtained imply major changes is some types of theoretical model calculations and can resolve some important current problems concerning energy and mass balance in the solar atmosphere. It is shown that energy balance in the lower solar transition region can be fully explained by means of radiation losses and conductive flux.

Spacecraft self-contamination due to back-scattering of outgas products

NASA Technical Reports Server (NTRS)

Robertson, S. J.

1976-01-01

The back-scattering of outgas contamination near an orbiting spacecraft due to intermolecular collisions was analyzed. Analytical tools were developed for making reasonably accurate quantitative estimates of the outgas contamination return flux, given a knowledge of the pertinent spacecraft and orbit conditions. Two basic collision mechanisms were considered: (1) collisions involving only outgas molecules (self-scattering) and (2) collisions between outgas molecules and molecules in the ambient atmosphere (ambient-scattering). For simplicity, the geometry was idealized to a uniformly outgassing sphere and to a disk oriented normal to the freestream. The method of solution involved an integration of an approximation of the Boltzmann kinetic equation known as the BGK (or Krook) model equation. Results were obtained in the form of simple equations relating outgas return flux to spacecraft and orbit parameters. Results were compared with previous analyses based on more simplistic models of the collision processes.

Triggering Excimer Lasers by Photoionization from Corona Discharges

NASA Astrophysics Data System (ADS)

Xiong, Zhongmin; Duffey, Thomas; Brown, Daniel; Kushner, Mark

2009-10-01

High repetition rate ArF (192 nm) excimer lasers are used for photolithography sources in microelectronics fabrication. In highly attaching gas mixtures, preionization is critical to obtaining stable, reproducible glow discharges. Photoionization from a separate corona discharge is one technique for preionization which triggers the subsequent electron avalanche between the main electrodes. Photoionization triggering of an ArF excimer laser sustained in multi-atmosphere Ne/Ar/F2/Xe gas mixtures has been investigated using a 2-dimensional plasma hydrodynamics model including radiation transport. Continuity equations for charged and neutral species, and Poisson's equation are solved coincident with the electron temperature with transport coefficients obtained from solutions of Boltzmann's equation. Photoionizing radiation is produced by a surface discharge which propagates along a corona-bar located adjacent to the discharge electrodes. The consequences of pulse power waveform, corona bar location, capacitance and gas mixture on uniformity, symmetry and gain of the avalanche discharge will be discussed.

The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

NASA Astrophysics Data System (ADS)

Latyshev, A. V.; Gordeeva, N. M.

2017-09-01

We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov-Boltzmann equation with the Bhatnagar-Gross-Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi-Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.

Mass and heat transfer between evaporation and condensation surfaces: Atomistic simulation and solution of Boltzmann kinetic equation.

PubMed

Zhakhovsky, Vasily V; Kryukov, Alexei P; Levashov, Vladimir Yu; Shishkova, Irina N; Anisimov, Sergey I

2018-04-16

Boundary conditions required for numerical solution of the Boltzmann kinetic equation (BKE) for mass/heat transfer between evaporation and condensation surfaces are analyzed by comparison of BKE results with molecular dynamics (MD) simulations. Lennard-Jones potential with parameters corresponding to solid argon is used to simulate evaporation from the hot side, nonequilibrium vapor flow with a Knudsen number of about 0.02, and condensation on the cold side of the condensed phase. The equilibrium density of vapor obtained in MD simulation of phase coexistence is used in BKE calculations for consistency of BKE results with MD data. The collision cross-section is also adjusted to provide a thermal flux in vapor identical to that in MD. Our MD simulations of evaporation toward a nonreflective absorbing boundary show that the velocity distribution function (VDF) of evaporated atoms has the nearly semi-Maxwellian shape because the binding energy of atoms evaporated from the interphase layer between bulk phase and vapor is much smaller than the cohesive energy in the condensed phase. Indeed, the calculated temperature and density profiles within the interphase layer indicate that the averaged kinetic energy of atoms remains near-constant with decreasing density almost until the interphase edge. Using consistent BKE and MD methods, the profiles of gas density, mass velocity, and temperatures together with VDFs in a gap of many mean free paths between the evaporation and condensation surfaces are obtained and compared. We demonstrate that the best fit of BKE results with MD simulations can be achieved with the evaporation and condensation coefficients both close to unity.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brull, S., E-mail: Stephane.Brull@math.u-bordeaux.fr; Charrier, P., E-mail: Pierre.Charrier@math.u-bordeaux.fr; Mieussens, L., E-mail: Luc.Mieussens@math.u-bordeaux.fr

It is well known that the roughness of the wall has an effect on microscale gas flows. This effect can be shown for large Knudsen numbers by using a numerical solution of the Boltzmann equation. However, when the wall is rough at a nanometric scale, it is necessary to use a very small mesh size which is much too expansive. An alternative approach is to incorporate the roughness effect in the scattering kernel of the boundary condition, such as the Maxwell-like kernel introduced by the authors in a previous paper. Here, we explain how this boundary condition can be implementedmore » in a discrete velocity approximation of the Boltzmann equation. Moreover, the influence of the roughness is shown by computing the structure scattering pattern of mono-energetic beams of the incident gas molecules. The effect of the angle of incidence of these molecules, of their mass, and of the morphology of the wall is investigated and discussed in a simplified two-dimensional configuration. The effect of the azimuthal angle of the incident beams is shown for a three-dimensional configuration. Finally, the case of non-elastic scattering is considered. All these results suggest that our approach is a promising way to incorporate enough physics of gas-surface interaction, at a reasonable computing cost, to improve kinetic simulations of micro- and nano-flows.« less

The Physical Mechanism for Retinal Discrete Dark Noise: Thermal Activation or Cellular Ultraweak Photon Emission?

PubMed Central

Salari, Vahid; Scholkmann, Felix; Bokkon, Istvan; Shahbazi, Farhad; Tuszynski, Jack

2016-01-01

For several decades the physical mechanism underlying discrete dark noise of photoreceptors in the eye has remained highly controversial and poorly understood. It is known that the Arrhenius equation, which is based on the Boltzmann distribution for thermal activation, can model only a part (e.g. half of the activation energy) of the retinal dark noise experimentally observed for vertebrate rod and cone pigments. Using the Hinshelwood distribution instead of the Boltzmann distribution in the Arrhenius equation has been proposed as a solution to the problem. Here, we show that the using the Hinshelwood distribution does not solve the problem completely. As the discrete components of noise are indistinguishable in shape and duration from those produced by real photon induced photo-isomerization, the retinal discrete dark noise is most likely due to ‘internal photons’ inside cells and not due to thermal activation of visual pigments. Indeed, all living cells exhibit spontaneous ultraweak photon emission (UPE), mainly in the optical wavelength range, i.e., 350–700 nm. We show here that the retinal discrete dark noise has a similar rate as UPE and therefore dark noise is most likely due to spontaneous cellular UPE and not due to thermal activation. PMID:26950936

Dust particle radial confinement in a dc glow discharge.

PubMed

Sukhinin, G I; Fedoseev, A V; Antipov, S N; Petrov, O F; Fortov, V E

2013-01-01

A self-consistent nonlocal model of the positive column of a dc glow discharge with dust particles is presented. Radial distributions of plasma parameters and the dust component in an axially homogeneous glow discharge are considered. The model is based on the solution of a nonlocal Boltzmann equation for the electron energy distribution function, drift-diffusion equations for ions, and the Poisson equation for a self-consistent electric field. The radial distribution of dust particle density in a dust cloud was fixed as a given steplike function or was chosen according to an equilibrium Boltzmann distribution. The balance of electron and ion production in argon ionization by an electron impact and their losses on the dust particle surface and on the discharge tube walls is taken into account. The interrelation of discharge plasma and the dust cloud is studied in a self-consistent way, and the radial distributions of the discharge plasma and dust particle parameters are obtained. It is shown that the influence of the dust cloud on the discharge plasma has a nonlocal behavior, e.g., density and charge distributions in the dust cloud substantially depend on the plasma parameters outside the dust cloud. As a result of a self-consistent evolution of plasma parameters to equilibrium steady-state conditions, ionization and recombination rates become equal to each other, electron and ion radial fluxes become equal to zero, and the radial component of electric field is expelled from the dust cloud.

Modulation Doped GaAs/Al sub xGA sub (1-x)As Layered Structures with Applications to Field Effect Transistors.

DTIC Science & Technology

1982-02-15

function of the doping density at 300 and 77 K for the classical Boltzmann statistics or depletion approximation (solid line) and for the approximate...Fermi-Dirac statistics (equation (19) dotted line)• This comparison demonstrates that the deviation from Boltzmann statistics is quite noticeable...tunneling Schottky barriers cannot be obtained at these doping levels. The dotted lines are obtained when Boltzmann statistics are used in the Al Ga

Calculation of the Maxwell stress tensor and the Poisson-Boltzmann force on a solvated molecular surface using hypersingular boundary integrals

NASA Astrophysics Data System (ADS)

Lu, Benzhuo; Cheng, Xiaolin; Hou, Tingjun; McCammon, J. Andrew

2005-08-01

The electrostatic interaction among molecules solvated in ionic solution is governed by the Poisson-Boltzmann equation (PBE). Here the hypersingular integral technique is used in a boundary element method (BEM) for the three-dimensional (3D) linear PBE to calculate the Maxwell stress tensor on the solvated molecular surface, and then the PB forces and torques can be obtained from the stress tensor. Compared with the variational method (also in a BEM frame) that we proposed recently, this method provides an even more efficient way to calculate the full intermolecular electrostatic interaction force, especially for macromolecular systems. Thus, it may be more suitable for the application of Brownian dynamics methods to study the dynamics of protein/protein docking as well as the assembly of large 3D architectures involving many diffusing subunits. The method has been tested on two simple cases to demonstrate its reliability and efficiency, and also compared with our previous variational method used in BEM.

Second Order Boltzmann-Gibbs Principle for Polynomial Functions and Applications

NASA Astrophysics Data System (ADS)

Gonçalves, Patrícia; Jara, Milton; Simon, Marielle

2017-01-01

In this paper we give a new proof of the second order Boltzmann-Gibbs principle introduced in Gonçalves and Jara (Arch Ration Mech Anal 212(2):597-644, 2014). The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards (1) a trivial process in case of super-diffusive systems, (2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation, as defined in Gubinelli and Jara (SPDEs Anal Comput (1):325-350, 2013) and Gubinelli and Perkowski (Arxiv:1508.07764, 2015), in case of weakly asymmetric diffusive systems. Examples and applications are presented for weakly and partial asymmetric exclusion processes, weakly asymmetric speed change exclusion processes and hamiltonian systems with exponential interactions.

Streaming potential generated by a pressure-driven flow over a super-hydrophobic surface

NASA Astrophysics Data System (ADS)

Zhao, Hui

2010-11-01

The streaming potential generated by a pressured-driven flow over a weakly charged striped slip-stick surface (the zeta potential of the surface is smaller than the thermal potential (25 mV) with an arbitrary double layer thickness is theoretically studied by solving the Poisson-Boltzmann equation and Stokes equation. A series solution of the streaming potential is derived. Approximate expressions for the streaming potential in the limits of thin double layers and thick double layers are also presented, in excellent agreement with the full solution. The streaming potential is compared against that over a homogenously charged smooth surface. Our results indicate that the streaming potential over a super-hydrophobic surface only can be enhanced when the liquid-gas interface is charged. In addition, as the double layer thickness increases, the advantage of the super-hydrophobic surface diminishes. The impact of a slip-stick surface on the streaming potential might provide guidance for designing novel and efficient microfludic energy conversion devices using a super-hydrophobic surface.

Diffusion of low-energy electrons in tissue-like liquids.

PubMed

Malamut, C; Paes-Leme, P J; Paschoa, A S

1992-11-01

The spatial-energetic distribution of low-energy electrons was studied for a source located in a liquid medium simulating biological tissue. A time-independent Boltzmann equation was used to model this distribution microscopically. Ionization was treated as a perturbation to a quasi-elastic collision process between the electron and the medium. A diffusion limit was obtained by using a scale parameter, leading to a sequence of recursive partial differential equations whose solutions, associated with a macroscopic scale, were obtained by numerical approximations. As an application, electron ranges were estimated based on these solutions and then compared with values reported in the open literature based on experimental results and on Monte Carlo calculation. Local dosimetry, i.e., the energy imparted to a volume of a sphere with radius equal to the range of low-energy electrons, of low-energy electrons from internal emitters can benefit by the knowledge of the ranges estimated for biological tissue. Auger electron emitters, for example, have been the object of a number of investigations because of their radiobiological significance.

Studies of HZE particle interactions and transport for space radiation protection purposes

NASA Technical Reports Server (NTRS)

Townsend, Lawrence W.; Wilson, John W.; Schimmerling, Walter; Wong, Mervyn

1987-01-01

The main emphasis is on developing general methods for accurately predicting high-energy heavy ion (HZE) particle interactions and transport for use by researchers in mission planning studies, in evaluating astronaut self-shielding factors, and in spacecraft shield design and optimization studies. The two research tasks are: (1) to develop computationally fast and accurate solutions to the Boltzmann (transport) equation; and (2) to develop accurate HZE interaction models, from fundamental physical considerations, for use as inputs into these transport codes. Accurate solutions to the HZE transport problem have been formulated through a combination of analytical and numerical techniques. In addition, theoretical models for the input interaction parameters are under development: stopping powers, nuclear absorption cross sections, and fragmentation parameters.

Generalized Spencer-Lewis equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Filippone, W.L.

The Spencer-Lewis equation, which describes electron transport in homogeneous media when continuous slowing down theory is valid, is derived from the Boltzmann equation. Also derived is a time-dependent generalized Spencer-Lewis equation valid for inhomogeneous media. An independent verification of this last equation is obtained for the one-dimensional case using particle balance considerations.

Lattice Boltzmann model for three-phase viscoelastic fluid flow

NASA Astrophysics Data System (ADS)

Xie, Chiyu; Lei, Wenhai; Wang, Moran

2018-02-01

A lattice Boltzmann (LB) framework is developed for simulation of three-phase viscoelastic fluid flows in complex geometries. This model is based on a Rothman-Keller type model for immiscible multiphase flows which ensures mass conservation of each component in porous media even for a high density ratio. To account for the viscoelastic effects, the Maxwell constitutive relation is correctly introduced into the momentum equation, which leads to a modified lattice Boltzmann evolution equation for Maxwell fluids by removing the normal but excess viscous term. Our simulation tests indicate that this excess viscous term may induce significant errors. After three benchmark cases, the displacement processes of oil by dispersed polymer are studied as a typical example of three-phase viscoelastic fluid flow. The results show that increasing either the polymer intrinsic viscosity or the elastic modulus will enhance the oil recovery.

A low noise discrete velocity method for the Boltzmann equation with quantized rotational and vibrational energy

NASA Astrophysics Data System (ADS)

Clarke, Peter; Varghese, Philip; Goldstein, David

2018-01-01

A discrete velocity method is developed for gas mixtures of diatomic molecules with both rotational and vibrational energy states. A full quantized model is described, and rotation-translation and vibration-translation energy exchanges are simulated using a Larsen-Borgnakke exchange model. Elastic and inelastic molecular interactions are modeled during every simulated collision to help produce smooth internal energy distributions. The method is verified by comparing simulations of homogeneous relaxation by our discrete velocity method to numerical solutions of the Jeans and Landau-Teller equations, and to direct simulation Monte Carlo. We compute the structure of a 1D shock using this method, and determine how the rotational energy distribution varies with spatial location in the shock and with position in velocity space.

Role of Various Entropies in the Black Hole Information Loss Problem

NASA Astrophysics Data System (ADS)

Nieuwenhuizen, Th. M.; Volovich, I. V.

2007-09-01

We discuss the current status of the black hole information loss paradox and propose a plan for its solution based on analogies with solid state physics and the irreversibility problem. In a recent paper Hawking has argued that there is no information loss in black holes in asymptotically AdS spacetimes. We remind that there are several types of information (entropy) in statistical physics - fine grained (microscopic) and coarse grained (macroscopic) ones which behave differently under unitary evolution. We suggest that the coarse grained information of the rest of the Universe is lost while fine grained information is preserved. A possibility to develop in quantum gravity an analogue of the Bogoliubov derivation of the irreversible Boltzmann and Navier - Stokes equations from the reversible mechanical equations is discussed.

Solvable Hydrodynamics of Quantum Integrable Systems

NASA Astrophysics Data System (ADS)

Bulchandani, Vir B.; Vasseur, Romain; Karrasch, Christoph; Moore, Joel E.

2017-12-01

The conventional theory of hydrodynamics describes the evolution in time of chaotic many-particle systems from local to global equilibrium. In a quantum integrable system, local equilibrium is characterized by a local generalized Gibbs ensemble or equivalently a local distribution of pseudomomenta. We study time evolution from local equilibria in such models by solving a certain kinetic equation, the "Bethe-Boltzmann" equation satisfied by the local pseudomomentum density. Explicit comparison with density matrix renormalization group time evolution of a thermal expansion in the XXZ model shows that hydrodynamical predictions from smooth initial conditions can be remarkably accurate, even for small system sizes. Solutions are also obtained in the Lieb-Liniger model for free expansion into vacuum and collisions between clouds of particles, which model experiments on ultracold one-dimensional Bose gases.

Accuracy of the lattice Boltzmann method for describing the behavior of a gas in the continuum limit.

PubMed

Kataoka, Takeshi; Tsutahara, Michihisa

2010-11-01

The accuracy of the lattice Boltzmann method (LBM) for describing the behavior of a gas in the continuum limit is systematically investigated. The asymptotic analysis for small Knudsen numbers is carried out to derive the corresponding fluid-dynamics-type equations, and the errors of the LBM are estimated by comparing them with the correct fluid-dynamics-type equations. We discuss the following three important cases: (I) the Mach number of the flow is much smaller than the Knudsen number, (II) the Mach number is of the same order as the Knudsen number, and (III) the Mach number is finite. From the von Karman relation, the above three cases correspond to the flows of (I) small Reynolds number, (II) finite Reynolds number, and (III) large Reynolds number, respectively. The analysis is made with the information only of the fundamental properties of the lattice Boltzmann models without stepping into their detailed form. The results are therefore applicable to various lattice Boltzmann models that satisfy the fundamental properties used in the analysis.

Capturing the Large Scale Behavior of Many Particle Systems Through Coarse-Graining

NASA Astrophysics Data System (ADS)

Punshon-Smith, Samuel

This dissertation is concerned with two areas of investigation: the first is understanding the mathematical structures behind the emergence of macroscopic laws and the effects of small scales fluctuations, the second involves the rigorous mathematical study of such laws and related questions of well-posedness. To address these areas of investigation the dissertation involves two parts: Part I concerns the theory of coarse-graining of many particle systems. We first investigate the mathematical structure behind the Mori-Zwanzig (projection operator) formalism by introducing two perturbative approaches to coarse-graining of systems that have an explicit scale separation. One concerns systems with little dissipation, while the other concerns systems with strong dissipation. In both settings we obtain an asymptotic series of `corrections' to the limiting description which are small with respect to the scaling parameter, these corrections represent the effects of small scales. We determine that only certain approximations give rise to dissipative effects in the resulting evolution. Next we apply this framework to the problem of coarse-graining the locally conserved quantities of a classical Hamiltonian system. By lumping conserved quantities into a collection of mesoscopic cells, we obtain, through a series of approximations, a stochastic particle system that resembles a discretization of the non-linear equations of fluctuating hydrodynamics. We study this system in the case that the transport coefficients are constant and prove well-posedness of the stochastic dynamics. Part II concerns the mathematical description of models where the underlying characteristics are stochastic. Such equations can model, for instance, the dynamics of a passive scalar in a random (turbulent) velocity field or the statistical behavior of a collection of particles subject to random environmental forces. First, we study general well-posedness properties of stochastic transport equation with rough diffusion coefficients. Our main result is strong existence and uniqueness under certain regularity conditions on the coefficients, and uses the theory of renormalized solutions of transport equations adapted to the stochastic setting. Next, in a work undertaken with collaborator Scott-Smith we study the Boltzmann equation with a stochastic forcing. The noise describing the forcing is white in time and colored in space and describes the effects of random environmental forces on a rarefied gas undergoing instantaneous, binary collisions. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. Tightness of the appropriate quantities is proved by an extension of the Skorohod theorem to non-metric spaces.

Hydration effects on the electrostatic potential around tuftsin.

PubMed

Valdeavella, C V; Blatt, H D; Yang, L; Pettitt, B M

1999-08-01

The electrostatic potential and component dielectric constants from molecular dynamics (MD) trajectories of tuftsin, a tetrapeptide with the amino acid sequence Thr-Lys-Pro-Arg in water and in saline solution are presented. The results obtained from the analysis of the MD trajectories for the total electrostatic potential at points on a grid using the Ewald technique are compared with the solution to the Poisson-Boltzmann (PB) equation. The latter was solved using several sets of dielectric constant parameters. The effects of structural averaging on the PB results were also considered. Solute conformational mobility in simulations gives rise to an electrostatic potential map around the solute dominated by the solute monopole (or lowest order multipole). The detailed spatial variation of the electrostatic potential on the molecular surface brought about by the compounded effects of the distribution of water and ions close to the peptide, solvent mobility, and solute conformational mobility are not qualitatively reproducible from a reparametrization of the input solute and solvent dielectric constants to the PB equation for a single structure or for structurally averaged PB calculations. Nevertheless, by fitting the PB to the MD electrostatic potential surfaces with the dielectric constants as fitting parameters, we found that the values that give the best fit are the values calculated from the MD trajectories. Implications of using such field calculations on the design of tuftsin peptide analogues are discussed.

Gas-kinetic unified algorithm for hypersonic flows covering various flow regimes solving Boltzmann model equation in nonequilibrium effect

DOE Office of Scientific and Technical Information (OSTI.GOV)

Li, Zhihui; Ma, Qiang; Wu, Junlin

2014-12-09

Based on the Gas-Kinetic Unified Algorithm (GKUA) directly solving the Boltzmann model equation, the effect of rotational non-equilibrium is investigated recurring to the kinetic Rykov model with relaxation property of rotational degrees of freedom. The spin movement of diatomic molecule is described by moment of inertia, and the conservation of total angle momentum is taken as a new Boltzmann collision invariant. The molecular velocity distribution function is integrated by the weight factor on the internal energy, and the closed system of two kinetic controlling equations is obtained with inelastic and elastic collisions. The optimization selection technique of discrete velocity ordinatemore » points and numerical quadrature rules for macroscopic flow variables with dynamic updating evolvement are developed to simulate hypersonic flows, and the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions. The gas-kinetic boundary conditions in thermodynamic non-equilibrium and numerical procedures are studied and implemented by directly acting on the velocity distribution function, and then the unified algorithm of Boltzmann model equation involving non-equilibrium effect is presented for the whole range of flow regimes. The hypersonic flows involving non-equilibrium effect are numerically simulated including the inner flows of shock wave structures in nitrogen with different Mach numbers of 1.5-Ma-25, the planar ramp flow with the whole range of Knudsen numbers of 0.0009-Kn-10 and the three-dimensional re-entering flows around tine double-cone body.« less

Collision partner selection schemes in DSMC: From micro/nano flows to hypersonic flows

NASA Astrophysics Data System (ADS)

Roohi, Ehsan; Stefanov, Stefan

2016-10-01

The motivation of this review paper is to present a detailed summary of different collision models developed in the framework of the direct simulation Monte Carlo (DSMC) method. The emphasis is put on a newly developed collision model, i.e., the Simplified Bernoulli trial (SBT), which permits efficient low-memory simulation of rarefied gas flows. The paper starts with a brief review of the governing equations of the rarefied gas dynamics including Boltzmann and Kac master equations and reiterates that the linear Kac equation reduces to a non-linear Boltzmann equation under the assumption of molecular chaos. An introduction to the DSMC method is provided, and principles of collision algorithms in the DSMC are discussed. A distinction is made between those collision models that are based on classical kinetic theory (time counter, no time counter (NTC), and nearest neighbor (NN)) and the other class that could be derived mathematically from the Kac master equation (pseudo-Poisson process, ballot box, majorant frequency, null collision, Bernoulli trials scheme and its variants). To provide a deeper insight, the derivation of both collision models, either from the principles of the kinetic theory or the Kac master equation, is provided with sufficient details. Some discussions on the importance of subcells in the DSMC collision procedure are also provided and different types of subcells are presented. The paper then focuses on the simplified version of the Bernoulli trials algorithm (SBT) and presents a detailed summary of validation of the SBT family collision schemes (SBT on transient adaptive subcells: SBT-TAS, and intelligent SBT: ISBT) in a broad spectrum of rarefied gas-flow test cases, ranging from low speed, internal micro and nano flows to external hypersonic flow, emphasizing first the accuracy of these new collision models and second, demonstrating that the SBT family scheme, if compared to other conventional and recent collision models, requires smaller number of particles per cell to obtain sufficiently accurate solutions.

Low energy e-Ar momentum transfer cross-section

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brennan, M.J.

1992-12-01

Recent work has shown that solutions of the Boltzmann equation which use the so called {open_quotes}two-term{close_quotes} approximation provide an inadequate description of the transverse diffusion of electrons in argon gas at low values of E/N, contrary to earlier evidence. Previous determinations of the momentum transfer cross section for argon from the analysis of transport data have used two-term codes in good faith. Progress towards the determination of a new cross section in the energy range O - 4 eV, including an analysis of the energy dependence of the uncertainty in the derived cross section is reported.

Computing Interactions Of Free-Space Radiation With Matter

NASA Technical Reports Server (NTRS)

Wilson, J. W.; Cucinotta, F. A.; Shinn, J. L.; Townsend, L. W.; Badavi, F. F.; Tripathi, R. K.; Silberberg, R.; Tsao, C. H.; Badwar, G. D.

1995-01-01

High Charge and Energy Transport (HZETRN) computer program computationally efficient, user-friendly package of software adressing problem of transport of, and shielding against, radiation in free space. Designed as "black box" for design engineers not concerned with physics of underlying atomic and nuclear radiation processes in free-space environment, but rather primarily interested in obtaining fast and accurate dosimetric information for design and construction of modules and devices for use in free space. Computational efficiency achieved by unique algorithm based on deterministic approach to solution of Boltzmann equation rather than computationally intensive statistical Monte Carlo method. Written in FORTRAN.

BRYNTRN: A baryon transport model

NASA Technical Reports Server (NTRS)

Wilson, John W.; Townsend, Lawrence W.; Nealy, John E.; Chun, Sang Y.; Hong, B. S.; Buck, Warren W.; Lamkin, S. L.; Ganapol, Barry D.; Khan, Ferdous; Cucinotta, Francis A.

1989-01-01

The development of an interaction data base and a numerical solution to the transport of baryons through an arbitrary shield material based on a straight ahead approximation of the Boltzmann equation are described. The code is most accurate for continuous energy boundary values, but gives reasonable results for discrete spectra at the boundary using even a relatively coarse energy grid (30 points) and large spatial increments (1 cm in H2O). The resulting computer code is self-contained, efficient and ready to use. The code requires only a very small fraction of the computer resources required for Monte Carlo codes.

