Hydrodynamics of Collisionless Boltzmann Equation for a Highly Flattened Galaxy
NASA Astrophysics Data System (ADS)
Aoki, S.
The collisionless Boltzmann equation is studied in order to be connected with hydrodynamic equations. Each of the later equations can be obtained by taking a moment of the former equation. The difficulty against the system of moment equations, called closure problem, can be overtaken by a trick of neglecting the terms of higher order moments under small-pressure assumption.
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.
Exact solutions of the nonlinear Boltzmann equation
NASA Astrophysics Data System (ADS)
Ernst, Matthieu H.
1984-03-01
A review is given of research activities since 1976 on the nonlinear Boltzmann equation and related equations of Boltzmann type, in which several rediscoveries have been made and several conjectures have been disproved. Subjects are (i) the BKW solution of the Boltzmann equation for Maxwell molecules, first discovered by Krupp in 1967, and the Krook-Wu conjecture concerning the universal significance of the BKW solution for the large (v, t) behavior of the velocity distribution function f (v, t); (ii) moment equations and polynomial expansions of f (v, t); (iii) model Boltzmann equation for a spatially uniform system of very hard particles, that can be solved in closed form for general initial conditions; (iv) for Maxwell and non-Maxwell-type molecules there exist solutions of the nonlinear Boltzmann equation with algebraic decrease at υ→∞; connections with nonuniqueness and violation of conservation laws; (v) conjectured super- H-theorem and the BKW solution; (vi) exactly soluble one-dimensional Boltzmann equation with spatial dependence.
Analytic solutions of the relativistic Boltzmann equation
NASA Astrophysics Data System (ADS)
Hatta, Yoshitaka; Martinez, Mauricio; Xiao, Bo-Wen
2015-04-01
We present new analytic solutions to the relativistic Boltzmann equation within the relaxation time approximation. We first obtain spherically expanding solutions which are the kinetic counterparts of the exact solutions of the Israel-Stewart equation in the literature. This allows us to compare the solutions of the kinetic and hydrodynamic equations at an analytical level. We then derive a novel boost-invariant solution of the Boltzmann equation which has an unconventional dependence on the proper time. The existence of such a solution is also suggested in second-order hydrodynamics and fluid-gravity correspondence.
The Boltzmann equation in molecular biology.
Dubois, Jean-Marc; Ouanounou, Gilles; Rouzaire-Dubois, Béatrice
2009-01-01
In the 1870's, Ludwig Boltzmann proposed a simple equation that was based on the notion of atoms and molecules and that defined the probability of finding a molecule in a given state. Several years later, the Boltzmann equation was developed and used to calculate the equilibrium potential of an ion species that is permeant through membrane channels and to describe conformational changes of biological molecules involved in different mechanisms including: open probability of ion channels, effect of molecular crowding on protein conformation, biochemical reactions and cell proliferation. The aim of this review is to trace the history of the developments of the Boltzmann equation that account for the behaviour of proteins involved in molecular biology and physiology.
The Boltzmann equation in the difference formulation
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.
Classical non-Markovian Boltzmann equation
Alexanian, Moorad
2014-08-01
The modeling of particle transport involves anomalous diffusion, (x²(t) ) ∝ t{sup α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.
Connection Between the Lattice Boltzmann Equation and the Beam Scheme
NASA Technical Reports Server (NTRS)
Xu, Kun; Luo, Li-Shi
1999-01-01
In this paper we analyze and compare the lattice Boltzmann equation with the beam scheme in details. We notice the similarity and differences between the lattice Boltzmann equation and the beam scheme. We show that the accuracy of the lattice Boltzmann equation is indeed second order in space. We discuss the advantages and limitations of lattice Boltzmann equation and the beam scheme. Based on our analysis, we propose an improved multi-dimensional beam scheme.
Time-dependent closure relations for relativistic collisionless fluid equations
Bendib-Kalache, K.; Bendib, A.; El Hadj, K. Mohammed
2010-11-15
Linear fluid equations for relativistic and collisionless plasmas are derived. Closure relations for the fluid equations are analytically computed from the relativistic Vlasov equation in the Fourier space ({omega},k), where {omega} and k are the conjugate variables of time t and space x variables, respectively. The mathematical method used is based on the projection operator techniques and the continued fraction mathematical tools. The generalized heat flux and stress tensor are calculated for arbitrary parameter {omega}/kc where c is the speed of light, and for arbitrary relativistic parameter z=mc{sup 2}/T, where m is the particle rest mass and T, the plasma temperature in energy units.
Vlasov equation and collisionless hydrodynamics adapted to curved spacetime
Dodin, I. Y.; Fisch, N. J.
2010-11-15
The modification of the Vlasov equation, in its standard form describing a charged particle distribution in the six-dimensional phase space, is derived explicitly within a formal Hamiltonian approach for arbitrarily curved spacetime. The equation accounts simultaneously for the Lorentz force and the effects of general relativity, with the latter appearing as the gravity force and an additional force due to the extrinsic curvature of spatial hypersurfaces. For an arbitrary spatial metric, the equations of collisionless hydrodynamics are also obtained in the usual three-vector form.
Time-dependent closure relations for relativistic collisionless fluid equations.
Bendib-Kalache, K; Bendib, A; El Hadj, K Mohammed
2010-11-01
Linear fluid equations for relativistic and collisionless plasmas are derived. Closure relations for the fluid equations are analytically computed from the relativistic Vlasov equation in the Fourier space (ω,k), where ω and k are the conjugate variables of time t and space x variables, respectively. The mathematical method used is based on the projection operator techniques and the continued fraction mathematical tools. The generalized heat flux and stress tensor are calculated for arbitrary parameter ω/kc where c is the speed of light, and for arbitrary relativistic parameter z=mc²/T , where m is the particle rest mass and T, the plasma temperature in energy units.
NASA Astrophysics Data System (ADS)
Held, M.; Kendl, A.
2015-10-01
A lattice Boltzmann method (LBM) approach to the Charney-Hasegawa-Mima (CHM) model for adiabatic drift wave turbulence in magnetised plasmas is implemented. The CHM-LBM model contains a barotropic equation of state for the potential, a force term including a cross-product analogous to the Coriolis force in quasigeostrophic models, and a density gradient source term. Expansion of the resulting lattice Boltzmann model equations leads to cold-ion fluid continuity and momentum equations, which resemble CHM dynamics under drift ordering. The resulting numerical solutions of standard test cases (monopole propagation, stable drift modes and decaying turbulence) are compared to results obtained by a conventional finite difference scheme that directly discretizes the CHM equation. The LB scheme resembles characteristic CHM dynamics apart from an additional shear in the density gradient direction. The occurring shear reduces with the drift ratio and is ascribed to the compressible limit of the underlying LBM.
Solving the Boltzmann equation on GPUs
NASA Astrophysics Data System (ADS)
Frezzotti, A.; Ghiroldi, G. P.; Gibelli, L.
2011-12-01
We show how to accelerate the direct solution of the Boltzmann equation using Graphics Processing Units (GPUs). In order to fully exploit the computational power of the GPU, we choose a method of solution which combines a finite difference discretization of the free-streaming term with a Monte Carlo evaluation of the collision integral. The efficiency of the code is demonstrated by solving the two-dimensional driven cavity flow. Computational results show that it is possible to cut down the computing time of the sequential code of two order of magnitude. This makes the proposed method of solution a viable alternative to particle simulations for studying unsteady low Mach number flows.
Asymptotic-preserving Boltzmann model equations for binary gas mixture.
Liu, Sha; Liang, Yihua
2016-02-01
An improved system of Boltzmann model equations is developed for binary gas mixture. This system of model equations has a complete asymptotic preserving property that can strictly recover the Navier-Stokes equations in the continuum limit with the correct constitutive relations and the correct viscosity, thermal conduction, diffusion, and thermal diffusion coefficients. In this equation system, the self- and cross-collision terms in Boltzmann equations are replaced by single relaxation terms. In monocomponent case, this system of equations can be reduced to the commonly used Shakhov equation. The conservation property and the H theorem which are important for model equations are also satisfied by this system of model equations. PMID:26986408
Analytic Solution of the Boltzmann Equation in an Expanding System.
Bazow, D; Denicol, G S; Heinz, U; Martinez, M; Noronha, J
2016-01-15
For a massless gas with a constant cross section in a homogeneous, isotropically expanding spacetime we reformulate the relativistic Boltzmann equation as a set of nonlinear coupled moment equations. For a particular initial condition this set can be solved exactly, yielding the first analytical solution of the Boltzmann equation for an expanding system. The nonequilibrium behavior of this relativistic gas can be mapped onto that of a homogeneous, static nonrelativistic gas of Maxwell molecules. PMID:26824535
Asymptotic behaviour of the Boltzmann equation as a cosmological model
NASA Astrophysics Data System (ADS)
Lee, Ho
2016-08-01
As a Newtonian cosmological model the Vlasov-Poisson-Boltzmann system is considered, and a slightly modified Boltzmann equation, which describes the stability of an expanding universe, is derived. Asymptotic behaviour of solutions turns out to depend on the expansion of the universe, and in this paper we consider the soft potential case and will obtain asymptotic behaviour.