On the accuracy of the rate coefficients used in plasma fluid models for breakdown in air

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kourtzanidis, Konstantinos, E-mail: kkourt@utexas.edu; Raja, Laxminarayan L., E-mail: lraja@mail.utexas.edu

2016-07-15

The electrical breakdown of air depends on the balance between creation and loss of charged particles. In fluid models, datasets of the rate coefficients used are obtained either from fits to experimental data or by solutions of the Boltzmann equation. Here, we study the accuracy of the commonly used models for ionization and attachment frequencies and their impact on the prediction of the breakdown threshold for air. We show that large errors can occur depending on the model and propose the most accurate dataset available for modeling of air breakdown phenomena.

An efficient annealing in Boltzmann machine in Hopfield neural network

NASA Astrophysics Data System (ADS)

Kin, Teoh Yeong; Hasan, Suzanawati Abu; Bulot, Norhisam; Ismail, Mohammad Hafiz

2012-09-01

This paper proposes and implements Boltzmann machine in Hopfield neural network doing logic programming based on the energy minimization system. The temperature scheduling in Boltzmann machine enhancing the performance of doing logic programming in Hopfield neural network. The finest temperature is determined by observing the ratio of global solution and final hamming distance using computer simulations. The study shows that Boltzmann Machine model is more stable and competent in term of representing and solving difficult combinatory problems.

Extension of lattice Boltzmann flux solver for simulation of compressible multi-component flows

NASA Astrophysics Data System (ADS)

Yang, Li-Ming; Shu, Chang; Yang, Wen-Ming; Wang, Yan

2018-05-01

The lattice Boltzmann flux solver (LBFS), which was presented by Shu and his coworkers for solving compressible fluid flow problems, is extended to simulate compressible multi-component flows in this work. To solve the two-phase gas-liquid problems, the model equations with stiffened gas equation of state are adopted. In this model, two additional non-conservative equations are introduced to represent the material interfaces, apart from the classical Euler equations. We first convert the interface equations into the full conservative form by applying the mass equation. After that, we calculate the numerical fluxes of the classical Euler equations by the existing LBFS and the numerical fluxes of the interface equations by the passive scalar approach. Once all the numerical fluxes at the cell interface are obtained, the conservative variables at cell centers can be updated by marching the equations in time and the material interfaces can be identified via the distributions of the additional variables. The numerical accuracy and stability of present scheme are validated by its application to several compressible multi-component fluid flow problems.

On the inclusion of mass source terms in a single-relaxation-time lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Aursjø, Olav; Jettestuen, Espen; Vinningland, Jan Ludvig; Hiorth, Aksel

2018-05-01

We present a lattice Boltzmann algorithm for incorporating a mass source in a fluid flow system. The proposed mass source/sink term, included in the lattice Boltzmann equation, maintains the Galilean invariance and the accuracy of the overall method, while introducing a mass source/sink term in the fluid dynamical equations. The method can, for instance, be used to inject or withdraw fluid from any preferred lattice node in a system. This suggests that injection and withdrawal of fluid does not have to be introduced through cumbersome, and sometimes less accurate, boundary conditions. The method also suggests that, through a chosen equation of state relating mass density to pressure, the proposed mass source term will render it possible to set a preferred pressure at any lattice node in a system. We demonstrate how this model handles injection and withdrawal of a fluid. And we show how it can be used to incorporate pressure boundaries. The accuracy of the algorithm is identified through a Chapman-Enskog expansion of the model and supported by the numerical simulations.

Accuracy assessment of the linear Poisson-Boltzmann equation and reparametrization of the OBC generalized Born model for nucleic acids and nucleic acid-protein complexes.

PubMed

Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro

2015-04-05

The generalized Born model in the Onufriev, Bashford, and Case (Onufriev et al., Proteins: Struct Funct Genet 2004, 55, 383) implementation has emerged as one of the best compromises between accuracy and speed of computation. For simulations of nucleic acids, however, a number of issues should be addressed: (1) the generalized Born model is based on a linear model and the linearization of the reference Poisson-Boltmann equation may be questioned for highly charged systems as nucleic acids; (2) although much attention has been given to potentials, solvation forces could be much less sensitive to linearization than the potentials; and (3) the accuracy of the Onufriev-Bashford-Case (OBC) model for nucleic acids depends on fine tuning of parameters. Here, we show that the linearization of the Poisson Boltzmann equation has mild effects on computed forces, and that with optimal choice of the OBC model parameters, solvation forces, essential for molecular dynamics simulations, agree well with those computed using the reference Poisson-Boltzmann model. © 2015 Wiley Periodicals, Inc.

Computational Algorithms or Identification of Distributed Parameter Systems

DTIC Science & Technology

1993-04-24

delay-differential equations, Volterra integral equations, and partial differential equations with memory terms . In particular we investigated a...tested for estimating parameters in a Volterra integral equation arising from a viscoelastic model of a flexible structure with Boltzmann damping. In...particular, one of the parameters identified was the order of the derivative in Volterra integro-differential equations containing fractional

Lattice Boltzmann method for weakly ionized isothermal plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Li Huayu; Ki, Hyungson

2007-12-15

In this paper, a lattice Boltzmann method (LBM) for weakly ionized isothermal plasmas is presented by introducing a rescaling scheme for the Boltzmann transport equation. Without using this rescaling, we found that the nondimensional relaxation time used in the LBM is too large and the LBM does not produce physically realistic results. The developed model was applied to the electrostatic wave problem and the diffusion process of singly ionized helium plasmas with a 1-3% degree of ionization under an electric field. The obtained results agree well with theoretical values.

Comparison between Smoluchowski and Boltzmann approaches for self-propelled rods.

PubMed

Bertin, Eric; Baskaran, Aparna; Chaté, Hugues; Marchetti, M Cristina

2015-10-01

Considering systems of self-propelled polar particles with nematic interactions ("rods"), we compare the continuum equations describing the evolution of polar and nematic order parameters, derived either from Smoluchowski or Boltzmann equations. Our main goal is to understand the discrepancies between the continuum equations obtained so far in both frameworks. We first show that, in the simple case of point-like particles with only alignment interactions, the continuum equations obtained have the same structure in both cases. We further study, in the Smoluchowski framework, the case where an interaction force is added on top of the aligning torque. This clarifies the origin of the additional terms obtained in previous works. Our observations lead us to emphasize the need for a more involved closure scheme than the standard normal form of the distribution when dealing with active systems.

Optical display for radar sensing

NASA Astrophysics Data System (ADS)

Szu, Harold; Hsu, Charles; Willey, Jefferson; Landa, Joseph; Hsieh, Minder; Larsen, Louis V.; Krzywicki, Alan T.; Tran, Binh Q.; Hoekstra, Philip; Dillard, John T.; Krapels, Keith A.; Wardlaw, Michael; Chu, Kai-Dee

2015-05-01

Boltzmann headstone S = kB Log W turns out to be the Rosette stone for Greek physics translation optical display of the microwave sensing hieroglyphics. The LHS is the molecular entropy S measuring the degree of uniformity scattering off the sensing cross sections. The RHS is the inverse relationship (equation) predicting the Planck radiation spectral distribution parameterized by the Kelvin temperature T. Use is made of the conservation energy law of the heat capacity of Reservoir (RV) change T Δ S = -ΔE equals to the internal energy change of black box (bb) subsystem. Moreover, an irreversible thermodynamics Δ S > 0 for collision mixing toward totally larger uniformity of heat death, asserted by Boltzmann, that derived the so-called Maxwell-Boltzmann canonical probability. Given the zero boundary condition black box, Planck solved a discrete standing wave eigenstates (equation). Together with the canonical partition function (equation) an average ensemble average of all possible internal energy yielded the celebrated Planck radiation spectral (equation) where the density of states (equation). In summary, given the multispectral sensing data (equation), we applied Lagrange Constraint Neural Network (LCNN) to solve the Blind Sources Separation (BSS) for a set of equivalent bb target temperatures. From the measurements of specific value, slopes and shapes we can fit a set of Kelvin temperatures T's for each bb targets. As a result, we could apply the analytical continuation for each entropy sources along the temperature-unique Planck spectral curves always toward the RGB color temperature display for any sensing probing frequency.

Stability of Nonlinear Wave Patterns to the Bipolar Vlasov-Poisson-Boltzmann System

NASA Astrophysics Data System (ADS)

Li, Hailiang; Wang, Yi; Yang, Tong; Zhong, Mingying

2018-04-01

The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction waves for the bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in Liu and Yu (Commun Math Phys 246:133-179, 2004) and Liu et al. (Physica D 188:178-192, 2004), we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction waves are proved for the 1D bipolar VPB system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of the rarefaction wave fan to compressible Euler equations is proved for the 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in Li et al. (Indiana Univ Math J 65(2):665-725, 2016) sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.

Project SQUID: The Foundations of Nonequilibrium Statistical Mechanics. Volume 1

DTIC Science & Technology

1963-06-01

equations available (Boltzmann, Landau,, Bogolubov- Balescu -Lenard) are essentially exact and cannot be improved. That isv for kinetic gases (those...effects) as well as to the newly obtained kinetic equation for plasmas (8) (Bogolubov- Balescu -Lenard equation). The hope of ob- taining correctly

Rarefied gas flows through a curved channel: Application of a diffusion-type equation

NASA Astrophysics Data System (ADS)

Aoki, Kazuo; Takata, Shigeru; Tatsumi, Eri; Yoshida, Hiroaki

2010-11-01

Rarefied gas flows through a curved two-dimensional channel, caused by a pressure or a temperature gradient, are investigated numerically by using a macroscopic equation of convection-diffusion type. The equation, which was derived systematically from the Bhatnagar-Gross-Krook model of the Boltzmann equation and diffuse-reflection boundary condition in a previous paper [K. Aoki et al., "A diffusion model for rarefied flows in curved channels," Multiscale Model. Simul. 6, 1281 (2008)], is valid irrespective of the degree of gas rarefaction when the channel width is much shorter than the scale of variations of physical quantities and curvature along the channel. Attention is also paid to a variant of the Knudsen compressor that can produce a pressure raise by the effect of the change of channel curvature and periodic temperature distributions without any help of moving parts. In the process of analysis, the macroscopic equation is (partially) extended to the case of the ellipsoidal-statistical model of the Boltzmann equation.

From Newton's Law to the Linear Boltzmann Equation Without Cut-Off

NASA Astrophysics Data System (ADS)

Ayi, Nathalie

2017-03-01

We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.

Slits, plates, and Poisson-Boltzmann theory in a local formulation of nonlocal electrostatics

NASA Astrophysics Data System (ADS)

Paillusson, Fabien; Blossey, Ralf

2010-11-01

Polar liquids like water carry a characteristic nanometric length scale, the correlation length of orientation polarizations. Continuum theories that can capture this feature commonly run under the name of “nonlocal” electrostatics since their dielectric response is characterized by a scale-dependent dielectric function ɛ(q) , where q is the wave vector; the Poisson(-Boltzmann) equation then turns into an integro-differential equation. Recently, “local” formulations have been put forward for these theories and applied to water, solvated ions, and proteins. We review the local formalism and show how it can be applied to a structured liquid in slit and plate geometries, and solve the Poisson-Boltzmann theory for a charged plate in a structured solvent with counterions. Our results establish a coherent picture of the local version of nonlocal electrostatics and show its ease of use when compared to the original formulation.

Great moments in kinetic theory: 150 years of Maxwell’s (other) equations

NASA Astrophysics Data System (ADS)

Robson, Robert E.; Mehrling, Timon J.; Osterhoff, Jens

2017-11-01

In 1867, just two years after laying the foundations of electromagnetism, J. Clerk Maxwell presented a fundamental paper on kinetic gas theory, in which he described the evolution of the gas in terms of certain ‘moments’ of its velocity distribution function. This inspired Ludwig Boltzmann to formulate his famous kinetic equation, from which followed the H-theorem and the connection with entropy. On the occasion of the 150th anniversary of publication of Maxwell's paper, we review the Maxwell-Boltzmann formalism and discuss how its generality and adaptability enable it to play a key role in describing the behaviour of a variety of systems of current interest, in both gaseous and condensed matter, and in modern-day physics and technologies which Maxwell and Boltzmann could not possibly have foreseen. In particular, we illustrate the relevance and applicability of Maxwell's formalism to the dynamic field of plasma-wakefield acceleration.

Lattice Boltzmann model for numerical relativity.

PubMed

Ilseven, E; Mendoza, M

2016-02-01

In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.

Droplet flow along the wall of rectangular channel with gradient of wettability

NASA Astrophysics Data System (ADS)

Kupershtokh, A. L.

2018-03-01

The lattice Boltzmann equations (LBE) method (LBM) is applicable for simulating the multiphysics problems of fluid flows with free boundaries, taking into account the viscosity, surface tension, evaporation and wetting degree of a solid surface. Modeling of the nonstationary motion of a drop of liquid along a solid surface with a variable level of wettability is carried out. For the computer simulation of such a problem, the three-dimensional lattice Boltzmann equations method D3Q19 is used. The LBE method allows us to parallelize the calculations on multiprocessor graphics accelerators using the CUDA programming technology.

Influence of nonelectrostatic ion-ion interactions on double-layer capacitance

NASA Astrophysics Data System (ADS)

Zhao, Hui

2012-11-01

Recently a Poisson-Helmholtz-Boltzmann (PHB) model [Bohinc , Phys. Rev. EPLEEE81539-375510.1103/PhysRevE.85.031130 85, 031130 (2012)] was developed by accounting for solvent-mediated nonelectrostatic ion-ion interactions. Nonelectrostatic interactions are described by a Yukawa-like pair potential. In the present work, we modify the PHB model by adding steric effects (finite ion size) into the free energy to derive governing equations. The modified PHB model is capable of capturing both ion specificity and ion crowding. This modified model is then employed to study the capacitance of the double layer. More specifically, we focus on the influence of nonelectrostatic ion-ion interactions on charging a double layer near a flat surface in the presence of steric effects. We numerically compute the differential capacitance as a function of the voltage under various conditions. At small voltages and low salt concentrations (dilute solution), we find out that the predictions from the modified PHB model are the same as those from the classical Poisson-Boltzmann theory, indicating that nonelectrostatic ion-ion interactions and steric effects are negligible. At moderate voltages, nonelectrostatic ion-ion interactions play an important role in determining the differential capacitance. Generally speaking, nonelectrostatic interactions decrease the capacitance because of additional nonelectrostatic repulsion among excess counterions inside the double layer. However, increasing the voltage gradually favors steric effects, which induce a condensed layer with crowding of counterions near the electrode. Accordingly, the predictions from the modified PHB model collapse onto those computed by the modified Poisson-Boltzmann theory considering steric effects alone. Finally, theoretical predictions are compared and favorably agree with experimental data, in particular, in concentrated solutions, leading one to conclude that the modified PHB model adequately predicts the diffuse-charge dynamics of the double layer with ion specificity and steric effects.

A modified Poisson-Boltzmann equation applied to protein adsorption.

PubMed

Gama, Marlon de Souza; Santos, Mirella Simões; Lima, Eduardo Rocha de Almeida; Tavares, Frederico Wanderley; Barreto, Amaro Gomes Barreto

2018-01-05

Ion-exchange chromatography has been widely used as a standard process in purification and analysis of protein, based on the electrostatic interaction between the protein and the stationary phase. Through the years, several approaches are used to improve the thermodynamic description of colloidal particle-surface interaction systems, however there are still a lot of gaps specifically when describing the behavior of protein adsorption. Here, we present an improved methodology for predicting the adsorption equilibrium constant by solving the modified Poisson-Boltzmann (PB) equation in bispherical coordinates. By including dispersion interactions between ions and protein, and between ions and surface, the modified PB equation used can describe the Hofmeister effects. We solve the modified Poisson-Boltzmann equation to calculate the protein-surface potential of mean force, treated as spherical colloid-plate system, as a function of process variables. From the potential of mean force, the Henry constants of adsorption, for different proteins and surfaces, are calculated as a function of pH, salt concentration, salt type, and temperature. The obtained Henry constants are compared with experimental data for several isotherms showing excellent agreement. We have also performed a sensitivity analysis to verify the behavior of different kind of salts and the Hofmeister effects. Copyright © 2017 Elsevier B.V. All rights reserved.

On the Rate of Relaxation for the Landau Kinetic Equation and Related Models

NASA Astrophysics Data System (ADS)

Bobylev, Alexander; Gamba, Irene M.; Zhang, Chenglong

2017-08-01

We study the rate of relaxation to equilibrium for Landau kinetic equation and some related models by considering the relatively simple case of radial solutions of the linear Landau-type equations. The well-known difficulty is that the evolution operator has no spectral gap, i.e. its spectrum is not separated from zero. Hence we do not expect purely exponential relaxation for large values of time t>0. One of the main goals of our work is to numerically identify the large time asymptotics for the relaxation to equilibrium. We recall the work of Strain and Guo (Arch Rat Mech Anal 187:287-339 2008, Commun Partial Differ Equ 31:17-429 2006), who rigorously show that the expected law of relaxation is \\exp (-ct^{2/3}) with some c > 0. In this manuscript, we find an heuristic way, performed by asymptotic methods, that finds this "law of two thirds", and then study this question numerically. More specifically, the linear Landau equation is approximated by a set of ODEs based on expansions in generalized Laguerre polynomials. We analyze the corresponding quadratic form and the solution of these ODEs in detail. It is shown that the solution has two different asymptotic stages for large values of time t and maximal order of polynomials N: the first one focus on intermediate asymptotics which agrees with the "law of two thirds" for moderately large values of time t and then the second one on absolute, purely exponential asymptotics for very large t, as expected for linear ODEs. We believe that appearance of intermediate asymptotics in finite dimensional approximations must be a generic behavior for different classes of equations in functional spaces (some PDEs, Boltzmann equations for soft potentials, etc.) and that our methods can be applied to related problems.

Are electrostatic potentials between regions of different chemical composition measurable? The Gibbs-Guggenheim Principle reconsidered, extended and its consequences revisited.

PubMed

Pethica, Brian A

2007-12-21

As indicated by Gibbs and made explicit by Guggenheim, the electrical potential difference between two regions of different chemical composition cannot be measured. The Gibbs-Guggenheim Principle restricts the use of classical electrostatics in electrochemical theories as thermodynamically unsound with some few approximate exceptions, notably for dilute electrolyte solutions and concomitant low potentials where the linear limit for the exponential of the relevant Boltzmann distribution applies. The Principle invalidates the widespread use of forms of the Poisson-Boltzmann equation which do not include the non-electrostatic components of the chemical potentials of the ions. From a thermodynamic analysis of the parallel plate electrical condenser, employing only measurable electrical quantities and taking into account the chemical potentials of the components of the dielectric and their adsorption at the surfaces of the condenser plates, an experimental procedure to provide exceptions to the Principle has been proposed. This procedure is now reconsidered and rejected. No other related experimental procedures circumvent the Principle. Widely-used theoretical descriptions of electrolyte solutions, charged surfaces and colloid dispersions which neglect the Principle are briefly discussed. MD methods avoid the limitations of the Poisson-Bolzmann equation. Theoretical models which include the non-electrostatic components of the inter-ion and ion-surface interactions in solutions and colloid systems assume the additivity of dispersion and electrostatic forces. An experimental procedure to test this assumption is identified from the thermodynamics of condensers at microscopic plate separations. The available experimental data from Kelvin probe studies are preliminary, but tend against additivity. A corollary to the Gibbs-Guggenheim Principle is enunciated, and the Principle is restated that for any charged species, neither the difference in electrostatic potential nor the sum of the differences in the non-electrostatic components of the thermodynamic potential difference between regions of different chemical compositions can be measured.

Analytical study of mixed electroosmotic-pressure-driven flow in rectangular micro-channels

NASA Astrophysics Data System (ADS)

Movahed, Saeid; Kamali, Reza; Eghtesad, Mohammad; Khosravifard, Amir

2013-09-01

Operational state of many miniaturized devices deals with flow field in microchannels. Pressure-driven flow (PDF) and electroosmotic flow (EOF) can be recognized as the two most important types of the flow field in such channels. EOF has many advantages in comparison with PDF, such as being vibration free and not requiring any external mechanical pumps or moving parts. However, the disadvantages of this type of flow such as Joule heating, electrophoresis demixing, and not being suitable for mobile devices must be taken into consideration carefully. By using mixed electroosmotic/pressure-driven flow, the role of EOF in producing desired velocity profile will be reduced. In this way, the advantages of EOF can be exploited, and its disadvantages can be prevented. Induced pressure gradient can be utilized in order to control the separation in the system. Furthermore, in many complicated geometries such as T-shape microchannels, turns may induce pressure gradient to the electroosmotic velocity. While analytical formulas are completely essential for analysis and control of any industrial and laboratory microdevices, lack of such formulas in the literature for solving Poisson-Boltzmann equation and predicting electroosmotic velocity field in rectangular domains is evident. In the present study, first a novel method is proposed to solve Poisson-Boltzmann equation (PBE). Subsequently, this solution is utilized to find the electroosmotic and the mixed electroosmotic/pressure-driven velocity profile in a rectangular domain of the microchannels. To demonstrate the accuracy of the presented analytical method in solving PBE and finding electroosmotic velocity, a general nondimensional example is analyzed, and the results are compared with the solution of boundary element method. Additionally, the effects of different nondimensional parameters and also aspect ratio of channels on the electroosmotic part of the flow field will be investigated.

Collision group and renormalization of the Boltzmann collision integral.

PubMed

Saveliev, V L; Nanbu, K

2002-05-01

On the basis of a recently discovered collision group [V. L. Saveliev, in Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. Gallis, AIP Conf. Proc. No. 585 (AIP, Melville, NY, 2001), p. 101], the Boltzmann collision integral is exactly rewritten in two parts. The first part describes the scattering of particles with small angles. In this part the infinity due to the infinite cross sections is extracted from the Boltzmann collision integral. Moreover, the Boltzmann collision integral is represented as a divergence of the flow in velocity space. Owing to this, the role of collisions in the kinetic equation can be interpreted in terms of the nonlocal friction force that depends on the distribution function.

Collision group and renormalization of the Boltzmann collision integral

NASA Astrophysics Data System (ADS)

Saveliev, V. L.; Nanbu, K.

2002-05-01

On the basis of a recently discovered collision group [V. L. Saveliev, in Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. Gallis, AIP Conf. Proc. No. 585 (AIP, Melville, NY, 2001), p. 101], the Boltzmann collision integral is exactly rewritten in two parts. The first part describes the scattering of particles with small angles. In this part the infinity due to the infinite cross sections is extracted from the Boltzmann collision integral. Moreover, the Boltzmann collision integral is represented as a divergence of the flow in velocity space. Owing to this, the role of collisions in the kinetic equation can be interpreted in terms of the nonlocal friction force that depends on the distribution function.

Computational analysis of fluid dynamics in pharmaceutical freeze-drying.

PubMed

Alexeenko, Alina A; Ganguly, Arnab; Nail, Steven L

2009-09-01

Analysis of water vapor flows encountered in pharmaceutical freeze-drying systems, laboratory-scale and industrial, is presented based on the computational fluid dynamics (CFD) techniques. The flows under continuum gas conditions are analyzed using the solution of the Navier-Stokes equations whereas the rarefied flow solutions are obtained by the direct simulation Monte Carlo (DSMC) method for the Boltzmann equation. Examples of application of CFD techniques to laboratory-scale and industrial scale freeze-drying processes are discussed with an emphasis on the utility of CFD for improvement of design and experimental characterization of pharmaceutical freeze-drying hardware and processes. The current article presents a two-dimensional simulation of a laboratory scale dryer with an emphasis on the importance of drying conditions and hardware design on process control and a three-dimensional simulation of an industrial dryer containing a comparison of the obtained results with analytical viscous flow solutions. It was found that the presence of clean in place (CIP)/sterilize in place (SIP) piping in the duct lead to significant changes in the flow field characteristics. The simulation results for vapor flow rates in an industrial freeze-dryer have been compared to tunable diode laser absorption spectroscopy (TDLAS) and gravimetric measurements.

GroPBS: Fast Solver for Implicit Electrostatics of Biomolecules

PubMed Central

Bertelshofer, Franziska; Sun, Liping; Greiner, Günther; Böckmann, Rainer A.

2015-01-01

Knowledge about the electrostatic potential on the surface of biomolecules or biomembranes under physiological conditions is an important step in the attempt to characterize the physico-chemical properties of these molecules and, in particular, also their interactions with each other. Additionally, knowledge about solution electrostatics may also guide the design of molecules with specified properties. However, explicit water models come at a high computational cost, rendering them unsuitable for large design studies or for docking purposes. Implicit models with the water phase treated as a continuum require the numerical solution of the Poisson–Boltzmann equation (PBE). Here, we present a new flexible program for the numerical solution of the PBE, allowing for different geometries, and the explicit and implicit inclusion of membranes. It involves a discretization of space and the computation of the molecular surface. The PBE is solved using finite differences, the resulting set of equations is solved using a Gauss–Seidel method. It is shown for the example of the sucrose transporter ScrY that the implicit inclusion of a surrounding membrane has a strong effect also on the electrostatics within the pore region and, thus, needs to be carefully considered, e.g., in design studies on membrane proteins. PMID:26636074

Large eddy simulation of rotating turbulent flows and heat transfer by the lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Liou, Tong-Miin; Wang, Chun-Sheng

2018-01-01

Due to its advantage in parallel efficiency and wall treatment over conventional Navier-Stokes equation-based methods, the lattice Boltzmann method (LBM) has emerged as an efficient tool in simulating turbulent heat and fluid flows. To properly simulate the rotating turbulent flow and heat transfer, which plays a pivotal role in tremendous engineering devices such as gas turbines, wind turbines, centrifugal compressors, and rotary machines, the lattice Boltzmann equations must be reformulated in a rotating coordinate. In this study, a single-rotating reference frame (SRF) formulation of the Boltzmann equations is newly proposed combined with a subgrid scale model for the large eddy simulation of rotating turbulent flows and heat transfer. The subgrid scale closure is modeled by a shear-improved Smagorinsky model. Since the strain rates are also locally determined by the non-equilibrium part of the distribution function, the calculation process is entirely local. The pressure-driven turbulent channel flow with spanwise rotation and heat transfer is used for validating the approach. The Reynolds number characterized by the friction velocity and channel half height is fixed at 194, whereas the rotation number in terms of the friction velocity and channel height ranges from 0 to 3.0. A working fluid of air is chosen, which corresponds to a Prandtl number of 0.71. Calculated results are demonstrated in terms of mean velocity, Reynolds stress, root mean square (RMS) velocity fluctuations, mean temperature, RMS temperature fluctuations, and turbulent heat flux. Good agreement is found between the present LBM predictions and previous direct numerical simulation data obtained by solving the conventional Navier-Stokes equations, which confirms the capability of the proposed SRF LBM and subgrid scale relaxation time formulation for the computation of rotating turbulent flows and heat transfer.

A fast Laplace solver approach to pore scale permeability

NASA Astrophysics Data System (ADS)

Arns, Christoph; Adler, Pierre

2017-04-01

The permeability of a porous medium can be derived by solving the Stokes equations in the pore space with no slip at the walls. The resulting velocity averaged over the pore volume yields the permeability KS by application of the Darcy law. The Stokes equations can be solved by a number of different techniques such as finite differences, finite volume, Lattice Boltzmann, but whatever the technique it remains a heavy task since there are four unknowns at each node (the three velocity components and the pressure) which necessitate the solution of four equations (the projection of Newton's law on each axis and mass conservation). By comparison, the Laplace equation is scalar with a single unknown at each node. The objective of this work is to replace the Stokes equations by an elliptical equation with a space dependent permeability. More precisely, the local permeability k is supposed to be proportional to (r-alpha)**2 where r is the distance of the voxel to the closest wall, and alpha a constant; k is zero in the solid phase. The elliptical equation is div(k gradp)=0. A macroscopic pressure gradient is assumed to be exerted on the medium and again the resulting velocity averaged over space yields a permeability K_L. In order to validate this method, systematic calculations have been performed. First, elementary shapes (plane channel, circular pipe, rectangular channels) were studied for which flow occurs along parallel lines in which case KL is the arithmetic average of the k's. KL was calculated for various discretizations of the pore space and various values of alpha. For alpha=0.5, the agreement with the exact analytical value of KS is excellent for the plane and rectangular channels while it is only approximate for circular pipes. Second, the permeability KL of channels with sinusoidal walls was calculated and compared with analytical results and numerical ones provided by a Lattice Boltzmann algorithm. Generally speaking, the discrepancy does not exceed 25% when alpha=0.5. Third, the most important test was performed on two types of real media that were used for previous studies. A fracture network measured by FIB/SEM in a low permeability sandstone was used for that purpose; the two dimensionless permeabilities KS and KL are equal to 9.3d-3 and 8.5d-3. Similar calculations were performed on 256 samples of Fontainebleau sandstones and the agreement was in general excellent, except may be for very low permeabilities. To conclude, the Laplace solver is significantly more stable than the lattice Boltzmann approach, uses less memory, and is significantly faster. Permeabilities are in excellent agreement over a wide range of porosities.

Effect of electron-electron scattering on the conductance of a quantum wire studied with the Boltzman transport equation

NASA Astrophysics Data System (ADS)

Lyo, S. K.; Huang, Danhong

2006-05-01

Electron-electron scattering conserves total momentum and does not dissipate momentum directly in a low-density system where the umklapp process is forbidden. However, it can still affect the conductance through the energy relaxation of the electrons. We show here that this effect can be studied with arbitrary accuracy in a multisublevel one-dimensional (1D) single quantum wire system in the presence of roughness and phonon scattering using a formally exact solution of the Boltzmann transport equation. The intrasubband electron-electron scattering is found to yield no net effect on the transport of electrons in 1D with only one sublevel occupied. For a system with a multilevel occupation, however, we find a significant effect of intersublevel electron-electron scattering on the temperature and density dependence of the resistance at low temperatures.

First-principles analysis of anharmonic nuclear motion and thermal transport in thermoelectric materials

NASA Astrophysics Data System (ADS)

Tadano, Terumasa; Tsuneyuki, Shinji

2015-12-01

We show a first-principles approach for analyzing anharmonic properties of lattice vibrations in solids. We firstly extract harmonic and anharmonic force constants from accurate first-principles calculations based on the density functional theory. Using the many-body perturbation theory of phonons, we then estimate the phonon scattering probability due to anharmonic phonon-phonon interactions. We show the validity of the approach by computing the lattice thermal conductivity of Si, a typical covalent semiconductor, and selected thermoelectric materials PbTe and Bi2Te3 based on the Boltzmann transport equation. We also show that the phonon lifetime and the lattice thermal conductivity of the high-temperature phase of SrTiO3 can be estimated by employing the perturbation theory on top of the solution of the self-consistent phonon equation.

Interaction with the lower ionosphere of electromagnetic pulses from lightning - Heating, attachment, and ionization

NASA Technical Reports Server (NTRS)

Taranenko, Y. N.; Inan, U. S.; Bell, T. F.

1993-01-01

A Boltzmann formulation of the electron distribution function and Maxwell's equations for the EM fields are used to simulate the interaction of lightning radiated EM pulses with the lower ionosphere. Ionization and dissociative attachment induced by the heated electrons cause significant changes in the local electron density, N(e). Due to 'slow' field changes of typical lightning EM pulses over time scales of tens of microsec, the distribution function follows the quasi-equilibrium solution of the Boltzmann equation in the altitude range of interest (70 to 100 km). The EM pulse is simulated as a planar 100 microsec long single period oscillation of a 10 kHz wave injected at 70 km. Under nighttime conditions, individual pulses of intensity 10-20 V/m (normalized to 100 km horizontal distance) produce changes in N(e) of 1-30 percent while a sequence of pulses leads to strong modification of N(e) at altitudes less than 95 km. The N(e) changes produce a 'sharpening' of the lower ionospheric boundary by causing a reduction in electron density at 75-85 km (due to attachment) and a substantial increase at 85-95 km (due to ionization) (e.g., the scale height decreases by a factor of about 2 at about 85 km for a single 20 V/m EM pulse). No substantial N(e) changes occur during daytime.

2D Lattice Boltzmann Simulation Of Chemical Reactions Within Rayleigh-Bénard And Poiseuille-Bénard Convection Systems

NASA Astrophysics Data System (ADS)

Amaya-Ventura, Gilberto; Rodríguez-Romo, Suemi

2011-09-01

This paper deals with the computational simulation of the reaction-diffusion-advection phenomena emerging in Rayleigh-Bénard (RB) and Poiseuille-Bénard reactive convection systems. We use the Boussinesq's approximation for buoyancy forces and the Lattice Boltzmann method (LBM). The first kinetic mesoscopic model proposed here is based on the discrete Boltzmann equation needed to solve the momentum balance coupled with buoyancy forces. Then, a second lattice Boltzmann algorithm is applied to solve the reaction-diffusion-advection equation to calculate the evolution of the chemical species concentration. We use a reactive system composed by nitrous oxide (so call laughing gas) in air as an example; its spatio-temporal decomposition is calculated. Two cases are considered, a rectangular enclosed cavity and an open channel. The simulations are performed at low Reynolds numbers and in a steady state between the first and second thermo-hydrodynamic instabilities. The results presented here, for the thermo-hydrodynamic behavior, are in good agreement with experimental data; while our| chemical kinetics simulation yields expected results. Some applications of our approach are related to chemical reactors and atmospheric phenomena, among others.

Deformation of a Capsule in a Power-Law Shear Flow

PubMed Central

2016-01-01

An immersed boundary-lattice Boltzmann method is developed for fluid-structure interactions involving non-Newtonian fluids (e.g., power-law fluid). In this method, the flexible structure (e.g., capsule) dynamics and the fluid dynamics are coupled by using the immersed boundary method. The incompressible viscous power-law fluid motion is obtained by solving the lattice Boltzmann equation. The non-Newtonian rheology is achieved by using a shear rate-dependant relaxation time in the lattice Boltzmann method. The non-Newtonian flow solver is then validated by considering a power-law flow in a straight channel which is one of the benchmark problems to validate an in-house solver. The numerical results present a good agreement with the analytical solutions for various values of power-law index. Finally, we apply this method to study the deformation of a capsule in a power-law shear flow by varying the Reynolds number from 0.025 to 0.1, dimensionless shear rate from 0.004 to 0.1, and power-law index from 0.2 to 1.8. It is found that the deformation of the capsule increases with the power-law index for different Reynolds numbers and nondimensional shear rates. In addition, the Reynolds number does not have significant effect on the capsule deformation in the flow regime considered. Moreover, the power-law index effect is stronger for larger dimensionless shear rate compared to smaller values. PMID:27840656

Coupling lattice Boltzmann and continuum equations for flow and reactive transport in porous media.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Coon, Ethan; Porter, Mark L.; Kang, Qinjun

2012-06-18

In spatially and temporally localized instances, capturing sub-reservoir scale information is necessary. Capturing sub-reservoir scale information everywhere is neither necessary, nor computationally possible. The lattice Boltzmann Method for solving pore-scale systems. At the pore-scale, LBM provides an extremely scalable, efficient way of solving Navier-Stokes equations on complex geometries. Coupling pore-scale and continuum scale systems via domain decomposition. By leveraging the interpolations implied by pore-scale and continuum scale discretizations, overlapping Schwartz domain decomposition is used to ensure continuity of pressure and flux. This approach is demonstrated on a fractured medium, in which Navier-Stokes equations are solved within the fracture while Darcy'smore » equation is solved away from the fracture Coupling reactive transport to pore-scale flow simulators allows hybrid approaches to be extended to solve multi-scale reactive transport.« less

Optimal preconditioning of lattice Boltzmann methods

NASA Astrophysics Data System (ADS)

Izquierdo, Salvador; Fueyo, Norberto

2009-09-01

A preconditioning technique to accelerate the simulation of steady-state problems using the single-relaxation-time (SRT) lattice Boltzmann (LB) method was first proposed by Guo et al. [Z. Guo, T. Zhao, Y. Shi, Preconditioned lattice-Boltzmann method for steady flows, Phys. Rev. E 70 (2004) 066706-1]. The key idea in this preconditioner is to modify the equilibrium distribution function in such a way that, by means of a Chapman-Enskog expansion, a time-derivative preconditioner of the Navier-Stokes (NS) equations is obtained. In the present contribution, the optimal values for the free parameter γ of this preconditioner are searched both numerically and theoretically; the later with the aid of linear-stability analysis and with the condition number of the system of NS equations. The influence of the collision operator, single- versus multiple-relaxation-times (MRT), is also studied. Three steady-state laminar test cases are used for validation, namely: the two-dimensional lid-driven cavity, a two-dimensional microchannel and the three-dimensional backward-facing step. Finally, guidelines are suggested for an a priori definition of optimal preconditioning parameters as a function of the Reynolds and Mach numbers. The new optimally preconditioned MRT method derived is shown to improve, simultaneously, the rate of convergence, the stability and the accuracy of the lattice Boltzmann simulations, when compared to the non-preconditioned methods and to the optimally preconditioned SRT one. Additionally, direct time-derivative preconditioning of the LB equation is also studied.

ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSON-BOLTZMANN EQUATION

PubMed Central

HOLST, MICHAEL; MCCAMMON, JAMES ANDREW; YU, ZEYUN; ZHOU, YOUNGCHENG; ZHU, YUNRONG

2011-01-01

We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L∞ estimates to establish quasi-orthogonality. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein. PMID:21949541

Large scale atomistic approaches to thermal transport and phonon scattering in nanostructured materials

NASA Astrophysics Data System (ADS)

Savic, Ivana

2012-02-01

Decreasing the thermal conductivity of bulk materials by nanostructuring and dimensionality reduction, or by introducing some amount of disorder represents a promising strategy in the search for efficient thermoelectric materials [1]. For example, considerable improvements of the thermoelectric efficiency in nanowires with surface roughness [2], superlattices [3] and nanocomposites [4] have been attributed to a significantly reduced thermal conductivity. In order to accurately describe thermal transport processes in complex nanostructured materials and directly compare with experiments, the development of theoretical and computational approaches that can account for both anharmonic and disorder effects in large samples is highly desirable. We will first summarize the strengths and weaknesses of the standard atomistic approaches to thermal transport (molecular dynamics [5], Boltzmann transport equation [6] and Green's function approach [7]) . We will then focus on the methods based on the solution of the Boltzmann transport equation, that are computationally too demanding, at present, to treat large scale systems and thus to investigate realistic materials. We will present a Monte Carlo method [8] to solve the Boltzmann transport equation in the relaxation time approximation [9], that enables computation of the thermal conductivity of ordered and disordered systems with a number of atoms up to an order of magnitude larger than feasible with straightforward integration. We will present a comparison between exact and Monte Carlo Boltzmann transport results for small SiGe nanostructures and then use the Monte Carlo method to analyze the thermal properties of realistic SiGe nanostructured materials. This work is done in collaboration with Davide Donadio, Francois Gygi, and Giulia Galli from UC Davis.[4pt] [1] See e.g. A. J. Minnich, M. S. Dresselhaus, Z. F. Ren, and G. Chen, Energy Environ. Sci. 2, 466 (2009).[0pt] [2] A. I. Hochbaum et al, Nature 451, 163 (2008).[0pt] [3] R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, Nature 413, 597 (2001).[0pt] [4] B. Poudel et al, Science 320, 634 (2008).[0pt] [5] See e.g. Y. He, D. Donadio, and G. Galli, Nano Lett. 11, 3608 (2011).[0pt] [6] See e.g. A. Ward and D. A. Broido, Phys. Rev. B 81, 085205 (2010).[0pt] [7] See e.g. I. Savic, N. Mingo, and D. A. Stewart, Phys. Rev. Lett. 101, 165502 (2008).[0pt] [8] I. Savic, D.Donadio, F.Gygi, and G.Galli (in preparation).[0pt] [9] See e.g. J. E. Turney, E. S. Landry, A. J. H. McGaughey, and C. H. Amon, Phys. Rev. B, 79, 064301 (2009).

Towards a solution of the puzzle posed by superconducting SrTiO3

NASA Astrophysics Data System (ADS)

Malik, G. P.

2015-09-01

Suitably doped SrTiO3 was found in 1964 to undergo a superconducting transition below 1 K with a dome-like Tc versus n (electron concentration) plot. The apex of the dome — a point of inflection — corresponds to the point (n≈9 ×1019cm-3, Tc≈0.30 K). On either side of it, Tc goes down to ≈0.1 K for the extreme values between which n was varied. A single value of Tc is thus observed for two different values of n. The puzzle for the theory has been to explain this result. Treating the problem in all its generality, we present here three equations: the μ1-incorporated BCS equation for Tc, the μ0-incorporated equation for the T = 0 gap Δ0, where μ1 and μ0 are the chemical potentials at T = Tc and T = 0 respectively, and an equation that relates the interaction parameters λ1 and λ0 at these temperatures. Because there are five unknowns in the problem, we tackle these equations via an approximation scheme that includes setting μ1 = μ0 and λ1 = λ0. The latter of these is factually a basic tenet of the BCS theory. Salient features of our findings are: (i) the solutions for Tc and Δ0 on the RHS (LHS) of the dome correspond to μ> kBθD(μ < kBθD); kB = Boltzmann constant, θD = Debye temperature, (ii) for solutions on the LHS the limits of the integrals in the equations need to be curtailed to obtain real solutions and (iii) the point μ = kBθD is a point of inflection in the Tc versus μ plot. Since the puzzle has remained unsolved for a long time, we also offer here a purely mathematical model for λ(μ) — sans physical justification — which leads to a Tc versus μ plot qualitatively in agreement with experiment.

Modified Chapman-Enskog moment approach to diffusive phonon heat transport.

PubMed

Banach, Zbigniew; Larecki, Wieslaw

2008-12-01

A detailed treatment of the Chapman-Enskog method for a phonon gas is given within the framework of an infinite system of moment equations obtained from Callaway's model of the Boltzmann-Peierls equation. Introducing no limitations on the magnitudes of the individual components of the drift velocity or the heat flux, this method is used to derive various systems of hydrodynamic equations for the energy density and the drift velocity. For one-dimensional flow problems, assuming that normal processes dominate over resistive ones, it is found that the first three levels of the expansion (i.e., the zeroth-, first-, and second-order approximations) yield the equations of hydrodynamics which are linearly stable at all wavelengths. This result can be achieved either by examining the dispersion relations for linear plane waves or by constructing the explicit quadratic Lyapunov entropy functionals for the linear perturbation equations. The next order in the Chapman-Enskog expansion leads to equations which are unstable to some perturbations. Precisely speaking, the linearized equations of motion that describe the propagation of small disturbances in the flow have unstable plane-wave solutions in the short-wavelength limit of the dispersion relations. This poses no problem if the equations are used in their proper range of validity.

Electro-osmotic mobility of non-Newtonian fluids

PubMed Central

Zhao, Cunlu; Yang, Chun

2011-01-01

Electrokinetically driven microfluidic devices are usually used to analyze and process biofluids which can be classified as non-Newtonian fluids. Conventional electrokinetic theories resulting from Newtonian hydrodynamics then fail to describe the behaviors of these fluids. In this study, a theoretical analysis of electro-osmotic mobility of non-Newtonian fluids is reported. The general Cauchy momentum equation is simplified by incorporation of the Gouy–Chapman solution to the Poisson–Boltzmann equation and the Carreau fluid constitutive model. Then a nonlinear ordinary differential equation governing the electro-osmotic velocity of Carreau fluids is obtained and solved numerically. The effects of the Weissenberg number (Wi), the surface zeta potential (ψ¯s), the power-law exponent(n), and the transitional parameter (β) on electro-osmotic mobility are examined. It is shown that the results presented in this study for the electro-osmotic mobility of Carreau fluids are quite general so that the electro-osmotic mobility for the Newtonian fluids and the power-law fluids can be obtained as two limiting cases. PMID:21503161

Simulation of the effects of sub-breakdown electric fields on the chemical kinetics in nonpremixed counterflow methane/air flames

NASA Astrophysics Data System (ADS)

Belhi, Memdouh; Im, Hong; Computational Reacting Flows Laboratory, Clean Combustion Research Center Team

2017-11-01

The effects of an electric field on the combustion kinetics in nonpremixed counterflow methane/air flames were investigated via one-dimensional numerical simulations. A classical fluid model coupling Poison's equation with transport equations for combustion species and electric field-induced particles was used. A methane-air reaction mechanism accounting for the natural ionization in flames was combined with a set of reactions that describe the formation of active particles induced by the electric field. Kinetic parameters for electron-impact reactions and transport coefficients of electrons were modeled as functions of reduced electric field via solutions to the Boltzmann kinetic equation using the BOLSIG code. Mobility of ions was computed based on the (n,6,4) and coulomb interaction potentials, while the diffusion coefficient was approximated from the mobility using Einstein relation. Contributions of electron dissociation, excitation and ionization processes were characterized quantitatively. An analysis to identify the plasma regime where the electric field can alter the combustion kinetic was proposed.

Vertical distribution of vibrational energy of molecular nitrogen in a stable auroral red arc and its effect on ionospheric electron densities. Ph.D. Thesis - Catholic Univ. of Am.

NASA Technical Reports Server (NTRS)

Newton, G. P.

1973-01-01

Previous solutions of the problem of the distribution of vibrationally excited molecular nitrogen in the thermosphere have either assumed a Boltzmann distribution and considered diffusion as one of the loss processes or solved for the energy level populations and neglected diffusion. Both of the previous approaches are combined by solving the time dependent continuity equations, including the diffusion process, for the first six energy levels of molecular nitrogen for conditions in the thermosphere corresponding to a stable auroral red arc. The primary source of molecular nitrogen excitation was subexcitation, and inelastic collisions between thermal electrons and molecular nitrogen. The reaction rates for this process were calculated from published cross section calculations. The loss processes for vibrational energy were electron and atomic oxygen quenching and vibrational energy exchange. The coupled sets of nonlinear, partial differential equations were solved numerically by employing finite difference equations.

Thermodynamic aspect in using modified Boltzmann model as an acoustic probe for URu2Si2

NASA Astrophysics Data System (ADS)

Kwang-Hua, Chu Rainer

2018-05-01

The approximate system of equations describing ultrasonic attenuation propagating in many electrons of the heavy-fermion materials URu2Si2 under high magnetic fields were firstly derived and then calculated based on the modified Boltzmann model considering the microscopic contributions due to electronic fluids. A system of nonlinear partial differential coupled with integral equations were linearized firstly and approximately solved considering the perturbed thermodynamic equilibrium states. Our numerical data were compared with previous measurements using non-dimensional or normalized physical values. The rather good fit of our numerical calculations with experimental measurements confirms our present approach.

Isotropic matrix elements of the collision integral for the Boltzmann equation

NASA Astrophysics Data System (ADS)

Ender, I. A.; Bakaleinikov, L. A.; Flegontova, E. Yu.; Gerasimenko, A. B.

2017-09-01

We have proposed an algorithm for constructing matrix elements of the collision integral for the nonlinear Boltzmann equation isotropic in velocities. These matrix elements have been used to start the recurrent procedure for calculating matrix elements of the velocity-nonisotropic collision integral described in our previous publication. In addition, isotropic matrix elements are of independent interest for calculating isotropic relaxation in a number of physical kinetics problems. It has been shown that the coefficients of expansion of isotropic matrix elements in Ω integrals are connected by the recurrent relations that make it possible to construct the procedure of their sequential determination.

Hydrodynamics beyond Navier-Stokes: the slip flow model.

PubMed

Yudistiawan, Wahyu P; Ansumali, Santosh; Karlin, Iliya V

2008-07-01

Recently, analytical solutions for the nonlinear Couette flow demonstrated the relevance of the lattice Boltzmann (LB) models to hydrodynamics beyond the continuum limit [S. Ansumali, Phys. Rev. Lett. 98, 124502 (2007)]. In this paper, we present a systematic study of the simplest LB kinetic equation-the nine-bit model in two dimensions--in order to quantify it as a slip flow approximation. Details of the aforementioned analytical solution are presented, and results are extended to include a general shear- and force-driven unidirectional flow in confined geometry. Exact solutions for the velocity, as well as for pertinent higher-order moments of the distribution functions, are obtained in both Couette and Poiseuille steady-state flows for all values of rarefaction parameter (Knudsen number). Results are compared with the slip flow solution by Cercignani, and a good quantitative agreement is found for both flow situations. Thus, the standard nine-bit LB model is characterized as a valid and self-consistent slip flow model for simulations beyond the Navier-Stokes approximation.

Kinetics of wealth and the Pareto law

NASA Astrophysics Data System (ADS)

Boghosian, Bruce M.

2014-04-01

An important class of economic models involve agents whose wealth changes due to transactions with other agents. Several authors have pointed out an analogy with kinetic theory, which describes molecules whose momentum and energy change due to interactions with other molecules. We pursue this analogy and derive a Boltzmann equation for the time evolution of the wealth distribution of a population of agents for the so-called Yard-Sale Model of wealth exchange. We examine the solutions to this equation by a combination of analytical and numerical methods and investigate its long-time limit. We study an important limit of this equation for small transaction sizes and derive a partial integrodifferential equation governing the evolution of the wealth distribution in a closed economy. We then describe how this model can be extended to include features such as inflation, production, and taxation. In particular, we show that the model with taxation exhibits the basic features of the Pareto law, namely, a lower cutoff to the wealth density at small values of wealth, and approximate power-law behavior at large values of wealth.

Molecular description of steady supersonic free jets

NASA Astrophysics Data System (ADS)

Montero, S.

2017-09-01

A detailed analysis of the non-local thermal equilibrium (n-LTE) problem in the paraxial zone of silence of supersonic free jets is reported. The study is based on a hybrid approach that combines Navier-Stokes equations with a kinetic equation derived from the generalized Boltzmann (Waldmann-Snider) equation. The resulting system is solved for those flow quantities not easily amenable to experimental measure (translational temperature, flow velocity, and entropy) in terms of the quantities that can be measured accurately (distance, number density, population of rotational states, and their gradients). The reported solutions are essentially exact and are formulated in terms of macroscopic quantities, as well as in terms of elementary collision processes. Emphasis is made on the influence of dissipative effects onto the flow (viscous and diabatic) and of the breakdown of thermal equilibrium onto the evolution of entropy and translational temperature. The influence of inelastic collisions onto these effects is analysed in depth. The reported equations are aimed at optimizing the experimental knowledge of the n-LTE problem and its quantitative interpretation in terms of state-to-state rates for inelastic collisions.

Stable spin domains in a nondegenerate ultracold gas

NASA Astrophysics Data System (ADS)

Graham, S. D.; Niroomand, D.; Ragan, R. J.; McGuirk, J. M.

2018-05-01

We study the stability of two-domain spin structures in an ultracold gas of magnetically trapped 87Rb atoms above quantum degeneracy. Adding a small effective magnetic field gradient stabilizes the domains via coherent collective spin rotation effects, despite negligibly perturbing the potential energy relative to the thermal energy. We demonstrate that domain stabilization is accomplished through decoupling the dynamics of longitudinal magnetization, which remains in time-independent domains, from transverse magnetization, which undergoes a purely transverse spin wave trapped within the domain wall. We explore the effect of temperature and density on the steady-state domains, and compare our results to a hydrodynamic solution to a quantum Boltzmann equation.

Lattice Boltzmann heat transfer model for permeable voxels

NASA Astrophysics Data System (ADS)

Pereira, Gerald G.; Wu, Bisheng; Ahmed, Shakil

2017-12-01

We develop a gray-scale lattice Boltzmann (LB) model to study fluid flow combined with heat transfer for flow through porous media where voxels may be partially solid (or void). Heat transfer in rocks may lead to deformation, which in turn can modulate the fluid flow and so has significant contribution to rock permeability. The LB temperature field is compared to a finite difference solution of the continuum partial differential equations for fluid flow in a channel. Excellent quantitative agreement is found for both Poiseuille channel flow and Brinkman flow. The LB model is then applied to sample porous media such as packed beds and also more realistic sandstone rock sample, and both the convective and diffusive regimes are recovered when varying the thermal diffusivity. It is found that while the rock permeability can be comparatively small (order milli-Darcy), the temperature field can show significant variation depending on the thermal convection of the fluid. This LB method has significant advantages over other numerical methods such as finite and boundary element methods in dealing with coupled fluid flow and heat transfer in rocks which have irregular and nonsmooth pore spaces.

Nonlocal Electrostatics in Spherical Geometries Using Eigenfunction Expansions of Boundary-Integral Operators.

PubMed

Bardhan, Jaydeep P; Knepley, Matthew G; Brune, Peter

2015-01-01

In this paper, we present an exact, infinite-series solution to Lorentz nonlocal continuum electrostatics for an arbitrary charge distribution in a spherical solute. Our approach relies on two key steps: (1) re-formulating the PDE problem using boundary-integral equations, and (2) diagonalizing the boundary-integral operators using the fact that their eigenfunctions are the surface spherical harmonics. To introduce this uncommon approach for calculations in separable geometries, we first re-derive Kirkwood's classic results for a protein surrounded concentrically by a pure-water ion-exclusion (Stern) layer and then a dilute electrolyte, which is modeled with the linearized Poisson-Boltzmann equation. The eigenfunction-expansion approach provides a computationally efficient way to test some implications of nonlocal models, including estimating the reasonable range of the nonlocal length-scale parameter λ. Our results suggest that nonlocal solvent response may help to reduce the need for very high dielectric constants in calculating pH-dependent protein behavior, though more sophisticated nonlocal models are needed to resolve this question in full. An open-source MATLAB implementation of our approach is freely available online.