Lattice Boltzmann equation method for the Cahn-Hilliard equation.
Zheng, Lin; Zheng, Song; Zhai, Qinglan
2015-01-01
In this paper a lattice Boltzmann equation (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard equation (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010)]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results. PMID:25679741
Boltzmann-like and Boltzmann-Fokker-Planck equations as a foundation of behavioral models
NASA Astrophysics Data System (ADS)
Helbing, Dirk
1993-07-01
It is shown that the Boltzmann-like equations allow the formulation of a very general model for behavioral changes. This model takes into account spontaneous (or externally induced) behavioral changes and behavioral changes by pair interactions. As most important social pair interactions, imitative and avoidance processes are distinguished. The resulting model turns out to include as special cases many theoretical concepts of the social sciences. A Kramers-Moyal expansion of the Boltzmann-like equations leads to the Boltzmann- Fokker-Planck equations, which allows the introduction of “social forces” and “social fields”. A social field reflects the influence of the public opinion, social norms and trends on behavioural changes. It is not only given by external factors (the environment) but also by the interactions of the individuals. Variations of the individual behavior are taken into account by diffusion coefficients.
Lattice Boltzmann model for nonlinear convection-diffusion equations.
Shi, Baochang; Guo, Zhaoli
2009-01-01
A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonlinear heat conduction equation, and sine-Gordon equation, by using a real and complex-valued distribution function and relaxation time. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.
Properties of the Boltzmann equation in the classical approximation
Epelbaum, Thomas; Gelis, François; Tanji, Naoto; Wu, Bin
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 one has also access to the non-approximated result for comparison.
Properties of the Boltzmann equation in the classical approximation
Epelbaum, Thomas; Gelis, François; Tanji, Naoto; Wu, Bin
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
A Class of Lattice Boltzmann Models with the Energy Equation
NASA Astrophysics Data System (ADS)
Li, Yuanxiang; Xiong, Shengwu; Zou, Xiufen
In this paper a class of lattice Boltzmann models with the energy equation for simulating fluid thermodynamics are studied. The features of this class of models are that the discrete velocity set consists of multi-speed velocities and the internal energy of fluid is introduced by a multi-speed. Therefore, the energy term appears in the local equilibrium distribution functions of these models. Two examples are given in this paper. One is a 1D model and the other is a 2D model, which are used to model a shock wave tube problem and the Benard convection problem, respectively. Keywords: lattice Boltzmann model, energy equation, shock wave tube, Benard convection
Stochastic simulation algorithm for the quantum linear Boltzmann equation.
Busse, Marc; Pietrulewicz, Piotr; Breuer, Heinz-Peter; Hornberger, Klaus
2010-08-01
We develop a Monte Carlo wave function algorithm for the quantum linear Boltzmann equation, a Markovian master equation describing the quantum motion of a test particle interacting with the particles of an environmental background gas. The algorithm leads to a numerically efficient stochastic simulation procedure for the most general form of this integrodifferential equation, which involves a five-dimensional integral over microscopically defined scattering amplitudes that account for the gas interactions in a nonperturbative fashion. The simulation technique is used to assess various limiting forms of the quantum linear Boltzmann equation, such as the limits of pure collisional decoherence and quantum Brownian motion, the Born approximation, and the classical limit. Moreover, we extend the method to allow for the simulation of the dissipative and decohering dynamics of superpositions of spatially localized wave packets, which enables the study of many physically relevant quantum phenomena, occurring e.g., in the interferometry of massive particles.
Modeling adsorption with lattice Boltzmann equation
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-01-01
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied. PMID:27256325
Modeling adsorption with lattice Boltzmann equation.