Nonlocal Electrostatics in Spherical Geometries Using Eigenfunction Expansions of Boundary-Integral Operators

PubMed Central

Bardhan, Jaydeep P.; Knepley, Matthew G.; Brune, Peter

2015-01-01

In this paper, we present an exact, infinite-series solution to Lorentz nonlocal continuum electrostatics for an arbitrary charge distribution in a spherical solute. Our approach relies on two key steps: (1) re-formulating the PDE problem using boundary-integral equations, and (2) diagonalizing the boundary-integral operators using the fact that their eigenfunctions are the surface spherical harmonics. To introduce this uncommon approach for calculations in separable geometries, we first re-derive Kirkwood’s classic results for a protein surrounded concentrically by a pure-water ion-exclusion (Stern) layer and then a dilute electrolyte, which is modeled with the linearized Poisson–Boltzmann equation. The eigenfunction-expansion approach provides a computationally efficient way to test some implications of nonlocal models, including estimating the reasonable range of the nonlocal length-scale parameter λ. Our results suggest that nonlocal solvent response may help to reduce the need for very high dielectric constants in calculating pH-dependent protein behavior, though more sophisticated nonlocal models are needed to resolve this question in full. An open-source MATLAB implementation of our approach is freely available online. PMID:26273581

Electron distribution functions in electric field environments

NASA Technical Reports Server (NTRS)

Rudolph, Terence H.

1991-01-01

The amount of current carried by an electric discharge in its early stages of growth is strongly dependent on its geometrical shape. Discharges with a large number of branches, each funnelling current to a common stem, tend to carry more current than those with fewer branches. The fractal character of typical discharges was simulated using stochastic models based on solutions of the Laplace equation. Extension of these models requires the use of electron distribution functions to describe the behavior of electrons in the undisturbed medium ahead of the discharge. These electrons, interacting with the electric field, determine the propagation of branches in the discharge and the way in which further branching occurs. The first phase in the extension of the referenced models , the calculation of simple electron distribution functions in an air/electric field medium, is discussed. Two techniques are investigated: (1) the solution of the Boltzmann equation in homogeneous, steady state environments, and (2) the use of Monte Carlo simulations. Distribution functions calculated from both techniques are illustrated. Advantages and disadvantages of each technique are discussed.

Application of closed-form solutions to a mesh point field in silicon solar cells

NASA Technical Reports Server (NTRS)

Lamorte, M. F.

1985-01-01

A computer simulation method is discussed that provides for equivalent simulation accuracy, but that exhibits significantly lower CPU running time per bias point compared to other techniques. This new method is applied to a mesh point field as is customary in numerical integration (NI) techniques. The assumption of a linear approximation for the dependent variable, which is typically used in the finite difference and finite element NI methods, is not required. Instead, the set of device transport equations is applied to, and the closed-form solutions obtained for, each mesh point. The mesh point field is generated so that the coefficients in the set of transport equations exhibit small changes between adjacent mesh points. Application of this method to high-efficiency silicon solar cells is described; and the method by which Auger recombination, ambipolar considerations, built-in and induced electric fields, bandgap narrowing, carrier confinement, and carrier diffusivities are treated. Bandgap narrowing has been investigated using Fermi-Dirac statistics, and these results show that bandgap narrowing is more pronounced and that it is temperature-dependent in contrast to the results based on Boltzmann statistics.

Molecular representation of molar domain (volume), evolution equations, and linear constitutive relations for volume transport.

PubMed

Eu, Byung Chan

2008-09-07

In the traditional theories of irreversible thermodynamics and fluid mechanics, the specific volume and molar volume have been interchangeably used for pure fluids, but in this work we show that they should be distinguished from each other and given distinctive statistical mechanical representations. In this paper, we present a general formula for the statistical mechanical representation of molecular domain (volume or space) by using the Voronoi volume and its mean value that may be regarded as molar domain (volume) and also the statistical mechanical representation of volume flux. By using their statistical mechanical formulas, the evolution equations of volume transport are derived from the generalized Boltzmann equation of fluids. Approximate solutions of the evolution equations of volume transport provides kinetic theory formulas for the molecular domain, the constitutive equations for molar domain (volume) and volume flux, and the dissipation of energy associated with volume transport. Together with the constitutive equation for the mean velocity of the fluid obtained in a previous paper, the evolution equations for volume transport not only shed a fresh light on, and insight into, irreversible phenomena in fluids but also can be applied to study fluid flow problems in a manner hitherto unavailable in fluid dynamics and irreversible thermodynamics. Their roles in the generalized hydrodynamics will be considered in the sequel.

Lattice dynamics and thermal conductivity of lithium fluoride via first-principles calculations

NASA Astrophysics Data System (ADS)

Liang, Ting; Chen, Wen-Qi; Hu, Cui-E.; Chen, Xiang-Rong; Chen, Qi-Feng

2018-04-01

The lattice thermal conductivity of lithium fluoride (LiF) is accurately computed from a first-principles approach based on an iterative solution of the Boltzmann transport equation. Real-space finite-difference supercell approach is employed to generate the second- and third-order interatomic force constants. The related physical quantities of LiF are calculated by the second- and third- order potential interactions at 30 K-1000 K. The calculated lattice thermal conductivity 13.89 W/(m K) for LiF at room temperature agrees well with the experimental value, demonstrating that the parameter-free approach can furnish precise descriptions of the lattice thermal conductivity for this material. Besides, the Born effective charges, dielectric constants and phonon spectrum of LiF accord well with the existing data. The lattice thermal conductivities for the iterative solution of BTE are also presented.

A shallow water model for the propagation of tsunami via Lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Zergani, Sara; Aziz, Z. A.; Viswanathan, K. K.

2015-01-01

An efficient implementation of the lattice Boltzmann method (LBM) for the numerical simulation of the propagation of long ocean waves (e.g. tsunami), based on the nonlinear shallow water (NSW) wave equation is presented. The LBM is an alternative numerical procedure for the description of incompressible hydrodynamics and has the potential to serve as an efficient solver for incompressible flows in complex geometries. This work proposes the NSW equations for the irrotational surface waves in the case of complex bottom elevation. In recent time, equation involving shallow water is the current norm in modelling tsunami operations which include the propagation zone estimation. Several test-cases are presented to verify our model. Some implications to tsunami wave modelling are also discussed. Numerical results are found to be in excellent agreement with theory.

Self-consistent field model for strong electrostatic correlations and inhomogeneous dielectric media.

PubMed

Ma, Manman; Xu, Zhenli

2014-12-28

Electrostatic correlations and variable permittivity of electrolytes are essential for exploring many chemical and physical properties of interfaces in aqueous solutions. We propose a continuum electrostatic model for the treatment of these effects in the framework of the self-consistent field theory. The model incorporates a space- or field-dependent dielectric permittivity and an excluded ion-size effect for the correlation energy. This results in a self-energy modified Poisson-Nernst-Planck or Poisson-Boltzmann equation together with state equations for the self energy and the dielectric function. We show that the ionic size is of significant importance in predicting a finite self energy for an ion in an inhomogeneous medium. Asymptotic approximation is proposed for the solution of a generalized Debye-Hückel equation, which has been shown to capture the ionic correlation and dielectric self energy. Through simulating ionic distribution surrounding a macroion, the modified self-consistent field model is shown to agree with particle-based Monte Carlo simulations. Numerical results for symmetric and asymmetric electrolytes demonstrate that the model is able to predict the charge inversion at high correlation regime in the presence of multivalent interfacial ions which is beyond the mean-field theory and also show strong effect to double layer structure due to the space- or field-dependent dielectric permittivity.

Three-dimensional lattice Boltzmann model for compressible flows.

PubMed

Sun, Chenghai; Hsu, Andrew T

2003-07-01

A three-dimensional compressible lattice Boltzmann model is formulated on a cubic lattice. A very large particle-velocity set is incorporated in order to enable a greater variation in the mean velocity. Meanwhile, the support set of the equilibrium distribution has only six directions. Therefore, this model can efficiently handle flows over a wide range of Mach numbers and capture shock waves. Due to the simple form of the equilibrium distribution, the fourth-order velocity tensors are not involved in the formulation. Unlike the standard lattice Boltzmann model, no special treatment is required for the homogeneity of fourth-order velocity tensors on square lattices. The Navier-Stokes equations were recovered, using the Chapman-Enskog method from the Bhatnagar-Gross-Krook (BGK) lattice Boltzmann equation. The second-order discretization error of the fluctuation velocity in the macroscopic conservation equation was eliminated by means of a modified collision invariant. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Since the present scheme deals only with the equilibrium distribution that depends only on fluid density, velocity, and internal energy, boundary conditions on curved wall are easily implemented by an extrapolation of macroscopic variables. To verify the scheme for inviscid flows, we have successfully simulated a three-dimensional shock-wave propagation in a box and a normal shock of Mach number 10 over a wedge. As an application to viscous flows, we have simulated a flat plate boundary layer flow, flow over a cylinder, and a transonic flow over a NACA0012 airfoil cascade.

An Unsplit Monte-Carlo solver for the resolution of the linear Boltzmann equation coupled to (stiff) Bateman equations

NASA Astrophysics Data System (ADS)

Bernede, Adrien; Poëtte, Gaël

2018-02-01

In this paper, we are interested in the resolution of the time-dependent problem of particle transport in a medium whose composition evolves with time due to interactions. As a constraint, we want to use of Monte-Carlo (MC) scheme for the transport phase. A common resolution strategy consists in a splitting between the MC/transport phase and the time discretization scheme/medium evolution phase. After going over and illustrating the main drawbacks of split solvers in a simplified configuration (monokinetic, scalar Bateman problem), we build a new Unsplit MC (UMC) solver improving the accuracy of the solutions, avoiding numerical instabilities, and less sensitive to time discretization. The new solver is essentially based on a Monte Carlo scheme with time dependent cross sections implying the on-the-fly resolution of a reduced model for each MC particle describing the time evolution of the matter along their flight path.

The effect of viscous flow and thermal flux on the rate of chemical reaction in dilute gases

NASA Astrophysics Data System (ADS)

Cukrowski, A. S.; Popielawski, J.

1986-11-01

Expression for the corrections describing the effect of viscous flow and thermal flux on the rate of chemical reaction have been derived for the reaction A + A = B + C described by Prigogine-Xhrouet and Present. These corrections are calculated for the velocity distribution function up to the second-order approximation for the Chapman-Enskog solution of the Boltzmann equation. These corrections are shown to be the same as those which would follow after application of the method of linearized-moments equations described by Eu and Li. The effects of viscous flow and thermal flux are presented as functions of activation energy of chemical reaction, temperature, density, coefficients of shear viscosity of thermal conductivity, and relevant gradients of mean molecular velocity or temperature. It is pointed out that for very slow reactions and for very large gradients (e.g. in shock waves) these effects can be quite significant.

Counterion accumulation effects on a suspension of DNA molecules: Equation of state and pressure-driven denaturation

NASA Astrophysics Data System (ADS)

Nicasio-Collazo, Luz Adriana; Delgado-González, Alexandra; Hernández-Lemus, Enrique; Castañeda-Priego, Ramón

2017-04-01

The study of the effects associated with the electrostatic properties of DNA is of fundamental importance to understand both its molecular properties at the single molecule level, like the rigidity of the chain, and its interaction with other charged bio-molecules, including other DNA molecules; such interactions are crucial to maintain the thermodynamic stability of the intra-cellular medium. In the present work, we combine the Poisson-Boltzmann mean-field theory with an irreversible thermodynamic approximation to analyze the effects of counterion accumulation inside DNA on both the denaturation profile of the chain and the equation of state of the suspension. To this end, we model the DNA molecule as a porous charged cylinder immersed in an aqueous solution. These thermo-electrostatic effects are explicitly studied in the particular case of some genes for which damage in their sequence is associated with diffuse large B-cell lymphoma.

Theory of relativistic Brownian motion: the (1+3) -dimensional case.

PubMed

Dunkel, Jörn; Hänggi, Peter

2005-09-01

A theory for (1+3) -dimensional relativistic Brownian motion under the influence of external force fields is put forward. Starting out from a set of relativistically covariant, but multiplicative Langevin equations we describe the relativistic stochastic dynamics of a forced Brownian particle. The corresponding Fokker-Planck equations are studied in the laboratory frame coordinates. In particular, the stochastic integration prescription--i.e., the discretization rule dilemma--is elucidated (prepoint discretization rule versus midpoint discretization rule versus postpoint discretization rule). Remarkably, within our relativistic scheme we find that the postpoint rule (or the transport form) yields the only Fokker-Planck dynamics from which the relativistic Maxwell-Boltzmann statistics is recovered as the stationary solution. The relativistic velocity effects become distinctly more pronounced by going from one to three spatial dimensions. Moreover, we present numerical results for the asymptotic mean-square displacement of a free relativistic Brownian particle moving in 1+3 dimensions.

Adsorption of hard spheres via the non-uniform Percus-Yevick equation

NASA Astrophysics Data System (ADS)

Sokołowski, S.

We study the adsorption of hard spheres on solids interacting according to potentials whose Boltzmann functions contain a δ-function. The nonuniform Percus-Yevick equation is solved by using the method introduced by Lado to study two dimensional fluids.

Lattice Boltzmann formulation for conjugate heat transfer in heterogeneous media.

PubMed

Karani, Hamid; Huber, Christian

2015-02-01

In this paper, we propose an approach for studying conjugate heat transfer using the lattice Boltzmann method (LBM). The approach is based on reformulating the lattice Boltzmann equation for solving the conservative form of the energy equation. This leads to the appearance of a source term, which introduces the jump conditions at the interface between two phases or components with different thermal properties. The proposed source term formulation conserves conductive and advective heat flux simultaneously, which makes it suitable for modeling conjugate heat transfer in general multiphase or multicomponent systems. The simple implementation of the source term approach avoids any correction of distribution functions neighboring the interface and provides an algorithm that is independent from the topology of the interface. Moreover, our approach is independent of the choice of lattice discretization and can be easily applied to different advection-diffusion LBM solvers. The model is tested against several benchmark problems including steady-state convection-diffusion within two fluid layers with parallel and normal interfaces with respect to the flow direction, unsteady conduction in a three-layer stratified domain, and steady conduction in a two-layer annulus. The LBM results are in excellent agreement with analytical solution. Error analysis shows that our model is first-order accurate in space, but an extension to a second-order scheme is straightforward. We apply our LBM model to heat transfer in a two-component heterogeneous medium with a random microstructure. This example highlights that the method we propose is independent of the topology of interfaces between the different phases and, as such, is ideally suited for complex natural heterogeneous media. We further validate the present LBM formulation with a study of natural convection in a porous enclosure. The results confirm the reliability of the model in simulating complex coupled fluid and thermal dynamics in complex geometries.

Poisson-Boltzmann model for protein-surface electrostatic interactions and grid-convergence study using the PyGBe code

NASA Astrophysics Data System (ADS)

Cooper, Christopher D.; Barba, Lorena A.

2016-05-01

Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of proteins and charged nanosurfaces, via the Poisson-Boltzmann equation. We extended the implicit-solvent model used in the open-source code PyGBe to include surfaces of imposed charge or potential. This code solves the boundary integral formulation of the Poisson-Boltzmann equation, discretized with surface elements. PyGBe has at its core a treecode-accelerated Krylov iterative solver, resulting in O(N log N) scaling, with further acceleration on hardware via multi-threaded execution on GPUs. It computes solvation and surface free energies, providing a framework for studying the effect of electrostatics on adsorption. We derived an analytical solution for a spherical charged surface interacting with a spherical dielectric cavity, and used it in a grid-convergence study to build evidence on the correctness of our approach. The study showed the error decaying with the average area of the boundary elements, i.e., the method is O(1 / N) , which is consistent with our previous verification studies using PyGBe. We also studied grid-convergence using a real molecular geometry (protein G B1 D4‧), in this case using Richardson extrapolation (in the absence of an analytical solution) and confirmed the O(1 / N) scaling. With this work, we can now access a completely new family of problems, which no other major bioelectrostatics solver, e.g. APBS, is capable of dealing with. PyGBe is open-source under an MIT license and is hosted under version control at https://github.com/barbagroup/pygbe. To supplement this paper, we prepared ;reproducibility packages; consisting of running and post-processing scripts in Python for replicating the grid-convergence studies, all the way to generating the final plots, with a single command.

Simulating condensation on microstructured surfaces using Lattice Boltzmann Method

NASA Astrophysics Data System (ADS)

Alexeev, Alexander; Vasyliv, Yaroslav

2017-11-01

We simulate a single component fluid condensing on 2D structured surfaces with different wettability. To simulate the two phase fluid, we use the athermal Lattice Boltzmann Method (LBM) driven by a pseudopotential force. The pseudopotential force results in a non-ideal equation of state (EOS) which permits liquid-vapor phase change. To account for thermal effects, the athermal LBM is coupled to a finite volume discretization of the temperature evolution equation obtained using a thermal energy rate balance for the specific internal energy. We use the developed model to probe the effect of surface structure and surface wettability on the condensation rate in order to identify microstructure topographies promoting condensation. Financial support is acknowledged from Kimberly-Clark.

Modeling and simulation of ocean wave propagation using lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Nuraiman, Dian

2017-10-01

In this paper, we present on modeling and simulation of ocean wave propagation from the deep sea to the shoreline. This requires high computational cost for simulation with large domain. We propose to couple a 1D shallow water equations (SWE) model with a 2D incompressible Navier-Stokes equations (NSE) model in order to reduce the computational cost. The coupled model is solved using the lattice Boltzmann method (LBM) with the lattice Bhatnagar-Gross-Krook (BGK) scheme. Additionally, a special method is implemented to treat the complex behavior of free surface close to the shoreline. The result shows the coupled model can reduce computational cost significantly compared to the full NSE model.

Matrix-valued Boltzmann equation for the nonintegrable Hubbard chain.

PubMed

Fürst, Martin L R; Mendl, Christian B; Spohn, Herbert

2013-07-01

The standard Fermi-Hubbard chain becomes nonintegrable by adding to the nearest neighbor hopping additional longer range hopping amplitudes. We assume that the quartic interaction is weak and investigate numerically the dynamics of the chain on the level of the Boltzmann type kinetic equation. Only the spatially homogeneous case is considered. We observe that the huge degeneracy of stationary states in the case of nearest neighbor hopping is lost and the convergence to the thermal Fermi-Dirac distribution is restored. The convergence to equilibrium is exponentially fast. However for small next-nearest neighbor hopping amplitudes one has a rapid relaxation towards the manifold of quasistationary states and slow relaxation to the final equilibrium state.

Nonlinear response from transport theory and quantum field theory at finite temperature

NASA Astrophysics Data System (ADS)

Carrington, M. E.; Defu, Hou; Kobes, R.

2001-07-01

We study the nonlinear response in weakly coupled hot φ4 theory. We obtain an expression for a quadratic shear viscous response coefficient using two different formalisms: transport theory and response theory. The transport theory calculation is done by assuming a local equilibrium form for the distribution function and expanding in the gradient of the local four dimensional velocity field. By performing a Chapman-Enskog expansion on the Boltzmann equation we obtain a hierarchy of equations for the coefficients of the expanded distribution function. To do the response theory calculation we use Zubarev's techniques in nonequilibrium statistical mechanics to derive a generalized Kubo formula. Using this formula allows us to obtain the quadratic shear viscous response from the three-point retarded Green function of the viscous shear stress tensor. We use the closed time path formalism of real time finite temperature field theory to show that this three-point function can be calculated by writing it as an integral equation involving a four-point vertex. This four-point vertex can in turn be obtained from an integral equation which represents the resummation of an infinite series of ladder and extended-ladder diagrams. The connection between transport theory and response theory is made when we show that the integral equation for this four-point vertex has exactly the same form as the equation obtained from the Boltzmann equation for the coefficient of the quadratic term of the gradient expansion of the distribution function. We conclude that calculating the quadratic shear viscous response using transport theory and keeping terms that are quadratic in the gradient of the velocity field in the Chapman-Enskog expansion of the Boltzmann equation is equivalent to calculating the quadratic shear viscous response from response theory using the next-to-linear response Kubo formula, with a vertex given by an infinite resummation of ladder and extended-ladder diagrams.

Coupled Boltzmann computation of mixed axion neutralino dark matter in the SUSY DFSZ axion model

NASA Astrophysics Data System (ADS)

Bae, Kyu Jung; Baer, Howard; Lessa, Andre; Serce, Hasan

2014-10-01

The supersymmetrized DFSZ axion model is highly motivated not only because it offers solutions to both the gauge hierarchy and strong CP problems, but also because it provides a solution to the SUSY μ-problem which naturally allows for a Little Hierarchy. We compute the expected mixed axion-neutralino dark matter abundance for the SUSY DFSZ axion model in two benchmark cases—a natural SUSY model with a standard neutralino underabundance (SUA) and an mSUGRA/CMSSM model with a standard overabundance (SOA). Our computation implements coupled Boltzmann equations which track the radiation density along with neutralino, axion, axion CO (produced via coherent oscillations), saxion, saxion CO, axino and gravitino densities. In the SUSY DFSZ model, axions, axinos and saxions go through the process of freeze-in—in contrast to freeze-out or out-of-equilibrium production as in the SUSY KSVZ model—resulting in thermal yields which are largely independent of the re-heat temperature. We find the SUA case with suppressed saxion-axion couplings (ξ=0) only admits solutions for PQ breaking scale falesssim 6× 1012 GeV where the bulk of parameter space tends to be axion-dominated. For SUA with allowed saxion-axion couplings (ξ =1), then fa values up to ~ 1014 GeV are allowed. For the SOA case, almost all of SUSY DFSZ parameter space is disallowed by a combination of overproduction of dark matter, overproduction of dark radiation or violation of BBN constraints. An exception occurs at very large fa~ 1015-1016 GeV where large entropy dilution from CO-produced saxions leads to allowed models.

Radiation effects on bifurcation and dual solutions in transient natural convection in a horizontal annulus

DOE Office of Scientific and Technical Information (OSTI.GOV)

Luo, Kang; Yi, Hong-Liang, E-mail: yihongliang@hit.edu.cn; Tan, He-Ping, E-mail: tanheping@hit.edu.cn

2014-05-15

Transitions and bifurcations of transient natural convection in a horizontal annulus with radiatively participating medium are numerically investigated using the coupled lattice Boltzmann and direct collocation meshless (LB-DCM) method. As a hybrid approach based on a common multi-scale Boltzmann-type model, the LB-DCM scheme is easy to implement and has an excellent flexibility in dealing with the irregular geometries. Separate particle distribution functions in the LBM are used to calculate the density field, the velocity field and the thermal field. In the radiatively participating medium, the contribution of thermal radiation to natural convection must be taken into account, and it ismore » considered as a radiative term in the energy equation that is solved by the meshless method with moving least-squares (MLS) approximation. The occurrence of various instabilities and bifurcative phenomena is analyzed for different Rayleigh number Ra and Prandtl number Pr with and without radiation. Then, bifurcation diagrams and dual solutions are presented for relevant radiative parameters, such as convection-radiation parameter Rc and optical thickness τ. Numerical results show that the presence of volumetric radiation changes the static temperature gradient of the fluid, and generally results in an increase in the flow critical value. Besides, the existence and development of dual solutions of transient convection in the presence of radiation are greatly affected by radiative parameters. Finally, the advantage of LB-DCM combination is discussed, and the potential benefits of applying the LB-DCM method to multi-field coupling problems are demonstrated.« less

Acoustic equations of state for simple lattice Boltzmann velocity sets.

PubMed

Viggen, Erlend Magnus

2014-07-01

The lattice Boltzmann (LB) method typically uses an isothermal equation of state. This is not sufficient to simulate a number of acoustic phenomena where the equation of state cannot be approximated as linear and constant. However, it is possible to implement variable equations of state by altering the LB equilibrium distribution. For simple velocity sets with velocity components ξ(iα)∈(-1,0,1) for all i, these equilibria necessarily cause error terms in the momentum equation. These error terms are shown to be either correctable or negligible at the cost of further weakening the compressibility. For the D1Q3 velocity set, such an equilibrium distribution is found and shown to be unique. Its sound propagation properties are found for both forced and free waves, with some generality beyond D1Q3. Finally, this equilibrium distribution is applied to a nonlinear acoustics simulation where both mechanisms of nonlinearity are simulated with good results. This represents an improvement on previous such simulations and proves that the compressibility of the method is still sufficiently strong even for nonlinear acoustics.

Monte Carlo closure for moment-based transport schemes in general relativistic radiation hydrodynamic simulations

NASA Astrophysics Data System (ADS)

Foucart, Francois

2018-04-01

General relativistic radiation hydrodynamic simulations are necessary to accurately model a number of astrophysical systems involving black holes and neutron stars. Photon transport plays a crucial role in radiatively dominated accretion discs, while neutrino transport is critical to core-collapse supernovae and to the modelling of electromagnetic transients and nucleosynthesis in neutron star mergers. However, evolving the full Boltzmann equations of radiative transport is extremely expensive. Here, we describe the implementation in the general relativistic SPEC code of a cheaper radiation hydrodynamic method that theoretically converges to a solution of Boltzmann's equation in the limit of infinite numerical resources. The algorithm is based on a grey two-moment scheme, in which we evolve the energy density and momentum density of the radiation. Two-moment schemes require a closure that fills in missing information about the energy spectrum and higher order moments of the radiation. Instead of the approximate analytical closure currently used in core-collapse and merger simulations, we complement the two-moment scheme with a low-accuracy Monte Carlo evolution. The Monte Carlo results can provide any or all of the missing information in the evolution of the moments, as desired by the user. As a first test of our methods, we study a set of idealized problems demonstrating that our algorithm performs significantly better than existing analytical closures. We also discuss the current limitations of our method, in particular open questions regarding the stability of the fully coupled scheme.

Role of hydrodynamic viscosity on phonon transport in suspended graphene

NASA Astrophysics Data System (ADS)

Li, Xun; Lee, Sangyeop

2018-03-01

When phonon transport is in the hydrodynamic regime, the thermal conductivity exhibits peculiar dependences on temperatures (T ) and sample widths (W ). These features were used in the past to experimentally confirm the hydrodynamic phonon transport in three-dimensional bulk materials. Suspended graphene was recently predicted to exhibit strong hydrodynamic features in thermal transport at much higher temperature than the three-dimensional bulk materials, but its experimental confirmation requires quantitative guidance by theory and simulation. Here we quantitatively predict those peculiar dependences using the Monte Carlo solution of the Peierls-Boltzmann equation with an ab initio full three-phonon scattering matrix. Thermal conductivity is found to increase as Tα where α ranges from 1.89 to 2.49 depending on a sample width at low temperatures, much larger than 1.68 of the ballistic case. The thermal conductivity has a width dependence of W1.17 at 100 K, clearly distinguished from the sublinear dependence of the ballistic-diffusive regime. These peculiar features are explained with a phonon viscous damping effect of the hydrodynamic regime. We derive an expression for the phonon hydrodynamic viscosity from the Peierls-Boltzmann equation, and discuss the fact that the phonon viscous damping explains well those peculiar dependences of thermal conductivity at 100 K. The phonon viscous damping still causes significant thermal resistance when a temperature is 300 K and a sample width is around 1 µm, even though the hydrodynamic regime is not dominant over other regimes at this condition.

Large-Eddy/Lattice Boltzmann Simulations of Micro-blowing Strategies for Subsonic and Supersonic Drag Control

NASA Technical Reports Server (NTRS)

Menon, Suresh

2003-01-01

This report summarizes the progress made in the first 8 to 9 months of this research. The Lattice Boltzmann Equation (LBE) methodology for Large-eddy Simulations (LES) of microblowing has been validated using a jet-in-crossflow test configuration. In this study, the flow intake is also simulated to allow the interaction to occur naturally. The Lattice Boltzmann Equation Large-eddy Simulations (LBELES) approach is capable of capturing not only the flow features associated with the flow, such as hairpin vortices and recirculation behind the jet, but also is able to show better agreement with experiments when compared to previous RANS predictions. The LBELES is shown to be computationally very efficient and therefore, a viable method for simulating the injection process. Two strategies have been developed to simulate multi-hole injection process as in the experiment. In order to allow natural interaction between the injected fluid and the primary stream, the flow intakes for all the holes have to be simulated. The LBE method is computationally efficient but is still 3D in nature and therefore, there may be some computational penalty. In order to study a large number or holes, a new 1D subgrid model has been developed that will simulate a reduced form of the Navier-Stokes equation in these holes.

Electrokinetic flow in a capillary with a charge-regulating surface polymer layer.

PubMed

Keh, Huan J; Ding, Jau M

2003-07-15

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes/Brinkman equation. Explicit formulas for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of the potential-determining ions in the bulk solution.

Mass-conserved volumetric lattice Boltzmann method for complex flows with willfully moving boundaries.

PubMed

Yu, Huidan; Chen, Xi; Wang, Zhiqiang; Deep, Debanjan; Lima, Everton; Zhao, Ye; Teague, Shawn D

2014-06-01

In this paper, we develop a mass-conserved volumetric lattice Boltzmann method (MCVLBM) for numerically solving fluid dynamics with willfully moving arbitrary boundaries. In MCVLBM, fluid particles are uniformly distributed in lattice cells and the lattice Boltzmann equations deal with the time evolution of the particle distribution function. By introducing a volumetric parameter P(x,y,z,t) defined as the occupation of solid volume in the cell, we distinguish three types of lattice cells in the simulation domain: solid cell (pure solid occupation, P=1), fluid cell (pure fluid occupation, P=0), and boundary cell (partial solid and partial fluid, 0

Poisson–Boltzmann–Nernst–Planck model

PubMed Central

Zheng, Qiong; Wei, Guo-Wei

2011-01-01

The Poisson–Nernst–Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst–Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst–Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst–Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson–Boltzmann and Nernst–Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external voltages. Extensive numerical experiments show that there is an excellent consistency between the results predicted from the present PBNP model and those obtained from the PNP model in terms of the electrostatic potentials, ion concentration profiles, and current–voltage (I–V) curves. The present PBNP model is further validated by a comparison with experimental measurements of I–V curves under various ion bulk concentrations. Numerical experiments indicate that the proposed PBNP model is more efficient than the original PNP model in terms of simulation time. PMID:21599038

Electrostatic potential of B-DNA: effect of interionic correlations.

PubMed Central

Gavryushov, S; Zielenkiewicz, P

1998-01-01

Modified Poisson-Boltzmann (MPB) equations have been numerically solved to study ionic distributions and mean electrostatic potentials around a macromolecule of arbitrarily complex shape and charge distribution. Results for DNA are compared with those obtained by classical Poisson-Boltzmann (PB) calculations. The comparisons were made for 1:1 and 2:1 electrolytes at ionic strengths up to 1 M. It is found that ion-image charge interactions and interionic correlations, which are neglected by the PB equation, have relatively weak effects on the electrostatic potential at charged groups of the DNA. The PB equation predicts errors in the long-range electrostatic part of the free energy that are only approximately 1.5 kJ/mol per nucleotide even in the case of an asymmetrical electrolyte. In contrast, the spatial correlations between ions drastically affect the electrostatic potential at significant separations from the macromolecule leading to a clearly predicted effect of charge overneutralization. PMID:9826596

On Stable Wall Boundary Conditions for the Hermite Discretization of the Linearised Boltzmann Equation

NASA Astrophysics Data System (ADS)

Sarna, Neeraj; Torrilhon, Manuel

2018-01-01

We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331-407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.

Viscous dissipation effects on MHD slip flow and heat transfer in porous micro duct with LTNE assumptions using modified lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Rabhi, R.; Amami, B.; Dhahri, H.; Mhimid, A.

2017-11-01

This paper deals with heat transfer and fluid flow in a porous micro duct under local thermal non equilibrium conditions subjected to an external oriented magnetic field. The considered sample is a micro duct filled with porous media assumed to be homogenous, isotropic and saturated. The slip velocity and the temperature jump were uniformly imposed to the wall. In modeling the flow, the Brinkmann-Forchheimer extended Darcy model was incorporated into the momentum equations. In the energy equation, the local thermal non equilibrium between the two phases was adopted. A modified axisymmetric lattice Boltzmann method was used to solve the obtained governing equation system. Attention was focused on the influence of the emerging parameters such as Knudsen number, Kn, Hartmann number, Ha, Eckert number, Ec, Biot number, Bi and the magnetic field inclination γ on flow and heat transfer throughout this paper.

Role of electrostatic interactions during protein ultrafiltration.

PubMed

Rohani, Mahsa M; Zydney, Andrew L

2010-10-15

A number of studies over the last decade have clearly demonstrated the importance of electrostatic interactions on the transport of charged proteins through semipermeable ultrafiltration membranes. This paper provides a review of recent developments in this field with a focus on the role of both protein and membrane charge on the rate of protein transport. Experimental results are analyzed using available theoretical models developed from the solution of the Poisson-Boltzmann equation for the partitioning of a charged particle into a charged pore. The potential of exploiting these electrostatic interactions for selective protein separations and for the development of ultrafiltration membranes with enhanced performance characteristics is also examined. Copyright © 2010 Elsevier B.V. All rights reserved.

An implicit boundary integral method for computing electric potential of macromolecules in solvent

NASA Astrophysics Data System (ADS)

Zhong, Yimin; Ren, Kui; Tsai, Richard

2018-04-01

A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equation that arises in mathematical models for the electrostatics of molecules in solvent. The proposed method uses an implicit boundary integral formulation to derive a linear system defined on Cartesian nodes in a narrowband surrounding the closed surface that separates the molecule and the solvent. The needed implicit surface is constructed from the given atomic description of the molecules, by a sequence of standard level set algorithms. A fast multipole method is applied to accelerate the solution of the linear system. A few numerical studies involving some standard test cases are presented and compared to other existing results.

Extrinsic spin Nernst effect from first principles.

PubMed

Tauber, Katarina; Gradhand, Martin; Fedorov, Dmitry V; Mertig, Ingrid

2012-07-13

We present an ab initio description of the thermal transport phenomenon called the spin Nernst effect. It refers to generation of a spin accumulation or a pure spin current transverse to an applied temperature gradient. This is similar to the intensively studied spin Hall effect described by intrinsic and extrinsic mechanisms due to an applied electric field. Analogously, several contributions are present for the spin Nernst effect. Here we investigate the extrinsic skew scattering mechanism which is dominant in the limit of dilute alloys. Our calculations are based on a fully relativistic Korringa-Kohn-Rostoker method and a solution of the linearized Boltzmann equation. As a first application, we consider a Cu host with Au, Ti, and Bi impurities.

Chapman-Enskog Analyses on the Gray Lattice Boltzmann Equation Method for Fluid Flow in Porous Media

NASA Astrophysics Data System (ADS)

Chen, Chen; Li, Like; Mei, Renwei; Klausner, James F.

2018-03-01

The gray lattice Boltzmann equation (GLBE) method has recently been used to simulate fluid flow in porous media. It employs a partial bounce-back of populations (through a fractional coefficient θ, which represents the fraction of populations being reflected by the solid phase) in the evolution equation to account for the linear drag of the medium. Several particular GLBE schemes have been proposed in the literature and these schemes are very easy to implement; but there exists uncertainty about the need for redefining the macroscopic velocity as there has been no systematic analysis to recover the Brinkman equation from the various GLBE schemes. Rigorous Chapman-Enskog analyses are carried out to show that the momentum equation recovered from these schemes can satisfy Brinkman equation to second order in ɛ only if θ = O( ɛ ) in which ɛ is the ratio of the lattice spacing to the characteristic length of physical dimension. The need for redefining macroscopic velocity is shown to be scheme-dependent. When a body force is encountered such as the gravitational force or that caused by a pressure gradient, different forms of forcing redefinitions are required for each GLBE scheme.

Plane Poiseuille flow of a rarefied gas in the presence of strong gravitation.

PubMed

Doi, Toshiyuki

2011-02-01

Plane Poiseuille flow of a rarefied gas, which flows horizontally in the presence of strong gravitation, is studied based on the Boltzmann equation. Applying the asymptotic analysis for a small variation in the flow direction [Y. Sone, Molecular Gas Dynamics (Birkhäuser, 2007)], the two-dimensional problem is reduced to a one-dimensional problem, as in the case of a Poiseuille flow in the absence of gravitation, and the solution is obtained in a semianalytical form. The reduced one-dimensional problem is solved numerically for a hard sphere molecular gas over a wide range of the gas-rarefaction degree and the gravitational strength. The presence of gravitation reduces the mass flow rate, and the effect of gravitation is significant for large Knudsen numbers. To verify the validity of the asymptotic solution, a two-dimensional problem of a flow through a long channel is directly solved numerically, and the validity of the asymptotic solution is confirmed. ©2011 American Physical Society

Temperature-Dependent Implicit-Solvent Model of Polyethylene Glycol in Aqueous Solution.

PubMed

Chudoba, Richard; Heyda, Jan; Dzubiella, Joachim

2017-12-12

A temperature (T)-dependent coarse-grained (CG) Hamiltonian of polyethylene glycol/oxide (PEG/PEO) in aqueous solution is reported to be used in implicit-solvent material models in a wide temperature (i.e., solvent quality) range. The T-dependent nonbonded CG interactions are derived from a combined "bottom-up" and "top-down" approach. The pair potentials calculated from atomistic replica-exchange molecular dynamics simulations in combination with the iterative Boltzmann inversion are postrefined by benchmarking to experimental data of the radius of gyration. For better handling and a fully continuous transferability in T-space, the pair potentials are conveniently truncated and mapped to an analytic formula with three structural parameters expressed as explicit continuous functions of T. It is then demonstrated that this model without further adjustments successfully reproduces other experimentally known key thermodynamic properties of semidilute PEG solutions such as the full equation of state (i.e., T-dependent osmotic pressure) for various chain lengths as well as their cloud point (or collapse) temperature.

Mathematical and Numerical Aspects of the Adaptive Fast Multipole Poisson-Boltzmann Solver

DOE PAGES

Zhang, Bo; Lu, Benzhuo; Cheng, Xiaolin; ...

2013-01-01

This paper summarizes the mathematical and numerical theories and computational elements of the adaptive fast multipole Poisson-Boltzmann (AFMPB) solver. We introduce and discuss the following components in order: the Poisson-Boltzmann model, boundary integral equation reformulation, surface mesh generation, the nodepatch discretization approach, Krylov iterative methods, the new version of fast multipole methods (FMMs), and a dynamic prioritization technique for scheduling parallel operations. For each component, we also remark on feasible approaches for further improvements in efficiency, accuracy and applicability of the AFMPB solver to large-scale long-time molecular dynamics simulations. Lastly, the potential of the solver is demonstrated with preliminary numericalmore » results.« less

Gravitational instability of slowly rotating isothermal spheres

NASA Astrophysics Data System (ADS)

Chavanis, P. H.

2002-12-01

We discuss the statistical mechanics of rotating self-gravitating systems by allowing properly for the conservation of angular momentum. We study analytically the case of slowly rotating isothermal spheres by expanding the solutions of the Boltzmann-Poisson equation in a series of Legendre polynomials, adapting the procedure introduced by Chandrasekhar (1933) for distorted polytropes. We show how the classical spiral of Lynden-Bell & Wood (1967) in the temperature-energy plane is deformed by rotation. We find that gravitational instability occurs sooner in the microcanonical ensemble and later in the canonical ensemble. According to standard turning point arguments, the onset of the collapse coincides with the minimum energy or minimum temperature state in the series of equilibria. Interestingly, it happens to be close to the point of maximum flattening. We generalize the singular isothermal solution to the case of a slowly rotating configuration. We also consider slowly rotating configurations of the self-gravitating Fermi gas at non-zero temperature.

On Blackbody Radiation.

ERIC Educational Resources Information Center

Jain, Pushpendra K.

1991-01-01

The interrelationship between the various forms of the Planck radiation equation is discussed. A differential equation that gives intensity or energy density of radiation per unit wavelength or per unit frequency is emphasized. The Stefan-Boltzmann Law and the change in the glow of a hot body with temperature are also discussed. (KR)

Perturbative Out of Equilibrium Quantum Field Theory beyond the Gradient Approximation and Generalized Boltzmann Equation

NASA Astrophysics Data System (ADS)

Ozaki, H.

2004-01-01

Using the closed-time-path formalism, we construct perturbative frameworks, in terms of quasiparticle picture, for studying quasiuniform relativistic quantum field systems near equilibrium and non-equilibrium quasistationary systems. We employ the derivative expansion and take in up to the second-order term, i.e., one-order higher than the gradient approximation. After constructing self-energy resummed propagator, we formulated two kinds of mutually equivalent perturbative frameworks: The first one is formulated on the basis of the ``bare'' number density function, and the second one is formulated on the basis of ``physical'' number density function. In the course of construction of the second framework, the generalized Boltzmann equations directly come out, which describe the evolution of the system.

The electron Boltzmann equation in a plasma generated by fission fragments

NASA Technical Reports Server (NTRS)

Hassan, H. A.; Deese, J. E.

1976-01-01

A Boltzmann equation formulation is presented for the determination of the electron distribution function in a plasma generated by fission fragments. The formulation takes into consideration ambipolar diffusion, elastic and inelastic collisions, recombination and ionization, and allows for the fact that the primary electrons are not monoenergetic. Calculations for He in a tube coated with fissionable material show that, over a wide pressure and neutron flux range, the distribution function is non-Maxwellian, but the electrons are essentially thermal. Moreover, about a third of the energy of the primary electrons is transferred into the inelastic levels of He. This fraction of energy transfer is almost independent of pressure and neutron flux but increases sharply in the presence of a sustainer electric field.

Statistical mechanics in the context of special relativity. II.

PubMed

Kaniadakis, G

2005-09-01

The special relativity laws emerge as one-parameter (light speed) generalizations of the corresponding laws of classical physics. These generalizations, imposed by the Lorentz transformations, affect both the definition of the various physical observables (e.g., momentum, energy, etc.), as well as the mathematical apparatus of the theory. Here, following the general lines of [Phys. Rev. E 66, 056125 (2002)], we show that the Lorentz transformations impose also a proper one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits us to construct a coherent and self-consistent relativistic statistical theory, preserving the main features of the ordinary statistical theory, which is recovered in the classical limit. The predicted distribution function is a one-parameter continuous deformation of the classical Maxwell-Boltzmann distribution and has a simple analytic form, showing power law tails in accordance with the experimental evidence. Furthermore, this statistical mechanics can be obtained as the stationary case of a generalized kinetic theory governed by an evolution equation obeying the H theorem and reproducing the Boltzmann equation of the ordinary kinetics in the classical limit.

Discrete Boltzmann modeling of Rayleigh-Taylor instability in two-component compressible flows.

PubMed

Lin, Chuandong; Xu, Aiguo; Zhang, Guangcai; Luo, Kai Hong; Li, Yingjun

2017-11-01

A discrete Boltzmann model (DBM) is proposed to probe the Rayleigh-Taylor instability (RTI) in two-component compressible flows. Each species has a flexible specific-heat ratio and is described by one discrete Boltzmann equation (DBE). Independent discrete velocities are adopted for the two DBEs. The collision and force terms in the DBE account for the molecular collision and external force, respectively. Two types of force terms are exploited. In addition to recovering the modified Navier-Stokes equations in the hydrodynamic limit, the DBM has the capability of capturing detailed nonequilibrium effects. Furthermore, we use the DBM to investigate the dynamic process of the RTI. The invariants of tensors for nonequilibrium effects are presented and studied. For low Reynolds numbers, both global nonequilibrium manifestations and the growth rate of the entropy of mixing show three stages (i.e., the reducing, increasing, and then decreasing trends) in the evolution of the RTI. On the other hand, the early reducing tendency is suppressed and even eliminated for high Reynolds numbers. Relevant physical mechanisms are analyzed and discussed.

Implications of Analytical Investigations about the Semiconductor Equations on Device Modeling Programs.

DTIC Science & Technology

1983-04-01

34.. .. . ...- "- -,-. SIGNIFICANCE AND EXPLANATION Many different codes for the simulation of semiconductor devices such as transitors , diodes, thyristors are already circulated...partially take into account the consequences introduced by degenerate semiconductors (e.g. invalidity of Boltzmann’s statistics , bandgap narrowing). These...ft - ni p nep /Ut(2.10) Sni *e p nie 2.11) .7. (2.10) can be physically interpreted as the application of Boltzmann statistics . However (2.10) a.,zo

Multi-Subband Ensemble Monte Carlo simulations of scaled GAA MOSFETs

NASA Astrophysics Data System (ADS)

Donetti, L.; Sampedro, C.; Ruiz, F. G.; Godoy, A.; Gamiz, F.

2018-05-01

We developed a Multi-Subband Ensemble Monte Carlo simulator for non-planar devices, taking into account two-dimensional quantum confinement. It couples self-consistently the solution of the 3D Poisson equation, the 2D Schrödinger equation, and the 1D Boltzmann transport equation with the Ensemble Monte Carlo method. This simulator was employed to study MOS devices based on ultra-scaled Gate-All-Around Si nanowires with diameters in the range from 4 nm to 8 nm with gate length from 8 nm to 14 nm. We studied the output and transfer characteristics, interpreting the behavior in the sub-threshold region and in the ON state in terms of the spatial charge distribution and the mobility computed with the same simulator. We analyzed the results, highlighting the contribution of different valleys and subbands and the effect of the gate bias on the energy and velocity profiles. Finally the scaling behavior was studied, showing that only the devices with D = 4nm maintain a good control of the short channel effects down to the gate length of 8nm .

Combined electroosmotically and pressure driven flow in soft nanofluidics.

PubMed

Matin, Meisam Habibi; Ohshima, Hiroyuki

2015-12-15

The present study is devoted to the analysis of mixed electroosmotic and pressure driven flows through a soft charged nanochannel considering boundary slip and constant charge density on the walls of the slit channel. The sources of the fluid flow are the pressure gradient along the channel axis and the electrokinetic effects that trigger an electroosmotic flow under the influence of a uniformly applied electric field. The polyelectrolyte layer (PEL) is denoted as a fixed charge layer (FCL) and the electrolyte ions can be present both inside and outside the PEL i.e., the PEL-electrolyte interface acts as a semi-penetrable membrane. The Poisson-Boltzmann equation is solved assuming the Debye-Hückel linearization for the low electric potential to provide us with analytical closed form solutions for the conservation equations. The conservation equations are solved to obtain the electric potential and velocity distributions in terms of governing dimensionless parameters. The results for the dimensionless electric potential, the dimensionless velocity and Poiseuille number are presented graphically and discussed in detail. Copyright © 2015 Elsevier Inc. All rights reserved.

Second-order (2 +1 ) -dimensional anisotropic hydrodynamics

NASA Astrophysics Data System (ADS)

Bazow, Dennis; Heinz, Ulrich; Strickland, Michael

2014-11-01

We present a complete formulation of second-order (2 +1 ) -dimensional anisotropic hydrodynamics. The resulting framework generalizes leading-order anisotropic hydrodynamics by allowing for deviations of the one-particle distribution function from the spheroidal form assumed at leading order. We derive complete second-order equations of motion for the additional terms in the macroscopic currents generated by these deviations from their kinetic definition using a Grad-Israel-Stewart 14-moment ansatz. The result is a set of coupled partial differential equations for the momentum-space anisotropy parameter, effective temperature, the transverse components of the fluid four-velocity, and the viscous tensor components generated by deviations of the distribution from spheroidal form. We then perform a quantitative test of our approach by applying it to the case of one-dimensional boost-invariant expansion in the relaxation time approximation (RTA) in which case it is possible to numerically solve the Boltzmann equation exactly. We demonstrate that the second-order anisotropic hydrodynamics approach provides an excellent approximation to the exact (0+1)-dimensional RTA solution for both small and large values of the shear viscosity.

MPACT Theory Manual, Version 2.2.0

DOE Office of Scientific and Technical Information (OSTI.GOV)

Downar, Thomas; Collins, Benjamin S.; Gehin, Jess C.

2016-06-09

This theory manual describes the three-dimensional (3-D) whole-core, pin-resolved transport calculation methodology employed in the MPACT code. To provide sub-pin level power distributions with sufficient accuracy, MPACT employs the method of characteristics (MOC) solutions in the framework of a 3-D coarse mesh finite difference (CMFD) formulation. MPACT provides a 3D MOC solution, but also a 2D/1D solution in which the 2D planar solution is provided by MOC and the axial coupling is resolved by one-dimensional (1-D) lower order (diffusion or P3) solutions. In Chapter 2 of the manual, the MOC methodology is described for calculating the regional angular and scalarmore » fluxes from the Boltzmann transport equation. In Chapter 3, the 2D/1D methodology is described, together with the description of the CMFD iteration process involving dynamic homogenization and solution of the multigroup CMFD linear system. A description of the MPACT depletion algorithm is given in Chapter 4, followed by a discussion of the subgroup and ESSM resonance processing methods in Chapter 5. The final Chapter 6 describes a simplified thermal hydraulics model in MPACT.« less

An Immersed Boundary-Lattice Boltzmann Method for Simulating Particulate Flows

NASA Astrophysics Data System (ADS)

Zhang, Baili; Cheng, Ming; Lou, Jing

2013-11-01

A two-dimensional momentum exchange-based immersed boundary-lattice Boltzmann method developed by X.D. Niu et al. (2006) has been extended in three-dimensions for solving fluid-particles interaction problems. This method combines the most desirable features of the lattice Boltzmann method and the immersed boundary method by using a regular Eulerian mesh for the flow domain and a Lagrangian mesh for the moving particles in the flow field. The non-slip boundary conditions for the fluid and the particles are enforced by adding a force density term into the lattice Boltzmann equation, and the forcing term is simply calculated by the momentum exchange of the boundary particle density distribution functions, which are interpolated by the Lagrangian polynomials from the underlying Eulerian mesh. This method preserves the advantages of lattice Boltzmann method in tracking a group of particles and, at the same time, provides an alternative approach to treat solid-fluid boundary conditions. Numerical validations show that the present method is very accurate and efficient. The present method will be further developed to simulate more complex problems with particle deformation, particle-bubble and particle-droplet interactions.

Some New Results in Astrophysical Problems of Nonlinear Theory of Radiative Transfer

NASA Astrophysics Data System (ADS)

Pikichyan, H. V.

2017-07-01

In the interpretation of the observed astrophysical spectra, a decisive role is related to nonlinear problems of radiative transfer, because the processes of multiple interactions of matter of cosmic medium with the exciting intense radiation ubiquitously occur in astrophysical objects, and in their vicinities. Whereas, the intensity of the exciting radiation changes the physical properties of the original medium, and itself was modified, simultaneously, in a self-consistent manner under its influence. In the present report, we show that the consistent application of the principle of invariance in the nonlinear problem of bilateral external illumination of a scattering/absorbing one-dimensional anisotropic medium of finite geometrical thickness allows for simplifications that were previously considered as a prerogative only of linear problems. The nonlinear problem is analyzed through the three methods of the principle of invariance: (i) an adding of layers, (ii) its limiting form, described by differential equations of invariant imbedding, and (iii) a transition to the, so-called, functional equations of the "Ambartsumyan's complete invariance". Thereby, as an alternative to the Boltzmann equation, a new type of equations, so-called "kinetic equations of equivalence", are obtained. By the introduction of new functions - the so-called "linear images" of solution of nonlinear problem of radiative transfer, the linear structure of the solution of the nonlinear problem under study is further revealed. Linear images allow to convert naturally the statistical characteristics of random walk of a "single quantum" or their "beam of unit intensity", as well as widely known "probabilistic interpretation of phenomena of transfer", to the field of nonlinear problems. The structure of the equations obtained for determination of linear images is typical of linear problems.

Slip and barodiffusion phenomena in slow flows of a gas mixture

NASA Astrophysics Data System (ADS)

Zhdanov, V. M.

2017-03-01

The slip and barodiffusion problems for the slow flows of a gas mixture are investigated on the basis of the linearized moment equations following from the Boltzmann equation. We restrict ourselves to the set of the third-order moment equations and state two general relations (resembling conservation equations) for the moments of the distribution function similar to the conditions used by Loyalka [S. K. Loyalka, Phys. Fluids 14, 2291 (1971), 10.1063/1.1693331] in his approximation method (the modified Maxwell method). The expressions for the macroscopic velocities of the gas mixture species, the partial viscous stress tensors, and the reduced heat fluxes for the stationary slow flow of a gas mixture in the semi-infinite space over a plane wall are obtained as a result of the exact solution of the linearized moment equations in the 10- and 13-moment approximations. The general expression for the slip velocity and the simple and accurate expressions for the viscous, thermal, diffusion slip, and baroslip coefficients, which are given in terms of the basic transport coefficients, are derived by using the modified Maxwell method. The solutions of moment equations are also used for investigation of the flow and diffusion of a gas mixture in a channel formed by two infinite parallel plates. A fundamental result is that the barodiffusion factor in the cross-section-averaged expression for the diffusion flux contains contributions associated with the viscous transfer of momentum in the gas mixture and the effect of the Knudsen layer. Our study revealed that the barodiffusion factor is equal to the diffusion slip coefficient (correct to the opposite sign). This result is consistent with the Onsager's reciprocity relations for kinetic coefficients following from nonequilibrium thermodynamics of the discontinuous systems.

Multiphase Simulated Annealing Based on Boltzmann and Bose-Einstein Distribution Applied to Protein Folding Problem.