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-01-01
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied. PMID:27256325
Monitoring derivation of the quantum linear Boltzmann equation
Hornberger, Klaus; Vacchini, Bassano
2008-02-15
We show how the effective equation of motion for a distinguished quantum particle in an ideal gas environment can be obtained by means of the monitoring approach introduced by Hornberger [EPL 77, 50007 (2007)]. The resulting Lindblad master equation accounts for the quantum effects of the scattering dynamics in a nonperturbative fashion and it describes decoherence and dissipation in a unified framework. It incorporates various established equations as limiting cases and reduces to the classical linear Boltzmann equation once the state is diagonal in momentum.
From Boltzmann equations to steady wall velocities
Konstandin, Thomas; Nardini, Germano; Rues, Ingo E-mail: germano.nardini@desy.de
2014-09-01
By means of a relativistic microscopic approach we calculate the expansion velocity of bubbles generated during a first-order electroweak phase transition. In particular, we use the gradient expansion of the Kadanoff-Baym equations to set up the fluid system. This turns out to be equivalent to the one found in the semi-classical approach in the non-relativistic limit. Finally, by including hydrodynamic deflagration effects and solving the Higgs equations of motion in the fluid, we determine velocity and thickness of the bubble walls. Our findings are compared with phenomenological models of wall velocities. As illustrative examples, we apply these results to three theories providing first-order phase transitions with a particle content in the thermal plasma that resembles the Standard Model.
Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics.
García de Soria, María Isabel; Maynar, Pablo; Schehr, Grégory; Barrat, Alain; Trizac, Emmanuel
2008-05-01
We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzmann operator, which characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of molecular dynamics simulations that validate the theoretical predictions. PMID:18643046
Lattice Boltzmann model for generalized nonlinear wave equations
NASA Astrophysics Data System (ADS)
Lai, Huilin; Ma, Changfeng
2011-10-01
In this paper, a lattice Boltzmann model is developed to solve a class of the nonlinear wave equations. Through selecting equilibrium distribution function and an amending function properly, the governing evolution equation can be recovered correctly according to our proposed scheme, in which the Chapman-Enskog expansion is employed. We validate the algorithm on some problems where analytic solutions are available, including the second-order telegraph equation, the nonlinear Klein-Gordon equation, and the damped, driven sine-Gordon equation. It is found that the numerical results agree well with the analytic solutions, which indicates that the present algorithm is very effective and can be used to solve more general nonlinear problems.
Physical scales in the Wigner-Boltzmann equation
Nedjalkov, M.; Selberherr, S.; Ferry, D.K.; Vasileska, D.; Dollfus, P.; Querlioz, D.; Dimov, I.; Schwaha, P.
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. 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. - 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.
Numerical solution of Boltzmann equation using discrete velocity grids
NASA Astrophysics Data System (ADS)
Vedula, Prakash
2015-11-01
An importance sampling based approach for numerical solution of the (single species) Boltzmann equation using discrete velocity grids is proposed. This approach involves a stochastic method for evaluation of the collision integral based on sampling of depleting/replenishing collisions and is designed to preserve important symmetries of the collision operator, including collision invariants. The underlying particle distribution function is represented as a collection of delta functions with associated weights that are non-negative. A key feature in the construction of the proposed method is that it ensures that the weights associated with the distribution function remain non-negative during collisional relaxation, thereby satisfying an important realizability condition. Performance of the proposed approach will be studied using test problems involving spatially homogeneous collisional relaxation flow and microchannel flows. Results obtained from the proposed method will be compared with those obtained from the (deterministic) collisional Lattice Boltzmann Method (cLBM) and the traditional direct simulation Monte Carlo (DSMC) method for solution of Boltzmann equation. Extension of the proposed method using discrete velocity grids for multicomponent mixtures will also be discussed.
Lattice Boltzmann method for the fractional advection-diffusion equation.
Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Lattice Boltzmann method for the fractional advection-diffusion equation
NASA Astrophysics Data System (ADS)
Zhou, J. G.; Haygarth, P. M.; Withers, P. J. A.; Macleod, C. J. A.; Falloon, P. D.; Beven, K. J.; Ockenden, M. C.; Forber, K. J.; Hollaway, M. J.; Evans, R.; Collins, A. L.; Hiscock, K. M.; Wearing, C.; Kahana, R.; Villamizar Velez, M. L.
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β , the fractional order α , and the single relaxation time τ , the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Lattice Boltzmann method for the fractional advection-diffusion equation.
Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering. PMID:27176431
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
NASA Astrophysics Data System (ADS)
Molnár, Etele; Niemi, Harri; Rischke, Dirk H.