PubMed

Frausto-Solis, Juan; Liñán-García, Ernesto; Sánchez-Hernández, Juan Paulo; González-Barbosa, J Javier; González-Flores, Carlos; Castilla-Valdez, Guadalupe

2016-01-01

A new hybrid Multiphase Simulated Annealing Algorithm using Boltzmann and Bose-Einstein distributions (MPSABBE) is proposed. MPSABBE was designed for solving the Protein Folding Problem (PFP) instances. This new approach has four phases: (i) Multiquenching Phase (MQP), (ii) Boltzmann Annealing Phase (BAP), (iii) Bose-Einstein Annealing Phase (BEAP), and (iv) Dynamical Equilibrium Phase (DEP). BAP and BEAP are simulated annealing searching procedures based on Boltzmann and Bose-Einstein distributions, respectively. DEP is also a simulated annealing search procedure, which is applied at the final temperature of the fourth phase, which can be seen as a second Bose-Einstein phase. MQP is a search process that ranges from extremely high to high temperatures, applying a very fast cooling process, and is not very restrictive to accept new solutions. However, BAP and BEAP range from high to low and from low to very low temperatures, respectively. They are more restrictive for accepting new solutions. DEP uses a particular heuristic to detect the stochastic equilibrium by applying a least squares method during its execution. MPSABBE parameters are tuned with an analytical method, which considers the maximal and minimal deterioration of problem instances. MPSABBE was tested with several instances of PFP, showing that the use of both distributions is better than using only the Boltzmann distribution on the classical SA.

Modelling viscoacoustic wave propagation with the lattice Boltzmann method.

PubMed

Xia, Muming; Wang, Shucheng; Zhou, Hui; Shan, Xiaowen; Chen, Hanming; Li, Qingqing; Zhang, Qingchen

2017-08-31

In this paper, the lattice Boltzmann method (LBM) is employed to simulate wave propagation in viscous media. LBM is a kind of microscopic method for modelling waves through tracking the evolution states of a large number of discrete particles. By choosing different relaxation times in LBM experiments and using spectrum ratio method, we can reveal the relationship between the quality factor Q and the parameter τ in LBM. A two-dimensional (2D) homogeneous model and a two-layered model are tested in the numerical experiments, and the LBM results are compared against the reference solution of the viscoacoustic equations based on the Kelvin-Voigt model calculated by finite difference method (FDM). The wavefields and amplitude spectra obtained by LBM coincide with those by FDM, which demonstrates the capability of the LBM with one relaxation time. The new scheme is relatively simple and efficient to implement compared with the traditional lattice methods. In addition, through a mass of experiments, we find that the relaxation time of LBM has a quantitative relationship with Q. Such a novel scheme offers an alternative forward modelling kernel for seismic inversion and a new model to describe the underground media.

Tradeoff between mixing and transport for electroosmotic flow in heterogeneous microchannels with nonuniform surface potentials.

PubMed

Tian, Fuzhi; Li, Baoming; Kwok, Daniel Y

2005-02-01

Electroosmotic flow (EOF) is a phenomenon associated with the movement of an aqueous solution induced by the application of an electric field in microchannels. The characteristics of EOF depend on the nature of the surface potential, i.e., whether it is uniform or nonuniform. In this paper, a lattice Boltzmann model (LBM) combined with the Poisson-Boltzmann equation is used to simulate flow field in a rectangular microchannel with nonuniform (step change) surface potentials. The simulation results indicate that local circulations can occur near a heterogeneous region with nonuniform surface potentials, in agreement with those by other authors. Largest circulations, which imply a highest mixing efficiency due to convection and short-range diffusion, were found when the average surface potential is zero, regardless of whether the distribution of the heterogeneous patches is symmetric or asymmetric. In this work, we have illustrated that there is a trade-off between the mixing and liquid transport in EOF microfluidics. One should not simply focus on mixing and neglect liquid transport, as performed in the literature. Excellent mixing could lead to a poor transport of electroosmotic flow in microchannels.

Different Scalable Implementations of Collision and Streaming for Optimal Computational Performance of Lattice Boltzmann Simulations

NASA Astrophysics Data System (ADS)

Geneva, Nicholas; Wang, Lian-Ping

2015-11-01

In the past 25 years, the mesoscopic lattice Boltzmann method (LBM) has become an increasingly popular approach to simulate incompressible flows including turbulent flows. While LBM solves more solution variables compared to the conventional CFD approach based on the macroscopic Navier-Stokes equation, it also offers opportunities for more efficient parallelization. In this talk we will describe several different algorithms that have been developed over the past 10 plus years, which can be used to represent the two core steps of LBM, collision and streaming, more effectively than standard approaches. The application of these algorithms spans LBM simulations ranging from basic channel to particle laden flows. We will cover the essential detail on the implementation of each algorithm for simple 2D flows, to the challenges one faces when using a given algorithm for more complex simulations. The key is to explore the best use of data structure and cache memory. Two basic data structures will be discussed and the importance of effective data storage to maximize a CPU's cache will be addressed. The performance of a 3D turbulent channel flow simulation using these different algorithms and data structures will be compared along with important hardware related issues.

A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule

NASA Astrophysics Data System (ADS)

Ying, Jinyong; Xie, Dexuan

2015-10-01

The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model for calculating electrostatics of ionic solvated biomolecule. In this paper, a new finite element and finite difference hybrid method is presented to solve PBE efficiently based on a special seven-overlapped box partition with one central box containing the solute region and surrounded by six neighboring boxes. In particular, an efficient finite element solver is applied to the central box while a fast preconditioned conjugate gradient method using a multigrid V-cycle preconditioning is constructed for solving a system of finite difference equations defined on a uniform mesh of each neighboring box. Moreover, the PBE domain, the box partition, and an interface fitted tetrahedral mesh of the central box can be generated adaptively for a given PQR file of a biomolecule. This new hybrid PBE solver is programmed in C, Fortran, and Python as a software tool for predicting electrostatics of a biomolecule in a symmetric 1:1 ionic solvent. Numerical results on two test models with analytical solutions and 12 proteins validate this new software tool, and demonstrate its high performance in terms of CPU time and memory usage.

AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation.

PubMed

Koehl, Patrice; Delarue, Marc

2010-02-14

The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

Differential geometry based solvation model I: Eulerian formulation

NASA Astrophysics Data System (ADS)

Chen, Zhan; Baker, Nathan A.; Wei, G. W.

2010-11-01

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the solvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By optimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second-order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature.

Differential geometry based solvation model I: Eulerian formulation

PubMed Central

Chen, Zhan; Baker, Nathan A.; Wei, G. W.

2010-01-01

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature. PMID:20938489

Numerical and experimental investigation of a beveled trailing-edge flow field and noise emission

NASA Astrophysics Data System (ADS)

van der Velden, W. C. P.; Pröbsting, S.; van Zuijlen, A. H.; de Jong, A. T.; Guan, Y.; Morris, S. C.

2016-12-01

Efficient tools and methodology for the prediction of trailing-edge noise experience substantial interest within the wind turbine industry. In recent years, the Lattice Boltzmann Method has received increased attention for providing such an efficient alternative for the numerical solution of complex flow problems. Based on the fully explicit, transient, compressible solution of the Lattice Boltzmann Equation in combination with a Ffowcs-Williams and Hawking aeroacoustic analogy, an estimation of the acoustic radiation in the far field is obtained. To validate this methodology for the prediction of trailing-edge noise, the flow around a flat plate with an asymmetric 25° beveled trailing edge and obtuse corner in a low Mach number flow is analyzed. Flow field dynamics are compared to data obtained experimentally from Particle Image Velocimetry and Hot Wire Anemometry, and compare favorably in terms of mean velocity field and turbulent fluctuations. Moreover, the characteristics of the unsteady surface pressure, which are closely related to the acoustic emission, show good agreement between simulation and experiment. Finally, the prediction of the radiated sound is compared to the results obtained from acoustic phased array measurements in combination with a beamforming methodology. Vortex shedding results in a strong narrowband component centered at a constant Strouhal number in the acoustic spectrum. At higher frequency, a good agreement between simulation and experiment for the broadband noise component is obtained and a typical cardioid-like directivity is recovered.

Power Laws are Disguised Boltzmann Laws

NASA Astrophysics Data System (ADS)

Richmond, Peter; Solomon, Sorin

Using a previously introduced model on generalized Lotka-Volterra dynamics together with some recent results for the solution of generalized Langevin equations, we derive analytically the equilibrium mean field solution for the probability distribution of wealth and show that it has two characteristic regimes. For large values of wealth, it takes the form of a Pareto style power law. For small values of wealth, w<=wm, the distribution function tends sharply to zero. The origin of this law lies in the random multiplicative process built into the model. Whilst such results have been known since the time of Gibrat, the present framework allows for a stable power law in an arbitrary and irregular global dynamics, so long as the market is ``fair'', i.e., there is no net advantage to any particular group or individual. We further show that the dynamics of relative wealth is independent of the specific nature of the agent interactions and exhibits a universal character even though the total wealth may follow an arbitrary and complicated dynamics. In developing the theory, we draw parallels with conventional thermodynamics and derive for the system some new relations for the ``thermodynamics'' associated with the Generalized Lotka-Volterra type of stochastic dynamics. The power law that arises in the distribution function is identified with new additional logarithmic terms in the familiar Boltzmann distribution function for the system. These are a direct consequence of the multiplicative stochastic dynamics and are absent for the usual additive stochastic processes.

Revisiting Wiedemann-Franz law through Boltzmann transport equations and ab-initio density functional theory

NASA Astrophysics Data System (ADS)

Nag, Abhinav; Kumari, Anuja; Kumar, Jagdish

2018-05-01

We have investigated structural, electronic and transport properties of the alkali metals using ab-initio density functional theory. The electron energy dispersions are found parabolic free electron like which is expected for alkali metals. The lattice constants for all the studied metals are also in good agreement within 98% with experiments. We have further computed their transport properties using semi-classical Boltzmann transport equations with special focus on electrical and thermal conductivity. Our objective was to obtain Wiedemann-Franz law and hence Lorenz number. The motivation to do these calculations is to see that how the incorporation of different interactions such as electron-lattice, electron-electron interaction affect the Wiedeman-Franz law. By solving Boltzmann transport equations, we have obtained electrical conductivity (σ/τ) and thermal conductivity (κ0 /τ) at different temperatures and then calculated Lorenz number using L = κ0 /(σT). The obtained value of Lorenz number has been found to match with value derived for free electron Fermi gas 2.44× 10-8 WΩK-2. Our results prove that the Wiedemann-Franz law as derived for free electron gas does not change much for alkali metals, even when one incorporates interaction of electrons with atomic nuclei and other electrons. However, at lower temperatures, the Lorenz number, was found to be deviating from its theoretical value.

Thermal Transport in Crystals as a Kinetic Theory of Relaxons

NASA Astrophysics Data System (ADS)

Cepellotti, Andrea; Marzari, Nicola

2016-10-01

Thermal conductivity in dielectric crystals is the result of the relaxation of lattice vibrations described by the phonon Boltzmann transport equation. Remarkably, an exact microscopic definition of the heat carriers and their relaxation times is still missing: Phonons, typically regarded as the relevant excitations for thermal transport, cannot be identified as the heat carriers when most scattering events conserve momentum and do not dissipate heat flux. This is the case for two-dimensional or layered materials at room temperature, or three-dimensional crystals at cryogenic temperatures. In this work, we show that the eigenvectors of the scattering matrix in the Boltzmann equation define collective phonon excitations, which are termed here "relaxons". These excitations have well-defined relaxation times, directly related to heat-flux dissipation, and they provide an exact description of thermal transport as a kinetic theory of the relaxon gas. We show why Matthiessen's rule is violated, and we construct a procedure for obtaining the mean free paths and relaxation times of the relaxons. These considerations are general and would also apply to other semiclassical transport models, such as the electronic Boltzmann equation. For heat transport, they remain relevant even in conventional crystals like silicon, but they are of the utmost importance in the case of two-dimensional materials, where they can revise, by several orders of magnitude, the relevant time and length scales for thermal transport in the hydrodynamic regime.

Microcanonical entropy for classical systems

NASA Astrophysics Data System (ADS)

Franzosi, Roberto

2018-03-01

The entropy definition in the microcanonical ensemble is revisited. We propose a novel definition for the microcanonical entropy that resolve the debate on the correct definition of the microcanonical entropy. In particular we show that this entropy definition fixes the problem inherent the exact extensivity of the caloric equation. Furthermore, this entropy reproduces results which are in agreement with the ones predicted with standard Boltzmann entropy when applied to macroscopic systems. On the contrary, the predictions obtained with the standard Boltzmann entropy and with the entropy we propose, are different for small system sizes. Thus, we conclude that the Boltzmann entropy provides a correct description for macroscopic systems whereas extremely small systems should be better described with the entropy that we propose here.

Compressible fluids with Maxwell-type equations, the minimal coupling with electromagnetic field and the Stefan–Boltzmann law

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mendes, Albert C.R., E-mail: albert@fisica.ufjf.br; Takakura, Flavio I., E-mail: takakura@fisica.ufjf.br; Abreu, Everton M.C., E-mail: evertonabreu@ufrrj.br

In this work we have obtained a higher-derivative Lagrangian for a charged fluid coupled with the electromagnetic fluid and the Dirac’s constraints analysis was discussed. A set of first-class constraints fixed by noncovariant gauge condition were obtained. The path integral formalism was used to obtain the partition function for the corresponding higher-derivative Hamiltonian and the Faddeev–Popov ansatz was used to construct an effective Lagrangian. Through the partition function, a Stefan–Boltzmann type law was obtained. - Highlights: • Higher-derivative Lagrangian for a charged fluid. • Electromagnetic coupling and Dirac’s constraint analysis. • Partition function through path integral formalism. • Stefan–Boltzmann-kind lawmore » through the partition function.« less

Development of axisymmetric lattice Boltzmann flux solver for complex multiphase flows

NASA Astrophysics Data System (ADS)

Wang, Yan; Shu, Chang; Yang, Li-Ming; Yuan, Hai-Zhuan

2018-05-01

This paper presents an axisymmetric lattice Boltzmann flux solver (LBFS) for simulating axisymmetric multiphase flows. In the solver, the two-dimensional (2D) multiphase LBFS is applied to reconstruct macroscopic fluxes excluding axisymmetric effects. Source terms accounting for axisymmetric effects are introduced directly into the governing equations. As compared to conventional axisymmetric multiphase lattice Boltzmann (LB) method, the present solver has the kinetic feature for flux evaluation and avoids complex derivations of external forcing terms. In addition, the present solver also saves considerable computational efforts in comparison with three-dimensional (3D) computations. The capability of the proposed solver in simulating complex multiphase flows is demonstrated by studying single bubble rising in a circular tube. The obtained results compare well with the published data.

Lattice Boltzmann method for rain-induced overland flow

NASA Astrophysics Data System (ADS)

Ding, Yu; Liu, Haifei; Peng, Yong; Xing, Liming

2018-07-01

Complex rainfall situations can generate overland flow with complex hydrodynamic characteristics, affecting the surface configuration (i.e. sheet erosion) and environment to varying degrees. Reliable numerical simulations can provide a scientific method for the optimization of environmental management. A mesoscopic numerical method, the lattice Boltzmann method, was employed to simulate overland flows. To deal with complex rainfall, two schemes were introduced to improve the lattice Boltzmann equation and the local equilibrium function, respectively. Four typical cases with differences in rainfall, bed roughness, and slopes were selected to test the accuracy and applicability of the proposed schemes. It was found that the simulated results were in good agreement with the experimental data, analytical values, and the results produced by other models.

Generalized modification in the lattice Bhatnagar-Gross-Krook model for incompressible Navier-Stokes equations and convection-diffusion equations.

PubMed

Yang, Xuguang; Shi, Baochang; Chai, Zhenhua

2014-07-01

In this paper, two modified lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) models for incompressible Navier-Stokes equations and convection-diffusion equations are proposed via the addition of correction terms in the evolution equations. Utilizing this modification, the value of the dimensionless relaxation time in the LBGK model can be kept in a proper range, and thus the stability of the LBGK model can be improved. Although some gradient operators are included in the correction terms, they can be computed efficiently using local computational schemes such that the present LBGK models still retain the intrinsic parallelism characteristic of the lattice Boltzmann method. Numerical studies of the steady Poiseuille flow and unsteady Womersley flow show that the modified LBGK model has a second-order convergence rate in space, and the compressibility effect in the common LBGK model can be eliminated. In addition, to test the stability of the present models, we also performed some simulations of the natural convection in a square cavity, and we found that the results agree well with those reported in the previous work, even at a very high Rayleigh number (Ra = 10(12)).

Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows.

PubMed

Liang, H; Shi, B C; Guo, Z L; Chai, Z H

2014-05-01

In this paper, a phase-field-based multiple-relaxation-time lattice Boltzmann (LB) model is proposed for incompressible multiphase flow systems. In this model, one distribution function is used to solve the Chan-Hilliard equation and the other is adopted to solve the Navier-Stokes equations. Unlike previous phase-field-based LB models, a proper source term is incorporated in the interfacial evolution equation such that the Chan-Hilliard equation can be derived exactly and also a pressure distribution is designed to recover the correct hydrodynamic equations. Furthermore, the pressure and velocity fields can be calculated explicitly. A series of numerical tests, including Zalesak's disk rotation, a single vortex, a deformation field, and a static droplet, have been performed to test the accuracy and stability of the present model. The results show that, compared with the previous models, the present model is more stable and achieves an overall improvement in the accuracy of the capturing interface. In addition, compared to the single-relaxation-time LB model, the present model can effectively reduce the spurious velocity and fluctuation of the kinetic energy. Finally, as an application, the Rayleigh-Taylor instability at high Reynolds numbers is investigated.

Results from the OH-PT model: a Kinetic-MHD Model of the Outer Heliosphere within SWMF

NASA Astrophysics Data System (ADS)

Michael, A.; Opher, M.; Tenishev, V.; Borovikov, D.; Toth, G.

2017-12-01

We present an update of the OH-PT model, a kinetic-MHD model of the outer heliosphere. The OH-PT model couples the Outer Heliosphere (OH) and Particle Tracker (PT) components within the Space Weather Modeling Framework (SWMF). The OH component utilizes the Block-Adaptive Tree Solarwind Roe-type Upwind Scheme (BATS-R-US) MHD code, a highly parallel, 3D, and block-adaptive solver. As a stand-alone model, the OH component solves the ideal MHD equations for the plasma and a separate set of Euler's equations for the different populations of neutral atoms. The neutrals and plasma in the outer heliosphere are coupled through charge-exchange. While this provides an accurate solution for the plasma, it is an inaccurate description of the neutrals. The charge-exchange mean free path is on the order of the size of the heliosphere; therefore the neutrals cannot be described as a fluid. The PT component is based on the Adaptive Mesh Particle Simulator (AMPS) model, a 3D, direct simulation Monte Carlo model that solves the Boltzmann equation for the motion and interaction of multi-species plasma and is used to model the neutral distribution functions throughout the domain. The charge-exchange process occurs within AMPS, which handles each event on a particle-by-particle basis and calculates the resulting source terms to the MHD equations. The OH-PT model combines the MHD solution for the plasma with the kinetic solution for the neutrals to form a self-consistent model of the heliosphere. In this work, we present verification and validation of the model as well as demonstrate the codes capabilities. Furthermore we provide a comparison of the OH-PT model to our multi-fluid approximation and detail the differences between the models in both the plasma solution and neutral distribution functions.

Straight velocity boundaries in the lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Latt, Jonas; Chopard, Bastien; Malaspinas, Orestis; Deville, Michel; Michler, Andreas

2008-05-01

Various ways of implementing boundary conditions for the numerical solution of the Navier-Stokes equations by a lattice Boltzmann method are discussed. Five commonly adopted approaches are reviewed, analyzed, and compared, including local and nonlocal methods. The discussion is restricted to velocity Dirichlet boundary conditions, and to straight on-lattice boundaries which are aligned with the horizontal and vertical lattice directions. The boundary conditions are first inspected analytically by applying systematically the results of a multiscale analysis to boundary nodes. This procedure makes it possible to compare boundary conditions on an equal footing, although they were originally derived from very different principles. It is concluded that all five boundary conditions exhibit second-order accuracy, consistent with the accuracy of the lattice Boltzmann method. The five methods are then compared numerically for accuracy and stability through benchmarks of two-dimensional and three-dimensional flows. None of the methods is found to be throughout superior to the others. Instead, the choice of a best boundary condition depends on the flow geometry, and on the desired trade-off between accuracy and stability. From the findings of the benchmarks, the boundary conditions can be classified into two major groups. The first group comprehends boundary conditions that preserve the information streaming from the bulk into boundary nodes and complete the missing information through closure relations. Boundary conditions in this group are found to be exceptionally accurate at low Reynolds number. Boundary conditions of the second group replace all variables on boundary nodes by new values. They exhibit generally much better numerical stability and are therefore dedicated for use in high Reynolds number flows.

Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes II: Size Effects on Ionic Distributions and Diffusion-Reaction Rates

PubMed Central

Lu, Benzhuo; Zhou, Y.C.

2011-01-01

The effects of finite particle size on electrostatics, density profiles, and diffusion have been a long existing topic in the study of ionic solution. The previous size-modified Poisson-Boltzmann and Poisson-Nernst-Planck models are revisited in this article. In contrast to many previous works that can only treat particle species with a single uniform size or two sizes, we generalize the Borukhov model to obtain a size-modified Poisson-Nernst-Planck (SMPNP) model that is able to treat nonuniform particle sizes. The numerical tractability of the model is demonstrated as well. The main contributions of this study are as follows. 1), We show that an (arbitrarily) size-modified PB model is indeed implied by the SMPNP equations under certain boundary/interface conditions, and can be reproduced through numerical solutions of the SMPNP. 2), The size effects in the SMPNP effectively reduce the densities of highly concentrated counterions around the biomolecule. 3), The SMPNP is applied to the diffusion-reaction process for the first time, to our knowledge. In the case of low substrate density near the enzyme reactive site, it is observed that the rate coefficients predicted by SMPNP model are considerably larger than those by the PNP model, suggesting both ions and substrates are subject to finite size effects. 4), An accurate finite element method and a convergent Gummel iteration are developed for the numerical solution of the completely coupled nonlinear system of SMPNP equations. PMID:21575582

Path length differencing and energy conservation of the S[sub N] Boltzmann/Spencer-Lewis equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Filippone, W.L.; Monahan, S.P.

It is shown that the S[sub N] Boltzmann/Spencer-Lewis equations conserve energy locally if and only if they satisfy particle balance and diamond differencing is used in path length. In contrast, the spatial differencing schemes have no bearing on the energy balance. Energy is conserved globally if it is conserved locally and the multigroup cross sections are energy conserving. Although the coupled electron-photon cross sections generated by CEPXS conserve particles and charge, they do not precisely conserve energy. It is demonstrated that these cross sections can be adjusted such that particles, charge, and energy are conserved. Finally, since a conventional negativemore » flux fixup destroys energy balance when applied to path legend, a modified fixup scheme that does not is presented.« less

Discrete ellipsoidal statistical BGK model and Burnett equations

NASA Astrophysics Data System (ADS)

Zhang, Yu-Dong; Xu, Ai-Guo; Zhang, Guang-Cai; Chen, Zhi-Hua; Wang, Pei

2018-06-01

A new discrete Boltzmann model, the discrete ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) model, is proposed to simulate nonequilibrium compressible flows. Compared with the original discrete BGK model, the discrete ES-BGK has a flexible Prandtl number. For the discrete ES-BGK model in the Burnett level, two kinds of discrete velocity model are introduced and the relations between nonequilibrium quantities and the viscous stress and heat flux in the Burnett level are established. The model is verified via four benchmark tests. In addition, a new idea is introduced to recover the actual distribution function through the macroscopic quantities and their space derivatives. The recovery scheme works not only for discrete Boltzmann simulation but also for hydrodynamic ones, for example, those based on the Navier-Stokes or the Burnett equations.

Reacting gas mixtures in the state-to-state approach: The chemical reaction rates

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kustova, Elena V.; Kremer, Gilberto M.

2014-12-09

In this work chemically reacting mixtures of viscous flows are analyzed within the framework of Boltzmann equation. By applying a modified Chapman-Enskog method to the system of Boltzmann equations general expressions for the rates of chemical reactions and vibrational energy transitions are determined as functions of two thermodynamic forces: the velocity divergence and the affinity. As an application chemically reacting mixtures of N{sub 2} across a shock wave are studied, where the first lowest vibrational states are taken into account. Here we consider only the contributions from the first four single quantum vibrational-translational energy transitions. It is shown that themore » contribution to the chemical reaction rate related to the affinity is much larger than that of the velocity divergence.« less

Hydrodynamic equations for electrons in graphene obtained from the maximum entropy principle

DOE Office of Scientific and Technical Information (OSTI.GOV)

Barletti, Luigi, E-mail: luigi.barletti@unifi.it

2014-08-15

The maximum entropy principle is applied to the formal derivation of isothermal, Euler-like equations for semiclassical fermions (electrons and holes) in graphene. After proving general mathematical properties of the equations so obtained, their asymptotic form corresponding to significant physical regimes is investigated. In particular, the diffusive regime, the Maxwell-Boltzmann regime (high temperature), the collimation regime and the degenerate gas limit (vanishing temperature) are considered.

On the Development of a Deterministic Three-Dimensional Radiation Transport Code

NASA Technical Reports Server (NTRS)

Rockell, Candice; Tweed, John

2011-01-01

Since astronauts on future deep space missions will be exposed to dangerous radiations, there is a need to accurately model the transport of radiation through shielding materials and to estimate the received radiation dose. In response to this need a three dimensional deterministic code for space radiation transport is now under development. The new code GRNTRN is based on a Green's function solution of the Boltzmann transport equation that is constructed in the form of a Neumann series. Analytical approximations will be obtained for the first three terms of the Neumann series and the remainder will be estimated by a non-perturbative technique . This work discusses progress made to date and exhibits some computations based on the first two Neumann series terms.

Monte Carlo chord length sampling for d-dimensional Markov binary mixtures

NASA Astrophysics Data System (ADS)

Larmier, Coline; Lam, Adam; Brantley, Patrick; Malvagi, Fausto; Palmer, Todd; Zoia, Andrea

2018-01-01

The Chord Length Sampling (CLS) algorithm is a powerful Monte Carlo method that models the effects of stochastic media on particle transport by generating on-the-fly the material interfaces seen by the random walkers during their trajectories. This annealed disorder approach, which formally consists of solving the approximate Levermore-Pomraning equations for linear particle transport, enables a considerable speed-up with respect to transport in quenched disorder, where ensemble-averaging of the Boltzmann equation with respect to all possible realizations is needed. However, CLS intrinsically neglects the correlations induced by the spatial disorder, so that the accuracy of the solutions obtained by using this algorithm must be carefully verified with respect to reference solutions based on quenched disorder realizations. When the disorder is described by Markov mixing statistics, such comparisons have been attempted so far only for one-dimensional geometries, of the rod or slab type. In this work we extend these results to Markov media in two-dimensional (extruded) and three-dimensional geometries, by revisiting the classical set of benchmark configurations originally proposed by Adams, Larsen and Pomraning [1] and extended by Brantley [2]. In particular, we examine the discrepancies between CLS and reference solutions for scalar particle flux and transmission/reflection coefficients as a function of the material properties of the benchmark specifications and of the system dimensionality.

Monte Carlo chord length sampling for d-dimensional Markov binary mixtures

DOE PAGES

Larmier, Coline; Lam, Adam; Brantley, Patrick; ...

2017-09-27

The Chord Length Sampling (CLS) algorithm is a powerful Monte Carlo method that models the effects of stochastic media on particle transport by generating on-the-fly the material interfaces seen by the random walkers during their trajectories. This annealed disorder approach, which formally consists of solving the approximate Levermore–Pomraning equations for linear particle transport, enables a considerable speed-up with respect to transport in quenched disorder, where ensemble-averaging of the Boltzmann equation with respect to all possible realizations is needed. However, CLS intrinsically neglects the correlations induced by the spatial disorder, so that the accuracy of the solutions obtained by using thismore » algorithm must be carefully verified with respect to reference solutions based on quenched disorder realizations. When the disorder is described by Markov mixing statistics, such comparisons have been attempted so far only for one-dimensional geometries, of the rod or slab type. In this work we extend these results to Markov media in two-dimensional (extruded) and three-dimensional geometries, by revisiting the classical set of benchmark configurations originally proposed by Adams, Larsen and Pomraning and extended by Brantley. In particular, we examine the discrepancies between CLS and reference solutions for scalar particle flux and transmission/reflection coefficients as a function of the material properties of the benchmark specifications and of the system dimensionality.« less

Monte Carlo chord length sampling for d-dimensional Markov binary mixtures

DOE Office of Scientific and Technical Information (OSTI.GOV)

Larmier, Coline; Lam, Adam; Brantley, Patrick

The Chord Length Sampling (CLS) algorithm is a powerful Monte Carlo method that models the effects of stochastic media on particle transport by generating on-the-fly the material interfaces seen by the random walkers during their trajectories. This annealed disorder approach, which formally consists of solving the approximate Levermore–Pomraning equations for linear particle transport, enables a considerable speed-up with respect to transport in quenched disorder, where ensemble-averaging of the Boltzmann equation with respect to all possible realizations is needed. However, CLS intrinsically neglects the correlations induced by the spatial disorder, so that the accuracy of the solutions obtained by using thismore » algorithm must be carefully verified with respect to reference solutions based on quenched disorder realizations. When the disorder is described by Markov mixing statistics, such comparisons have been attempted so far only for one-dimensional geometries, of the rod or slab type. In this work we extend these results to Markov media in two-dimensional (extruded) and three-dimensional geometries, by revisiting the classical set of benchmark configurations originally proposed by Adams, Larsen and Pomraning and extended by Brantley. In particular, we examine the discrepancies between CLS and reference solutions for scalar particle flux and transmission/reflection coefficients as a function of the material properties of the benchmark specifications and of the system dimensionality.« less

Density-functional theory of spherical electric double layers and zeta potentials of colloidal particles in restricted-primitive-model electrolyte solutions.

PubMed

Yu, Yang-Xin; Wu, Jianzhong; Gao, Guang-Hua

2004-04-15

A density-functional theory is proposed to describe the density profiles of small ions around an isolated colloidal particle in the framework of the restricted primitive model where the small ions have uniform size and the solvent is represented by a dielectric continuum. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for the hard-sphere repulsion and a quadratic functional Taylor expansion for the electrostatic interactions. The theoretical predictions are in good agreement with the results from Monte Carlo simulations and from previous investigations using integral-equation theory for the ionic density profiles and the zeta potentials of spherical particles at a variety of solution conditions. Like the integral-equation approaches, the density-functional theory is able to capture the oscillatory density profiles of small ions and the charge inversion (overcharging) phenomena for particles with elevated charge density. In particular, our density-functional theory predicts the formation of a second counterion layer near the surface of highly charged spherical particle. Conversely, the nonlinear Poisson-Boltzmann theory and its variations are unable to represent the oscillatory behavior of small ion distributions and charge inversion. Finally, our density-functional theory predicts charge inversion even in a 1:1 electrolyte solution as long as the salt concentration is sufficiently high. (c) 2004 American Institute of Physics.

Discontinuous Galerkin Methods for NonLinear Differential Systems

NASA Technical Reports Server (NTRS)

Barth, Timothy; Mansour, Nagi (Technical Monitor)

2001-01-01

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Membrane protein properties revealed through data-rich electrostatics calculations

PubMed Central

Guerriero, Christopher J.; Brodsky, Jeffrey L.; Grabe, Michael

2015-01-01

SUMMARY The electrostatic properties of membrane proteins often reveal many of their key biophysical characteristics, such as ion channel selectivity and the stability of charged membrane-spanning segments. The Poisson-Boltzmann (PB) equation is the gold standard for calculating protein electrostatics, and the software APBSmem enables the solution of the PB equation in the presence of a membrane. Here, we describe significant advances to APBSmem including: full automation of system setup, per-residue energy decomposition, incorporation of PDB2PQR, calculation of membrane induced pKa shifts, calculation of non-polar energies, and command-line scripting for large scale calculations. We highlight these new features with calculations carried out on a number of membrane proteins, including the recently solved structure of the ion channel TRPV1 and a large survey of 1,614 membrane proteins of known structure. This survey provides a comprehensive list of residues with large electrostatic penalties for being embedded in the membrane potentially revealing interesting functional information. PMID:26118532

Membrane Protein Properties Revealed through Data-Rich Electrostatics Calculations.

PubMed

Marcoline, Frank V; Bethel, Neville; Guerriero, Christopher J; Brodsky, Jeffrey L; Grabe, Michael

2015-08-04

The electrostatic properties of membrane proteins often reveal many of their key biophysical characteristics, such as ion channel selectivity and the stability of charged membrane-spanning segments. The Poisson-Boltzmann (PB) equation is the gold standard for calculating protein electrostatics, and the software APBSmem enables the solution of the PB equation in the presence of a membrane. Here, we describe significant advances to APBSmem, including full automation of system setup, per-residue energy decomposition, incorporation of PDB2PQR, calculation of membrane-induced pKa shifts, calculation of non-polar energies, and command-line scripting for large-scale calculations. We highlight these new features with calculations carried out on a number of membrane proteins, including the recently solved structure of the ion channel TRPV1 and a large survey of 1,614 membrane proteins of known structure. This survey provides a comprehensive list of residues with large electrostatic penalties for being embedded in the membrane, potentially revealing interesting functional information. Copyright © 2015 Elsevier Ltd. All rights reserved.

Pre-Darcy flow in tight and shale formations

NASA Astrophysics Data System (ADS)

Dejam, Morteza; Hassanzadeh, Hassan; Chen, Zhangxin

2017-11-01

There are evidences that the fluid flow in tight and shale formations does not follow Darcy law, which is identified as pre-Darcy flow. Here, the unsteady linear flow of a slightly compressible fluid under the action of pre-Darcy flow is modeled and a generalized Boltzmann transformation technique is used to solve the corresponding highly nonlinear diffusivity equation analytically. The effect of pre-Darcy flow on the pressure diffusion in a homogenous formation is studied in terms of the nonlinear exponent, m, and the threshold pressure gradient, G1. In addition, the pressure gradient, flux, and cumulative production per unit area for different m and G1 are compared with the classical solution of the diffusivity equation based on Darcy flow. Department of Petroleum Engineering in College of Engineering and Applied Science at University of Wyoming and NSERC/AI-EES(AERI)/Foundation CMG and AITF (iCORE) Chairs in Department of Chemical and Petroleum Engineering at University of Calgary.

Dendritic polyelectrolytes as seen by the Poisson-Boltzmann-Flory theory.

PubMed

Kłos, J S; Milewski, J

2018-06-20

G3-G9 dendritic polyelectrolytes accompanied by counterions are investigated using the Poisson-Boltzmann-Flory theory. Within this approach we solve numerically the Poisson-Boltzmann equation for the mean electrostatic potential and minimize the Poisson-Boltzmann-Flory free energy with respect to the size of the molecules. Such a scheme enables us to inspect the conformational and electrostatic properties of the dendrimers in equilibrium based on their response to varying the dendrimer generation. The calculations indicate that the G3-G6 dendrimers exist in the polyelectrolyte regime where absorption of counterions into the volume of the molecules is minor. Trapping of ions in the interior region becomes significant for the G7-G9 dendrimers and signals the emergence of the osmotic regime. We find that the behavior of the dendritic polyelectrolytes corresponds with the degree of ion trapping. In particular, in both regimes the polyelectrolytes are swollen as compared to their neutral counterparts and the expansion factor is maximal at the crossover generation G7.

Differential geometry based solvation model II: Lagrangian formulation

PubMed Central

Chen, Zhan; Baker, Nathan A.; Wei, G. W.

2010-01-01

Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. This work presents a Lagrangian formulation of our differential geometry based solvation model. The Lagrangian representation of biomolecular surfaces has a few utilities/advantages. First, it provides an essential basis for biomolecular visualization, surface electrostatic potential map and visual perception of biomolecules. Additionally, it is consistent with the conventional setting of implicit solvent theories and thus, many existing theoretical algorithms and computational software packages can be directly employed. Finally, the Lagrangian representation does not need to resort to artificially enlarged van der Waals radii as often required by the Eulerian representation in solvation analysis. The main goal of the present work is to analyze the connection, similarity and difference between the Eulerian and Lagrangian formalisms of the solvation model. Such analysis is important to the understanding of the differential geometry based solvation model. The present model extends the scaled particle theory (SPT) of nonpolar solvation model with a solvent-solute interaction potential. The nonpolar solvation model is completed with a Poisson-Boltzmann (PB) theory based polar solvation model. The differential geometry theory of surfaces is employed to provide a natural description of solvent-solute interfaces. The minimization of the total free energy functional, which encompasses the polar and nonpolar contributions, leads to coupled potential driven geometric flow and Poisson-Boltzmann equations. Due to the development of singularities and nonsmooth manifolds in the Lagrangian representation, the resulting potential-driven geometric flow equation is embedded into the Eulerian representation for the purpose of computation, thanks to the equivalence of the Laplace-Beltrami operator in the two representations. The coupled partial differential equations (PDEs) are solved with an iterative procedure to reach a steady state, which delivers desired solvent-solute interface and electrostatic potential for problems of interest. These quantities are utilized to evaluate the solvation free energies and protein-protein binding affinities. A number of computational methods and algorithms are described for the interconversion of Lagrangian and Eulerian representations, and for the solution of the coupled PDE system. The proposed approaches have been extensively validated. We also verify that the mean curvature flow indeed gives rise to the minimal molecular surface (MMS) and the proposed variational procedure indeed offers minimal total free energy. Solvation analysis and applications are considered for a set of 17 small compounds and a set of 23 proteins. The salt effect on protein-protein binding affinity is investigated with two protein complexes by using the present model. Numerical results are compared to the experimental measurements and to those obtained by using other theoretical methods in the literature. PMID:21279359

Prediction of sound absorption in rigid porous media with the lattice Boltzmann method

NASA Astrophysics Data System (ADS)

da Silva, Andrey Ricardo; Mareze, Paulo; Brandão, Eric

2016-02-01

In this work, sound absorption phenomena associated with the viscous shear stress within rigid porous media is investigated with a simple isothermal lattice Boltzmann BGK model. Simulations are conducted for different macroscopic material properties such as sample thickness and porosity and the results are compared with the exact analytical solution for materials with slit-like structure in terms of acoustic impedance and sound absorption coefficient. The numerical results agree very well with the exact solution, particularly for the sound absorption coefficient. The small deviations found in the low frequency limit for the real part of the acoustic impedance are attributed to the ratio between the thicknesses of the slit and the viscous boundary layer. The results suggest that the lattice Boltzmann method can be a very compelling numerical tool for simulating viscous sound absorption phenomena in the time domain, particularly due to its computational simplicity when compared to traditional continuum based techniques.

Lattice Boltzmann simulations of liquid crystal particulate flow in a channel with finite anchoring boundary conditions

NASA Astrophysics Data System (ADS)

Zhang, Rui; Roberts, Tyler; de Pablo, Juan; dePablo Team

2014-11-01

Liquid crystals (LC) posses anisotropic viscoelastic properties, and, as such, LC flow can be incredibly complicated. Here we employ a hybrid lattice Boltzmann method (pioneered by Deniston, Yeomans and Cates) to systematically study the hydrodynamics of nematic liquid crystals (LCs) with and without solid particles. This method evolves the velocity field through lattice Boltzmann and the LC-order parameter via a finite-difference solver of the Beris-Edwards equation. The evolution equation of the boundary points with finite anchoring is obtained through Poisson bracket formulation. Our method has been validated by matching the Ericksen-Leslie theory. We demonstrate two applications in the flow alignment regime. We first investigate a hybrid channel flow in which the top and bottom walls have different anchoring directions. By measuring the apparent shear viscosity in terms of Couette flow, we achieve a viscosity inhomogeneous system which may be applicable to nano particle processing. In the other example, we introduce a homeotropic spherical particle to the channel, and focus on the deformations of the defect ring due to anchorings and flow. The results are then compared to the molecular dynamics simulations of a colloid particle in an LC modeled by a Gay-Berne potential.

Numerical investigation of the pseudopotential lattice Boltzmann modeling of liquid-vapor for multi-phase flows

NASA Astrophysics Data System (ADS)

Nemati, Maedeh; Shateri Najaf Abady, Ali Reza; Toghraie, Davood; Karimipour, Arash

2018-01-01

The incorporation of different equations of state into single-component multiphase lattice Boltzmann model is considered in this paper. The original pseudopotential model is first detailed, and several cubic equations of state, the Redlich-Kwong, Redlich-Kwong-Soave, and Peng-Robinson are then incorporated into the lattice Boltzmann model. A comparison of the numerical simulation achievements on the basis of density ratios and spurious currents is used for presentation of the details of phase separation in these non-ideal single-component systems. The paper demonstrates that the scheme for the inter-particle interaction force term as well as the force term incorporation method matters to achieve more accurate and stable results. The velocity shifting method is demonstrated as the force term incorporation method, among many, with accuracy and stability results. Kupershtokh scheme also makes it possible to achieve large density ratio (up to 104) and to reproduce the coexistence curve with high accuracy. Significant reduction of the spurious currents at vapor-liquid interface is another observation. High-density ratio and spurious current reduction resulted from the Redlich-Kwong-Soave and Peng-Robinson EOSs, in higher accordance with the Maxwell construction results.

Quantitative analysis of the correlations in the Boltzmann-Grad limit for hard spheres

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pulvirenti, M.

2014-12-09

In this contribution I consider the problem of the validity of the Boltzmann equation for a system of hard spheres in the Boltzmann-Grad limit. I briefly review the results available nowadays with a particular emphasis on the celebrated Lanford’s validity theorem. Finally I present some recent results, obtained in collaboration with S. Simonella, concerning a quantitative analysis of the propagation of chaos. More precisely we introduce a quantity (the correlation error) measuring how close a j-particle rescaled correlation function at time t (sufficiently small) is far from the full statistical independence. Roughly speaking, a correlation error of order k, measuresmore » (in the context of the BBKGY hierarchy) the event in which k tagged particles form a recolliding group.« less

Numerical Analysis of Plasma Transport in Tandem Volume Magnetic Multicusp Ion Sources

DTIC Science & Technology

1992-03-01

the results of the model are qualitatively correct. Boltzmann Equation, Ion Sources, Plasma Simulation, Electron Temperature, Plasma Density, Ion Temperature, Hydrogen Ions, Magnetic Filters, Hydrogen Plasma Chemistry .

Derivation of a generalized Schrödinger equation from the theory of scale relativity

NASA Astrophysics Data System (ADS)

Chavanis, Pierre-Henri

2017-06-01

Using Nottale's theory of scale relativity relying on a fractal space-time, we derive a generalized Schrödinger equation taking into account the interaction of the system with the external environment. This equation describes the irreversible evolution of the system towards a static quantum state. We first interpret the scale-covariant equation of dynamics stemming from Nottale's theory as a hydrodynamic viscous Burgers equation for a potential flow involving a complex velocity field and an imaginary viscosity. We show that the Schrödinger equation can be directly obtained from this equation by performing a Cole-Hopf transformation equivalent to the WKB transformation. We then introduce a friction force proportional and opposite to the complex velocity in the scale-covariant equation of dynamics in a way that preserves the local conservation of the normalization condition. We find that the resulting generalized Schrödinger equation, or the corresponding fluid equations obtained from the Madelung transformation, involve not only a damping term but also an effective thermal term. The friction coefficient and the temperature are related to the real and imaginary parts of the complex friction coefficient in the scale-covariant equation of dynamics. This may be viewed as a form of fluctuation-dissipation theorem. We show that our generalized Schrödinger equation satisfies an H-theorem for the quantum Boltzmann free energy. As a result, the probability distribution relaxes towards an equilibrium state which can be viewed as a Boltzmann distribution including a quantum potential. We propose to apply this generalized Schrödinger equation to dark matter halos in the Universe, possibly made of self-gravitating Bose-Einstein condensates.

Exploring a multi-scale method for molecular simulation in continuum solvent model: Explicit simulation of continuum solvent as an incompressible fluid.

PubMed

Xiao, Li; Luo, Ray

2017-12-07

We explored a multi-scale algorithm for the Poisson-Boltzmann continuum solvent model for more robust simulations of biomolecules. In this method, the continuum solvent/solute interface is explicitly simulated with a numerical fluid dynamics procedure, which is tightly coupled to the solute molecular dynamics simulation. There are multiple benefits to adopt such a strategy as presented below. At this stage of the development, only nonelectrostatic interactions, i.e., van der Waals and hydrophobic interactions, are included in the algorithm to assess the quality of the solvent-solute interface generated by the new method. Nevertheless, numerical challenges exist in accurately interpolating the highly nonlinear van der Waals term when solving the finite-difference fluid dynamics equations. We were able to bypass the challenge rigorously by merging the van der Waals potential and pressure together when solving the fluid dynamics equations and by considering its contribution in the free-boundary condition analytically. The multi-scale simulation method was first validated by reproducing the solute-solvent interface of a single atom with analytical solution. Next, we performed the relaxation simulation of a restrained symmetrical monomer and observed a symmetrical solvent interface at equilibrium with detailed surface features resembling those found on the solvent excluded surface. Four typical small molecular complexes were then tested, both volume and force balancing analyses showing that these simple complexes can reach equilibrium within the simulation time window. Finally, we studied the quality of the multi-scale solute-solvent interfaces for the four tested dimer complexes and found that they agree well with the boundaries as sampled in the explicit water simulations.

Geodesic curve-of-sight formulae for the cosmic microwave background: a unified treatment of redshift, time delay, and lensing

DOE Office of Scientific and Technical Information (OSTI.GOV)

Saito, Ryo; Naruko, Atsushi; Hiramatsu, Takashi

2014-10-01

In this paper, we introduce a new approach to a treatment of the gravitational effects (redshift, time delay and lensing) on the observed cosmic microwave background (CMB) anisotropies based on the Boltzmann equation. From the Liouville's theorem in curved spacetime, the intensity of photons is conserved along a photon geodesic when non-gravitational scatterings are absent. Motivated by this fact, we derive a second-order line-of-sight formula by integrating the Boltzmann equation along a perturbed geodesic (curve) instead of a background geodesic (line). In this approach, the separation of the gravitational and intrinsic effects are manifest. This approach can be considered asmore » a generalization of the remapping approach of CMB lensing, where all the gravitational effects can be treated on the same footing.« less

Thermalization of oscillator chains with onsite anharmonicity and comparison with kinetic theory

DOE PAGES

Mendl, Christian B.; Lu, Jianfeng; Lukkarinen, Jani

2016-12-02

We perform microscopic molecular dynamics simulations of particle chains with an onsite anharmonicity to study relaxation of spatially homogeneous states to equilibrium, and directly compare the simulations with the corresponding Boltzmann-Peierls kinetic theory. The Wigner function serves as a common interface between the microscopic and kinetic level. We demonstrate quantitative agreement after an initial transient time interval. In particular, besides energy conservation, we observe the additional quasiconservation of the phonon density, defined via an ensemble average of the related microscopic field variables and exactly conserved by the kinetic equations. On superkinetic time scales, density quasiconservation is lost while energy remainsmore » conserved, and we find evidence for eventual relaxation of the density to its canonical ensemble value. Furthermore, the precise mechanism remains unknown and is not captured by the Boltzmann-Peierls equations.« less

Thermalization of oscillator chains with onsite anharmonicity and comparison with kinetic theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mendl, Christian B.; Lu, Jianfeng; Lukkarinen, Jani

We perform microscopic molecular dynamics simulations of particle chains with an onsite anharmonicity to study relaxation of spatially homogeneous states to equilibrium, and directly compare the simulations with the corresponding Boltzmann-Peierls kinetic theory. The Wigner function serves as a common interface between the microscopic and kinetic level. We demonstrate quantitative agreement after an initial transient time interval. In particular, besides energy conservation, we observe the additional quasiconservation of the phonon density, defined via an ensemble average of the related microscopic field variables and exactly conserved by the kinetic equations. On superkinetic time scales, density quasiconservation is lost while energy remainsmore » conserved, and we find evidence for eventual relaxation of the density to its canonical ensemble value. Furthermore, the precise mechanism remains unknown and is not captured by the Boltzmann-Peierls equations.« less

Transition point prediction in a multicomponent lattice Boltzmann model: Forcing scheme dependencies

NASA Astrophysics Data System (ADS)

Küllmer, Knut; Krämer, Andreas; Joppich, Wolfgang; Reith, Dirk; Foysi, Holger

2018-02-01

Pseudopotential-based lattice Boltzmann models are widely used for numerical simulations of multiphase flows. In the special case of multicomponent systems, the overall dynamics are characterized by the conservation equations for mass and momentum as well as an additional advection diffusion equation for each component. In the present study, we investigate how the latter is affected by the forcing scheme, i.e., by the way the underlying interparticle forces are incorporated into the lattice Boltzmann equation. By comparing two model formulations for pure multicomponent systems, namely the standard model [X. Shan and G. D. Doolen, J. Stat. Phys. 81, 379 (1995), 10.1007/BF02179985] and the explicit forcing model [M. L. Porter et al., Phys. Rev. E 86, 036701 (2012), 10.1103/PhysRevE.86.036701], we reveal that the diffusion characteristics drastically change. We derive a generalized, potential function-dependent expression for the transition point from the miscible to the immiscible regime and demonstrate that it is shifted between the models. The theoretical predictions for both the transition point and the mutual diffusion coefficient are validated in simulations of static droplets and decaying sinusoidal concentration waves, respectively. To show the universality of our analysis, two common and one new potential function are investigated. As the shift in the diffusion characteristics directly affects the interfacial properties, we additionally show that phenomena related to the interfacial tension such as the modeling of contact angles are influenced as well.

Nonlocal and nonlinear electrostatics of a dipolar Coulomb fluid.

PubMed

Sahin, Buyukdagli; Ralf, Blossey

2014-07-16

We study a model Coulomb fluid consisting of dipolar solvent molecules of finite extent which generalizes the point-like dipolar Poisson-Boltzmann model (DPB) previously introduced by Coalson and Duncan (1996 J. Phys. Chem. 100 2612) and Abrashkin et al (2007 Phys. Rev. Lett. 99 077801). We formulate a nonlocal Poisson-Boltzmann equation (NLPB) and study both linear and nonlinear dielectric response in this model for the case of a single plane geometry. Our results shed light on the relevance of nonlocal versus nonlinear effects in continuum models of material electrostatics.

On thermalization of electron-positron-photon plasma

NASA Astrophysics Data System (ADS)

Siutsou, I. A.; Aksenov, A. G.; Vereshchagin, G. V.

2015-12-01

Recently a progress has been made in understanding thermalization mechanism of relativistic plasma starting from a non-equilibrium state. Relativistic Boltzmann equations were solved numerically for homogeneous isotropic plasma with collision integrals for two- and three-particle interactions calculated from the first principles by means of QED matrix elements. All particles were assumed to fulfill Boltzmann statistics. In this work we follow plasma thermalization by accounting for Bose enhancement and Pauli blocking in particle interactions. Our results show that particle in equilibrium reach Bose-Einstein distribution for photons, and Fermi-Dirac one for electrons, respectively.

Entropy inequality and hydrodynamic limits for the Boltzmann equation.

PubMed

Saint-Raymond, Laure

2013-12-28

Boltzmann brought a fundamental contribution to the understanding of the notion of entropy, by giving a microscopic formulation of the second principle of thermodynamics. His ingenious idea, motivated by the works of his contemporaries on the atomic nature of matter, consists of describing gases as huge systems of identical and indistinguishable elementary particles. The state of a gas can therefore be described in a statistical way. The evolution, which introduces couplings, loses part of the information, which is expressed by the decay of the so-called mathematical entropy (the opposite of physical entropy!).

Applications of the Lattice Boltzmann Method to Complex and Turbulent Flows

NASA Technical Reports Server (NTRS)

Luo, Li-Shi; Qi, Dewei; Wang, Lian-Ping; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

We briefly review the method of the lattice Boltzmann equation (LBE). We show the three-dimensional LBE simulation results for a non-spherical particle in Couette flow and 16 particles in sedimentation in fluid. We compare the LBE simulation of the three-dimensional homogeneous isotropic turbulence flow in a periodic cubic box of the size 1283 with the pseudo-spectral simulation, and find that the two results agree well with each other but the LBE method is more dissipative than the pseudo-spectral method in small scales, as expected.

A large eddy lattice Boltzmann simulation of magnetohydrodynamic turbulence

NASA Astrophysics Data System (ADS)

Flint, Christopher; Vahala, George

2018-02-01

Large eddy simulations (LES) of a lattice Boltzmann magnetohydrodynamic (LB-MHD) model are performed for the unstable magnetized Kelvin-Helmholtz jet instability. This algorithm is an extension of Ansumali et al. [1] to MHD in which one performs first an expansion in the filter width on the kinetic equations followed by the usual low Knudsen number expansion. These two perturbation operations do not commute. Closure is achieved by invoking the physical constraint that subgrid effects occur at transport time scales. The simulations are in very good agreement with direct numerical simulations.

Study of microvascular non-Newtonian blood flow modulated by electroosmosis.

PubMed

Tripathi, Dharmendra; Yadav, Ashu; Anwar Bég, O; Kumar, Rakesh

2018-05-01

An analytical study of microvascular non-Newtonian blood flow is conducted incorporating the electro-osmosis phenomenon. Blood is considered as a Bingham rheological aqueous ionic solution. An externally applied static axial electrical field is imposed on the system. The Poisson-Boltzmann equation for electrical potential distribution is implemented to accommodate the electrical double layer in the microvascular regime. With long wavelength, lubrication and Debye-Hückel approximations, the boundary value problem is rendered non-dimensional. Analytical solutions are derived for the axial velocity, volumetric flow rate, pressure gradient, volumetric flow rate, averaged volumetric flow rate along one time period, pressure rise along one wavelength and stream function. A plug swidth is featured in the solutions. Via symbolic software (Mathematica), graphical plots are generated for the influence of Bingham plug flow width parameter, electrical Debye length and Helmholtz-Smoluchowski velocity (maximum electro-osmotic velocity) on the key hydrodynamic variables. This study reveals that blood flow rate accelerates with decreasing the plug width (i.e. viscoplastic nature of fluids) and also with increasing the Debye length parameter. Copyright © 2018 Elsevier Inc. All rights reserved.