2016-06-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break down. Nevertheless, one can still derive a fluid-dynamical theory, called anisotropic dissipative fluid dynamics, in terms of an expansion around a single-particle distribution function, f^0 k, which incorporates (at least parts of) the momentum anisotropy via a suitable parametrization. We construct such an expansion in terms of polynomials in energy and momentum in the direction of the anisotropy and of irreducible tensors in the two-dimensional momentum subspace orthogonal to both the fluid velocity and the direction of the anisotropy. From the Boltzmann equation we then derive the set of equations of motion for the irreducible moments of the deviation of the single-particle distribution function from f^0 k. Truncating this set via the 14-moment approximation, we obtain the equations of motion of anisotropic dissipative fluid dynamics.
Lattice Boltzmann model for the convection-diffusion equation.
Chai, Zhenhua; Zhao, T S
2013-06-01
We propose a lattice Boltzmann (LB) model for the convection-diffusion equation (CDE) and show that the CDE can be recovered correctly from the model by the Chapman-Enskog analysis. The most striking feature of the present LB model is that it enables the collision process to be implemented locally, making it possible to retain the advantage of the lattice Boltzmann method in the study of the heat and mass transfer in complex geometries. A local scheme for computing the heat and mass fluxes is then proposed to replace conventional nonlocal finite-difference schemes. We further validate the present model and the local scheme for computing the flux against analytical solutions to several classical problems, and we show that both the model for the CDE and the computational scheme for the flux have a second-order convergence rate in space. It is also demonstrated the present model is more accurate than existing LB models for the CDE.
Lattice Boltzmann model for a steady radiative transfer equation.
Yi, Hong-Liang; Yao, Feng-Ju; Tan, He-Ping
2016-08-01
A complete lattice Boltzmann model (LBM) is proposed for the steady radiative transfer equation (RTE). The RTE can be regarded as a pure convection equation with a source term. To derive the expressions for the equilibrium distribution function and the relaxation time, an artificial isotropic diffusion term is introduced to form a convection-diffusion equation. When the dimensionless relaxation time has a value of 0.5, the lattice Boltzmann equation (LBE) is exactly applicable to the original steady RTE. We also perform a multiscale analysis based on the Chapman-Enskog expansion to recover the macroscopic RTE from the mesoscopic LBE. The D2Q9 model is used to solve the LBE, and the numerical results obtained by the LBM are comparable to the results obtained by other methods or analytical solutions, which demonstrates that the proposed model is highly accurate and stable in simulating multidimensional radiative transfer. In addition, we find that the convergence rate of the LBM depends on the transport properties of RTE: for diffusion-dominated RTE with a large optical thickness, the LBM shows a second-order convergence rate in space, while for convection-dominated RTE with a small optical thickness, a lower convergence rate is observed. PMID:27627417
Lattice Boltzmann model for a steady radiative transfer equation
NASA Astrophysics Data System (ADS)
Yi, Hong-Liang; Yao, Feng-Ju; Tan, He-Ping
2016-08-01
A complete lattice Boltzmann model (LBM) is proposed for the steady radiative transfer equation (RTE). The RTE can be regarded as a pure convection equation with a source term. To derive the expressions for the equilibrium distribution function and the relaxation time, an artificial isotropic diffusion term is introduced to form a convection-diffusion equation. When the dimensionless relaxation time has a value of 0.5, the lattice Boltzmann equation (LBE) is exactly applicable to the original steady RTE. We also perform a multiscale analysis based on the Chapman-Enskog expansion to recover the macroscopic RTE from the mesoscopic LBE. The D2Q9 model is used to solve the LBE, and the numerical results obtained by the LBM are comparable to the results obtained by other methods or analytical solutions, which demonstrates that the proposed model is highly accurate and stable in simulating multidimensional radiative transfer. In addition, we find that the convergence rate of the LBM depends on the transport properties of RTE: for diffusion-dominated RTE with a large optical thickness, the LBM shows a second-order convergence rate in space, while for convection-dominated RTE with a small optical thickness, a lower convergence rate is observed.
Matrix-valued Boltzmann equation for the Hubbard chain.
Fürst, Martin L R; Mendl, Christian B; Spohn, Herbert
2012-09-01
We study, both analytically and numerically, the Boltzmann transport equation for the Hubbard chain with nearest-neighbor hopping and spatially homogeneous initial condition. The time-dependent Wigner function is matrix-valued because of spin. The H theorem holds. The nearest-neighbor chain is integrable, which, on the kinetic level, is reflected by infinitely many additional conservation laws and linked to the fact that there are also nonthermal stationary states. We characterize all stationary solutions. Numerically, we observe an exponentially fast convergence to stationarity and investigate the convergence rate in dependence on the initial conditions.