Poisson-Box Sampling algorithms for three-dimensional Markov binary mixtures

NASA Astrophysics Data System (ADS)

Larmier, Coline; Zoia, Andrea; Malvagi, Fausto; Dumonteil, Eric; Mazzolo, Alain

2018-02-01

Particle transport in Markov mixtures can be addressed by the so-called Chord Length Sampling (CLS) methods, a family of Monte Carlo algorithms taking into account the effects of stochastic media on particle propagation by generating on-the-fly the material interfaces crossed by the random walkers during their trajectories. Such methods enable a significant reduction of computational resources as opposed to reference solutions obtained by solving the Boltzmann equation for a large number of realizations of random media. CLS solutions, which neglect correlations induced by the spatial disorder, are faster albeit approximate, and might thus show discrepancies with respect to reference solutions. In this work we propose a new family of algorithms (called 'Poisson Box Sampling', PBS) aimed at improving the accuracy of the CLS approach for transport in d-dimensional binary Markov mixtures. In order to probe the features of PBS methods, we will focus on three-dimensional Markov media and revisit the benchmark problem originally proposed by Adams, Larsen and Pomraning [1] and extended by Brantley [2]: for these configurations we will compare reference solutions, standard CLS solutions and the new PBS solutions for scalar particle flux, transmission and reflection coefficients. PBS will be shown to perform better than CLS at the expense of a reasonable increase in computational time.

Morphing Continuum Theory: A First Order Approximation to the Balance Laws

NASA Astrophysics Data System (ADS)

Wonnell, Louis; Cheikh, Mohamad Ibrahim; Chen, James

2017-11-01

Morphing Continuum Theory is constructed under the framework of Rational Continuum Mechanics (RCM) for fluid flows with inner structure. This multiscale theory has been successfully emplyed to model turbulent flows. The framework of RCM ensures the mathematical rigor of MCT, but contains new material constants related to the inner structure. The physical meanings of these material constants have yet to be determined. Here, a linear deviation from the zeroth-order Boltzmann-Curtiss distribution function is derived. When applied to the Boltzmann-Curtiss equation, a first-order approximation of the MCT governing equations is obtained. The integral equations are then related to the appropriate material constants found in the heat flux, Cauchy stress, and moment stress terms in the governing equations. These new material properties associated with the inner structure of the fluid are compared with the corresponding integrals, and a clearer physical interpretation of these coefficients emerges. The physical meanings of these material properties is determined by analyzing previous results obtained from numerical simulations of MCT for compressible and incompressible flows. The implications for the physics underlying the MCT governing equations will also be discussed. This material is based upon work supported by the Air Force Office of Scientific Research under Award Number FA9550-17-1-0154.

A two-component Matched Interface and Boundary (MIB) regularization for charge singularity in implicit solvation

NASA Astrophysics Data System (ADS)

Geng, Weihua; Zhao, Shan

2017-12-01

We present a new Matched Interface and Boundary (MIB) regularization method for treating charge singularity in solvated biomolecules whose electrostatics are described by the Poisson-Boltzmann (PB) equation. In a regularization method, by decomposing the potential function into two or three components, the singular component can be analytically represented by the Green's function, while other components possess a higher regularity. Our new regularization combines the efficiency of two-component schemes with the accuracy of the three-component schemes. Based on this regularization, a new MIB finite difference algorithm is developed for solving both linear and nonlinear PB equations, where the nonlinearity is handled by using the inexact-Newton's method. Compared with the existing MIB PB solver based on a three-component regularization, the present algorithm is simpler to implement by circumventing the work to solve a boundary value Poisson equation inside the molecular interface and to compute related interface jump conditions numerically. Moreover, the new MIB algorithm becomes computationally less expensive, while maintains the same second order accuracy. This is numerically verified by calculating the electrostatic potential and solvation energy on the Kirkwood sphere on which the analytical solutions are available and on a series of proteins with various sizes.

Highly efficient and exact method for parallelization of grid-based algorithms and its implementation in DelPhi

PubMed Central

Li, Chuan; Li, Lin; Zhang, Jie; Alexov, Emil

2012-01-01

The Gauss-Seidel method is a standard iterative numerical method widely used to solve a system of equations and, in general, is more efficient comparing to other iterative methods, such as the Jacobi method. However, standard implementation of the Gauss-Seidel method restricts its utilization in parallel computing due to its requirement of using updated neighboring values (i.e., in current iteration) as soon as they are available. Here we report an efficient and exact (not requiring assumptions) method to parallelize iterations and to reduce the computational time as a linear/nearly linear function of the number of CPUs. In contrast to other existing solutions, our method does not require any assumptions and is equally applicable for solving linear and nonlinear equations. This approach is implemented in the DelPhi program, which is a finite difference Poisson-Boltzmann equation solver to model electrostatics in molecular biology. This development makes the iterative procedure on obtaining the electrostatic potential distribution in the parallelized DelPhi several folds faster than that in the serial code. Further we demonstrate the advantages of the new parallelized DelPhi by computing the electrostatic potential and the corresponding energies of large supramolecular structures. PMID:22674480

Steady-state heat transport: Ballistic-to-diffusive with Fourier's law

DOE Office of Scientific and Technical Information (OSTI.GOV)

Maassen, Jesse, E-mail: jmaassen@purdue.edu; Lundstrom, Mark

2015-01-21

It is generally understood that Fourier's law does not describe ballistic phonon transport, which is important when the length of a material is similar to the phonon mean-free-path. Using an approach adapted from electron transport, we demonstrate that Fourier's law and the heat equation do capture ballistic effects, including temperature jumps at ideal contacts, and are thus applicable on all length scales. Local thermal equilibrium is not assumed, because allowing the phonon distribution to be out-of-equilibrium is important for ballistic and quasi-ballistic transport. The key to including the non-equilibrium nature of the phonon population is to apply the proper boundarymore » conditions to the heat equation. Simple analytical solutions are derived, showing that (i) the magnitude of the temperature jumps is simply related to the material properties and (ii) the observation of reduced apparent thermal conductivity physically stems from a reduction in the temperature gradient and not from a reduction in actual thermal conductivity. We demonstrate how our approach, equivalent to Fourier's law, easily reproduces results of the Boltzmann transport equation, in all transport regimes, even when using a full phonon dispersion and mean-free-path distribution.« less

Transient hydrodynamic finite-size effects in simulations under periodic boundary conditions

NASA Astrophysics Data System (ADS)

Asta, Adelchi J.; Levesque, Maximilien; Vuilleumier, Rodolphe; Rotenberg, Benjamin

2017-06-01

We use lattice-Boltzmann and analytical calculations to investigate transient hydrodynamic finite-size effects induced by the use of periodic boundary conditions. These effects are inevitable in simulations at the molecular, mesoscopic, or continuum levels of description. We analyze the transient response to a local perturbation in the fluid and obtain the local velocity correlation function via linear response theory. This approach is validated by comparing the finite-size effects on the steady-state velocity with the known results for the diffusion coefficient. We next investigate the full time dependence of the local velocity autocorrelation function. We find at long times a crossover between the expected t-3 /2 hydrodynamic tail and an oscillatory exponential decay, and study the scaling with the system size of the crossover time, exponential rate and amplitude, and oscillation frequency. We interpret these results from the analytic solution of the compressible Navier-Stokes equation for the slowest modes, which are set by the system size. The present work not only provides a comprehensive analysis of hydrodynamic finite-size effects in bulk fluids, which arise regardless of the level of description and simulation algorithm, but also establishes the lattice-Boltzmann method as a suitable tool to investigate such effects in general.

Simulation of Sound Waves Using the Lattice Boltzmann Method for Fluid Flow: Benchmark Cases for Outdoor Sound Propagation

PubMed Central

Salomons, Erik M.; Lohman, Walter J. A.; Zhou, Han

2016-01-01

Propagation of sound waves in air can be considered as a special case of fluid dynamics. Consequently, the lattice Boltzmann method (LBM) for fluid flow can be used for simulating sound propagation. In this article application of the LBM to sound propagation is illustrated for various cases: free-field propagation, propagation over porous and non-porous ground, propagation over a noise barrier, and propagation in an atmosphere with wind. LBM results are compared with solutions of the equations of acoustics. It is found that the LBM works well for sound waves, but dissipation of sound waves with the LBM is generally much larger than real dissipation of sound waves in air. To circumvent this problem it is proposed here to use the LBM for assessing the excess sound level, i.e. the difference between the sound level and the free-field sound level. The effect of dissipation on the excess sound level is much smaller than the effect on the sound level, so the LBM can be used to estimate the excess sound level for a non-dissipative atmosphere, which is a useful quantity in atmospheric acoustics. To reduce dissipation in an LBM simulation two approaches are considered: i) reduction of the kinematic viscosity and ii) reduction of the lattice spacing. PMID:26789631

Simulation of Sound Waves Using the Lattice Boltzmann Method for Fluid Flow: Benchmark Cases for Outdoor Sound Propagation.

PubMed

Salomons, Erik M; Lohman, Walter J A; Zhou, Han

2016-01-01

Propagation of sound waves in air can be considered as a special case of fluid dynamics. Consequently, the lattice Boltzmann method (LBM) for fluid flow can be used for simulating sound propagation. In this article application of the LBM to sound propagation is illustrated for various cases: free-field propagation, propagation over porous and non-porous ground, propagation over a noise barrier, and propagation in an atmosphere with wind. LBM results are compared with solutions of the equations of acoustics. It is found that the LBM works well for sound waves, but dissipation of sound waves with the LBM is generally much larger than real dissipation of sound waves in air. To circumvent this problem it is proposed here to use the LBM for assessing the excess sound level, i.e. the difference between the sound level and the free-field sound level. The effect of dissipation on the excess sound level is much smaller than the effect on the sound level, so the LBM can be used to estimate the excess sound level for a non-dissipative atmosphere, which is a useful quantity in atmospheric acoustics. To reduce dissipation in an LBM simulation two approaches are considered: i) reduction of the kinematic viscosity and ii) reduction of the lattice spacing.

Frequency-dependent laminar electroosmotic flow in a closed-end rectangular microchannel.

PubMed

Marcos; Yang, C; Ooi, K T; Wong, T N; Masliyah, J H

2004-07-15

This article presents an analysis of the frequency- and time-dependent electroosmotic flow in a closed-end rectangular microchannel. An exact solution to the modified Navier-Stokes equation governing the ac electroosmotic flow field is obtained by using the Green's function formulation in combination with a complex variable approach. An analytical expression for the induced backpressure gradient is derived. With the Debye-Hückel approximation, the electrical double-layer potential distribution in the channel is obtained by analytically solving the linearized two-dimensional Poisson-Boltzmann equation. Since the counterparts of the flow rate and the electrical current are shown to be linearly proportional to the applied electric field and the pressure gradient, Onsager's principle of reciprocity is demonstrated for transient and ac electroosmotic flows. The time evolution of the electroosmotic flow and the effect of a frequency-dependent ac electric field on the oscillating electroosmotic flow in a closed-end rectangular microchannel are examined. Specifically, the induced pressure gradient is analyzed under effects of the channel dimension and the frequency of electric field. In addition, based on the Stokes second problem, the solution of the slip velocity approximation is presented for comparison with the results obtained from the analytical scheme developed in this study. Copyright 2004 Elsevier Inc.

Origin of generalized entropies and generalized statistical mechanics for superstatistical multifractal systems

NASA Astrophysics Data System (ADS)

Gadjiev, Bahruz; Progulova, Tatiana

2015-01-01

We consider a multifractal structure as a mixture of fractal substructures and introduce a distribution function f (α), where α is a fractal dimension. Then we can introduce g(p)˜ ∫- ln p μe-yf(y)dy and show that the distribution functions f (α) in the form of f(α) = δ(α-1), f(α) = δ(α-θ) , f(α) = 1/α-1 , f(y)= y α-1 lead to the Boltzmann - Gibbs, Shafee, Tsallis and Anteneodo - Plastino entropies conformably. Here δ(x) is the Dirac delta function. Therefore the Shafee entropy corresponds to a fractal structure, the Tsallis entropy describes a multifractal structure with a homogeneous distribution of fractal substructures and the Anteneodo - Plastino entropy appears in case of a power law distribution f (y). We consider the Fokker - Planck equation for a fractal substructure and determine its stationary solution. To determine the distribution function of a multifractal structure we solve the two-dimensional Fokker - Planck equation and obtain its stationary solution. Then applying the Bayes theorem we obtain a distribution function for the entire system in the form of q-exponential function. We compare the results of the distribution functions obtained due to the superstatistical approach with the ones obtained according to the maximum entropy principle.

Relativistic anisotropic hydrodynamics

NASA Astrophysics Data System (ADS)

Alqahtani, Mubarak; Nopoush, Mohammad; Strickland, Michael

2018-07-01

In this paper we review recent progress in relativistic anisotropic hydrodynamics. We begin with a pedagogical introduction to the topic which takes into account the advances in our understanding of this topic since its inception. We consider both conformal and non-conformal systems and demonstrate how one can implement a realistic equation of state using a quasiparticle approach. We then consider the inclusion of non-spheroidal (non-ellipsoidal) corrections to leading-order anisotropic hydrodynamics and present the findings of the resulting second-order viscous anisotropic hydrodynamics framework. We compare the results obtained in both the conformal and non-conformal cases with exact solutions to the Boltzmann equation and demonstrate that, in all known cases, anisotropic hydrodynamics best reproduces the exact solutions. Based on this success, we then discuss the phenomenological application of anisotropic hydrodynamics. Along these lines, we review techniques which can be used to convert a momentum-space anisotropic fluid into hadronic degrees of freedom by generalizing the original idea of Cooper-Frye freeze-out to momentum-space anisotropic systems. And, finally, we present phenomenological results of 3 + 1 d quasiparticle anisotropic hydrodynamic simulations and compare them to experimental data produced in 2.76 TeV Pb-Pb collisions at the LHC. Our results indicate that anisotropic hydrodynamics provides a promising framework for describing the dynamics of the momentum-space anisotropic QGP created in heavy-ion collisions.

Kinetic theory for a mobile impurity in a degenerate Tonks-Girardeau gas.

PubMed

Gamayun, O; Lychkovskiy, O; Cheianov, V

2014-09-01

A kinetic theory describing the motion of an impurity particle in a degenerate Tonks-Girardeau gas is presented. The theory is based on the one-dimensional Boltzmann equation. An iterative procedure for solving this equation is proposed, leading to the exact solution in a number of special cases and to an approximate solution with the explicitly specified precision in a general case. Previously we reported that the impurity reaches a nonthermal steady state, characterized by an impurity momentum p(∞) depending on its initial momentum p(0) [E. Burovski, V. Cheianov, O. Gamayun, and O. Lychkovskiy, Phys. Rev. A 89, 041601(R) (2014)]. In the present paper the detailed derivation of p(∞)(p(0)) is provided. We also study the motion of an impurity under the action of a constant force F. It is demonstrated that if the impurity is heavier than the host particles, m(i)>m(h), damped oscillations of the impurity momentum develop, while in the opposite case, m(i)

Stability and accuracy of 3D neutron transport simulations using the 2D/1D method in MPACT

DOE Office of Scientific and Technical Information (OSTI.GOV)

Collins, Benjamin, E-mail: collinsbs@ornl.gov; Stimpson, Shane, E-mail: stimpsonsg@ornl.gov; Kelley, Blake W., E-mail: kelleybl@umich.edu

2016-12-01

A consistent “2D/1D” neutron transport method is derived from the 3D Boltzmann transport equation, to calculate fuel-pin-resolved neutron fluxes for realistic full-core Pressurized Water Reactor (PWR) problems. The 2D/1D method employs the Method of Characteristics to discretize the radial variables and a lower order transport solution to discretize the axial variable. This paper describes the theory of the 2D/1D method and its implementation in the MPACT code, which has become the whole-core deterministic neutron transport solver for the Consortium for Advanced Simulations of Light Water Reactors (CASL) core simulator VERA-CS. Several applications have been performed on both leadership-class and industry-classmore » computing clusters. Results are presented for whole-core solutions of the Watts Bar Nuclear Power Station Unit 1 and compared to both continuous-energy Monte Carlo results and plant data.« less

Stability and accuracy of 3D neutron transport simulations using the 2D/1D method in MPACT

DOE PAGES

Collins, Benjamin; Stimpson, Shane; Kelley, Blake W.; ...

2016-08-25

We derived a consistent “2D/1D” neutron transport method from the 3D Boltzmann transport equation, to calculate fuel-pin-resolved neutron fluxes for realistic full-core Pressurized Water Reactor (PWR) problems. The 2D/1D method employs the Method of Characteristics to discretize the radial variables and a lower order transport solution to discretize the axial variable. Our paper describes the theory of the 2D/1D method and its implementation in the MPACT code, which has become the whole-core deterministic neutron transport solver for the Consortium for Advanced Simulations of Light Water Reactors (CASL) core simulator VERA-CS. We also performed several applications on both leadership-class and industry-classmore » computing clusters. Results are presented for whole-core solutions of the Watts Bar Nuclear Power Station Unit 1 and compared to both continuous-energy Monte Carlo results and plant data.« less

Protein dielectric constants determined from NMR chemical shift perturbations.

PubMed

Kukic, Predrag; Farrell, Damien; McIntosh, Lawrence P; García-Moreno E, Bertrand; Jensen, Kristine Steen; Toleikis, Zigmantas; Teilum, Kaare; Nielsen, Jens Erik

2013-11-13

Understanding the connection between protein structure and function requires a quantitative understanding of electrostatic effects. Structure-based electrostatic calculations are essential for this purpose, but their use has been limited by a long-standing discussion on which value to use for the dielectric constants (ε(eff) and ε(p)) required in Coulombic and Poisson-Boltzmann models. The currently used values for ε(eff) and ε(p) are essentially empirical parameters calibrated against thermodynamic properties that are indirect measurements of protein electric fields. We determine optimal values for ε(eff) and ε(p) by measuring protein electric fields in solution using direct detection of NMR chemical shift perturbations (CSPs). We measured CSPs in 14 proteins to get a broad and general characterization of electric fields. Coulomb's law reproduces the measured CSPs optimally with a protein dielectric constant (ε(eff)) from 3 to 13, with an optimal value across all proteins of 6.5. However, when the water-protein interface is treated with finite difference Poisson-Boltzmann calculations, the optimal protein dielectric constant (ε(p)) ranged from 2 to 5 with an optimum of 3. It is striking how similar this value is to the dielectric constant of 2-4 measured for protein powders and how different it is from the ε(p) of 6-20 used in models based on the Poisson-Boltzmann equation when calculating thermodynamic parameters. Because the value of ε(p) = 3 is obtained by analysis of NMR chemical shift perturbations instead of thermodynamic parameters such as pK(a) values, it is likely to describe only the electric field and thus represent a more general, intrinsic, and transferable ε(p) common to most folded proteins.

Poisson-Boltzmann theory of the charge-induced adsorption of semi-flexible polyelectrolytes.

PubMed

Ubbink, Job; Khokhlov, Alexei R

2004-03-15

A model is suggested for the structure of an adsorbed layer of a highly charged semi-flexible polyelectrolyte on a weakly charged surface of opposite charge sign. The adsorbed phase is thin, owing to the effective reversal of the charge sign of the surface upon adsorption, and ordered, owing to the high surface density of polyelectrolyte strands caused by the generally strong binding between polyelectrolyte and surface. The Poisson-Boltzmann equation for the electrostatic interaction between the array of adsorbed polyelectrolytes and the charged surface is solved for a cylindrical geometry, both numerically, using a finite element method, and analytically within the weak curvature limit under the assumption of excess monovalent salt. For small separations, repulsive surface polarization and counterion osmotic pressure effects dominate over the electrostatic attraction and the resulting electrostatic interaction curve shows a minimum at nonzero separations on the Angstrom scale. The equilibrium density of the adsorbed phase is obtained by minimizing the total free energy under the condition of equality of chemical potential and osmotic pressure of the polyelectrolyte in solution and in the adsorbed phase. For a wide range of ionic conditions and charge densities of the charged surface, the interstrand separation as predicted by the Poisson-Boltzmann model and the analytical theory closely agree. For low to moderate charge densities of the adsorbing surface, the interstrand spacing decreases as a function of the charge density of the charged surface. Above about 0.1 M excess monovalent salt, it is only weakly dependent on the ionic strength. At high charge densities of the adsorbing surface, the interstrand spacing increases with increasing ionic strength, in line with the experiments by Fang and Yang [J. Phys. Chem. B 101, 441 (1997)]. (c) 2004 American Institute of Physics.

Lattice Boltzmann study on Kelvin-Helmholtz instability: roles of velocity and density gradients.

PubMed

Gan, Yanbiao; Xu, Aiguo; Zhang, Guangcai; Li, Yingjun

2011-05-01

A two-dimensional lattice Boltzmann model with 19 discrete velocities for compressible fluids is proposed. The fifth-order weighted essentially nonoscillatory (5th-WENO) finite difference scheme is employed to calculate the convection term of the lattice Boltzmann equation. The validity of the model is verified by comparing simulation results of the Sod shock tube with its corresponding analytical solutions [G. A. Sod, J. Comput. Phys. 27, 1 (1978).]. The velocity and density gradient effects on the Kelvin-Helmholtz instability (KHI) are investigated using the proposed model. Sharp density contours are obtained in our simulations. It is found that the linear growth rate γ for the KHI decreases by increasing the width of velocity transition layer D(v) but increases by increasing the width of density transition layer D(ρ). After the initial transient period and before the vortex has been well formed, the linear growth rates γ(v) and γ(ρ), vary with D(v) and D(ρ) approximately in the following way, lnγ(v)=a-bD(v) and γ(ρ)=c+elnD(ρ)(D(ρ)D(ρ)(E) the linear growth rate γ(ρ) does not vary significantly any more. One can use the hybrid effects of velocity and density transition layers to stabilize the KHI. Our numerical simulation results are in general agreement with the analytical results [L. F. Wang et al., Phys. Plasma 17, 042103 (2010)]. © 2011 American Physical Society

Variational Implicit Solvation with Solute Molecular Mechanics: From Diffuse-Interface to Sharp-Interface Models.

PubMed

Li, Bo; Zhao, Yanxiang

2013-01-01

Central in a variational implicit-solvent description of biomolecular solvation is an effective free-energy functional of the solute atomic positions and the solute-solvent interface (i.e., the dielectric boundary). The free-energy functional couples together the solute molecular mechanical interaction energy, the solute-solvent interfacial energy, the solute-solvent van der Waals interaction energy, and the electrostatic energy. In recent years, the sharp-interface version of the variational implicit-solvent model has been developed and used for numerical computations of molecular solvation. In this work, we propose a diffuse-interface version of the variational implicit-solvent model with solute molecular mechanics. We also analyze both the sharp-interface and diffuse-interface models. We prove the existence of free-energy minimizers and obtain their bounds. We also prove the convergence of the diffuse-interface model to the sharp-interface model in the sense of Γ-convergence. We further discuss properties of sharp-interface free-energy minimizers, the boundary conditions and the coupling of the Poisson-Boltzmann equation in the diffuse-interface model, and the convergence of forces from diffuse-interface to sharp-interface descriptions. Our analysis relies on the previous works on the problem of minimizing surface areas and on our observations on the coupling between solute molecular mechanical interactions with the continuum solvent. Our studies justify rigorously the self consistency of the proposed diffuse-interface variational models of implicit solvation.

Laser-beam scintillations for weak and moderate turbulence

NASA Astrophysics Data System (ADS)

Baskov, R. A.; Chumak, O. O.

2018-04-01

The scintillation index is obtained for the practically important range of weak and moderate atmospheric turbulence. To study this challenging range, the Boltzmann-Langevin kinetic equation, describing light propagation, is derived from first principles of quantum optics based on the technique of the photon distribution function (PDF) [Berman et al., Phys. Rev. A 74, 013805 (2006), 10.1103/PhysRevA.74.013805]. The paraxial approximation for laser beams reduces the collision integral for the PDF to a two-dimensional operator in the momentum space. Analytical solutions for the average value of PDF as well as for its fluctuating constituent are obtained using an iterative procedure. The calculated scintillation index is considerably greater than that obtained within the Rytov approximation even at moderate turbulence strength. The relevant explanation is proposed.

Laplace transform analysis of a multiplicative asset transfer model

NASA Astrophysics Data System (ADS)

Sokolov, Andrey; Melatos, Andrew; Kieu, Tien

2010-07-01

We analyze a simple asset transfer model in which the transfer amount is a fixed fraction f of the giver’s wealth. The model is analyzed in a new way by Laplace transforming the master equation, solving it analytically and numerically for the steady-state distribution, and exploring the solutions for various values of f∈(0,1). The Laplace transform analysis is superior to agent-based simulations as it does not depend on the number of agents, enabling us to study entropy and inequality in regimes that are costly to address with simulations. We demonstrate that Boltzmann entropy is not a suitable (e.g. non-monotonic) measure of disorder in a multiplicative asset transfer system and suggest an asymmetric stochastic process that is equivalent to the asset transfer model.

PCE: web tools to compute protein continuum electrostatics

PubMed Central

Miteva, Maria A.; Tufféry, Pierre; Villoutreix, Bruno O.

2005-01-01

PCE (protein continuum electrostatics) is an online service for protein electrostatic computations presently based on the MEAD (macroscopic electrostatics with atomic detail) package initially developed by D. Bashford [(2004) Front Biosci., 9, 1082–1099]. This computer method uses a macroscopic electrostatic model for the calculation of protein electrostatic properties, such as pKa values of titratable groups and electrostatic potentials. The MEAD package generates electrostatic energies via finite difference solution to the Poisson–Boltzmann equation. Users submit a PDB file and PCE returns potentials and pKa values as well as color (static or animated) figures displaying electrostatic potentials mapped on the molecular surface. This service is intended to facilitate electrostatics analyses of proteins and thereby broaden the accessibility to continuum electrostatics to the biological community. PCE can be accessed at . PMID:15980492

Simulation of diffuse-charge capacitance in electric double layer capacitors

NASA Astrophysics Data System (ADS)

Sun, Ning; Gersappe, Dilip

2017-01-01

We use a Lattice Boltzmann Model (LBM) in order to simulate diffuse-charge dynamics in Electric Double Layer Capacitors (EDLCs). Simulations are carried out for both the charge and the discharge processes on 2D systems of complex random electrode geometries (pure random, random spheres and random fibers). The steric effect of concentrated solutions is considered by using a Modified Poisson-Nernst-Planck (MPNP) equations and compared with regular Poisson-Nernst-Planck (PNP) systems. The effects of electrode microstructures (electrode density, electrode filler morphology, filler size, etc.) on the net charge distribution and charge/discharge time are studied in detail. The influence of applied potential during discharging process is also discussed. Our studies show how electrode morphology can be used to tailor the properties of supercapacitors.