Adaptive Mesh Enrichment for the Poisson-Boltzmann Equation
NASA Astrophysics Data System (ADS)
Dyshlovenko, Pavel
2001-09-01
An adaptive mesh enrichment procedure for a finite-element solution of the two-dimensional Poisson-Boltzmann equation is described. The mesh adaptation is performed by subdividing the cells using information obtained in the previous step of the solution and next rearranging the mesh to be a Delaunay triangulation. The procedure allows the gradual improvement of the quality of the solution and adjustment of the geometry of the problem. The performance of the proposed approach is illustrated by applying it to the problem of two identical colloidal particles in a symmetric electrolyte.
On the Krook-Wu model of the Boltzmann equation
NASA Astrophysics Data System (ADS)
Cornille, H.
1980-08-01
The distribution function of the Krook-Wu model of the nonlinear Boltzmann equation (elastic differential cross sections inversely proportional to the relative speed of the colliding particles) is obtained as a generalized Laguerre polynomial expansion where the only time dependence is provided by the coefficients. In a recent paper M. Barnsley and the present author have shown that these coefficients are recursively determined from the resolution of a nonlinear differential system. Here we explicitly show how to construct the solutions of the Krook-Wu model and study the properties of the corresponding Krook-Wu distribution functions.
Generalizing the Boltzmann equation in complex phase space.
Zadehgol, Abed
2016-08-01
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014)JCTPAH0021-999110.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015)PLEEE81539-375510.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others. PMID:27627421
Generalizing the Boltzmann equation in complex phase space
NASA Astrophysics Data System (ADS)
Zadehgol, Abed
2016-08-01
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014), 10.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015), 10.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others.
Thermal Curved Boundary Treatment for the Thermal Lattice Boltzmann Equation
NASA Astrophysics Data System (ADS)
Huang, Haibo; Lee, T. S.; Shu, C.
In this paper, a recent curved non-slip wall boundary treatment for isothermal Lattice Boltzmann equation (LBE) [Z. Guo, C. Zheng and B. Shi, Phys. Fluids 14(6) (2002)] is extended to handle the thermal curved wall boundary for a double-population thermal lattice Boltzmann equation (TLBE). The unknown distribution population at a wall node which is necessary to fulfill streaming step is decomposed into its equilibrium and non-equilibrium parts. The equilibrium part is evaluated according to Dirichlet and Neumann boundary constraints, and the non-equilibrium part is obtained using a first-order extrapolation from fluid lattices. To validate the thermal boundary condition treatment, we carry out numerical simulations of Couette flow between two circular cylinders, the natural convection in a square cavity, and the natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder. The results agree very well with analytical solution or available data in the literature. Our numerical results also demonstrate that the TLBE together with the present boundary scheme is of second-order accuracy.
Boltzmann equation solver adapted to emergent chemical non-equilibrium
Birrell, Jeremiah; Wilkening, Jon; Rafelski, Johann
2015-01-15
We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow for emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature T(t) and phase space occupation factor ϒ(t). In this first paper we address (effectively) massless fermions and derive dynamical equations for T(t) and ϒ(t) such that the zeroth order term of the basis alone captures the particle number density and energy density of each particle distribution. We validate our method and illustrate the reduced computational cost and the ability to easily represent final state chemical non-equilibrium by studying a model problem that is motivated by the physics of the neutrino freeze-out processes in the early Universe, where the essential physical characteristics include reheating from another disappearing particle component (e{sup ±}-annihilation)
NASA Astrophysics Data System (ADS)
Saveliev, V. L.
2011-05-01
Pair collisions is the main interaction process in the Boltzmann gas dynamics. By making use of exactly the same physical assumptions as was used by Ludwig Boltzmann we write the kinetic equation for two-particle distribution function of molecules in the gas mixtures. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. Because the scattering operator is more simple then Boltzmann collision integral this equation opens new opportunities for mathematical description of the Boltzmann gas dynamics.
Global Solutions of the Boltzmann Equation Over {{R}^D} Near Global Maxwellians with Small Mass
NASA Astrophysics Data System (ADS)
Bardos, Claude; Gamba, Irene M.; Golse, François; Levermore, C. David
2016-09-01
We study the dynamics defined by the Boltzmann equation set in the Euclidean space {{R}^D} in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory.
Global Solutions of the Boltzmann Equation Over {{R}^D} Near Global Maxwellians with Small Mass
NASA Astrophysics Data System (ADS)
Bardos, Claude; Gamba, Irene M.; Golse, François; Levermore, C. David
2016-07-01
We study the dynamics defined by the Boltzmann equation set in the Euclidean space {{R}^D} in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory.
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.
Conservation laws and exact solutions of the Boltzmann equation
Mattis, D.C.; Szpilka, A.M.; Chen, H.
1989-03-10
The distribution function f which satisfies the time-dependent Boltzmann equation (BE) for a Lorentz model with perfectly elastic random scatterers is proved nonnegative, and is computed exactly when backscattering dominates. Joule heating and Ohm's law are recovered, although f has no steady-state limit, contrary to the relaxation-time approximation. (The conventional approximation to the time-independent BE also yields OHm's law but not the Joule heating and, worse, it unphysically predicts f < O.) The exact solution is compared with various effective-temperature approximations, and is shown to remain very nearly unchanged over a wide range of times even in the presence of a small amount of inelastic scattering.
Lattice Boltzmann equation for microscale gas flows of binary mixtures.
Guo, Zhaoli; Asinari, Pietro; Zheng, Chuguang
2009-02-01
Modeling and simulating gas flows in and around microdevices are a challenging task in both science and engineering. In practical applications, a gas is usually a mixture made of different components. In this paper we propose a lattice Boltzmann equation (LBE) model for microscale flows of a binary mixture based on a recently developed LBE model for continuum mixtures [P. Asinari and L.-S. Luo, J. Comput. Phys. 227, 3878 (2008)]. A consistent boundary condition for gas-solid interactions is proposed and analyzed. The LBE is validated and compared with theoretical results or other reported data. The results show that the model can serve as a potential method for flows of binary mixture in the microscale.
Nanowire Conductivity: A Numerical Solution of the Boltzmann Equation
NASA Astrophysics Data System (ADS)
Sundaram, Venkat; Mizel, Ari
2002-03-01
The conduction properties of large nanowires (diameter 100nm), are calculated by a direct numerical solution of the Boltzmann equation. Nanowires are modelled by distinguishing a bulk-like interior region from a surface region of finite width and enhanced scattering. Of particular interest is the investigation of a negative longitudinal magnetoresistance in such a model, a result expected from kinetic-theoretical considerations. These calculations can be performed both using the relaxation time approximation and the collision integral, with a view to relating model parameters in either approach to experimental results. Such a direct numerical solution offers the advantage of significant freedom in incorporating nanowire characteristics and conditions such as the equilibrium electron density and temperature profile, or the presence of defects and impurities.
Boltzmann equation analysis of spatiotemporal electron swarm development
NASA Astrophysics Data System (ADS)
Ould Mohamed Mahmoud, M.; Yousfi, M.
1997-05-01
A powerful and a stable numerical method is developed to solve the Boltzmann equation of electrons moving under the action of an electric field in weakly ionized gases involving space and time gradients. It is based on the classical two term development of the distribution function and on a strongly implicit procedure following position and energy axis and an explicit approach along the time axis. This numerical algorithm is successfully applied to determine the spatiotemporal variation of the electron distribution function and the associated swarm parameters (mean energy, drift velocity, ionization and attachment coefficients, etc.) in the case of nonthermal electrical discharges in different gases (He, Ar and O2) under different applied electric fields and initial and boundary conditions. The transient phase, the following steady state phase and also the electrode effects are clearly emphasized and analyzed for each gas discharge studied.
Lin, X.
1991-01-01
This paper reports the development of an object-oriented programming methodology for particle simulations. It is established on the (m reductionist) view that many physical phenomena cana be reduced to many-body problems. By doing the reduction, many seemly unrelated physical phenomena can be simulated in a systematic way and a high-level programming system can be constructed to facilitate the programming and the solution of the simulations. In the object-oriented particle simulation methodology, a hierarchy of abstract particles is defined to represent a variety of characteristics in physical system simulations. A simulation program is constructed from particles derived from the abstract particles. The object- oriented particle simulation methodology provides a unifying modeling and simulation framework for a variety of simulation applications with the use of particle methods. It allows easy composition of simulation programs from predefined software modules and facilitates software reusability. It greatly increase the productivity of simulation program constructions. Boltzmann (after Ludwig Boltzmann, 1844-1906) is a prototype programming system in the object-oriented particle simulation methodology. Boltzmann is implemented in C++ and the X Window System. It contains a library of data types and functions that support simulations in particle methods. Moreover, it provides a visualization window to support friendly user-computer interaction. Examples of the application of the Boltzmann programming system are presented. The effectiveness of the object-oriented particle simulation methodology is demonstrated. A user's manual is included in the appendix.
The role of electron equation of state in heating partition of protons in a collisionless plasma
Parashar, Tulasi N.; Vasquez, Bernard J.; Markovskii, Sergei A.
2014-02-15
One of the outstanding questions related to the solar wind is the heating of solar wind plasma. Addressing this question requires a self consistent treatment of the kinetic physics of a collisionless plasma. A hybrid code (with particle ions and fluid electrons) is one of the most convenient computational tools, which allows us to explore self consistent ion kinetics, while saving us computational time as compared to the full particle in cell codes. A common assumption used in hybrid codes is that of isothermal electrons. In this paper, we discuss the role that the equation of state for electrons could potentially play in determining the ion kinetics.
Multigrid lattice Boltzmann method for accelerated solution of elliptic equations
NASA Astrophysics Data System (ADS)
Patil, Dhiraj V.; Premnath, Kannan N.; Banerjee, Sanjoy
2014-05-01
A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice Boltzmann method (LBM) and the multigrid (MG) technique is presented. Several benchmark elliptic equations are solved numerically with the inclusion of multiple grid-levels in two-dimensional domains at an optimal computational cost within the LB framework. The results are compared with the corresponding analytical solutions and numerical solutions obtained using the Stone's strongly implicit procedure. The classical PDEs considered in this article include the Laplace and Poisson equations with Dirichlet boundary conditions, with the latter involving both constant and variable coefficients. A detailed analysis of solution accuracy, convergence and computational efficiency of the proposed solver is given. It is observed that the use of a high-order stencil (for smoothing) improves convergence and accuracy for an equivalent number of smoothing sweeps. The effect of the type of scheduling cycle (V- or W-cycle) on the performance of the MG-LBM is analyzed. Next, a parallel algorithm for the MG-LBM solver is presented and then its parallel performance on a multi-core cluster is analyzed. Lastly, a practical example is provided wherein the proposed elliptic PDE solver is used to compute the electro-static potential encountered in an electro-chemical cell, which demonstrates the effectiveness of this new solver in complex coupled systems. Several orders of magnitude gains in convergence and parallel scaling for the canonical problems, and a factor of 5 reduction for the multiphysics problem are achieved using the MG-LBM.
Multilevel Methods for the Poisson-Boltzmann Equation
NASA Astrophysics Data System (ADS)
Holst, Michael Jay
We consider the numerical solution of the Poisson -Boltzmann equation (PBE), a three-dimensional second order nonlinear elliptic partial differential equation arising in biophysics. This problem has several interesting features impacting numerical algorithms, including discontinuous coefficients representing material interfaces, rapid nonlinearities, and three spatial dimensions. Similar equations occur in various applications, including nuclear physics, semiconductor physics, population genetics, astrophysics, and combustion. In this thesis, we study the PBE, discretizations, and develop multilevel-based methods for approximating the solutions of these types of equations. We first outline the physical model and derive the PBE, which describes the electrostatic potential of a large complex biomolecule lying in a solvent. We next study the theoretical properties of the linearized and nonlinear PBE using standard function space methods; since this equation has not been previously studied theoretically, we provide existence and uniqueness proofs in both the linearized and nonlinear cases. We also analyze box-method discretizations of the PBE, establishing several properties of the discrete equations which are produced. In particular, we show that the discrete nonlinear problem is well-posed. We study and develop linear multilevel methods for interface problems, based on algebraic enforcement of Galerkin or variational conditions, and on coefficient averaging procedures. Using a stencil calculus, we show that in certain simplified cases the two approaches are equivalent, with different averaging procedures corresponding to different prolongation operators. We also develop methods for nonlinear problems based on a nonlinear multilevel method, and on linear multilevel methods combined with a globally convergent damped-inexact-Newton method. We derive a necessary and sufficient descent condition for the inexact-Newton direction, enabling the development of extremely
Biomolecular electrostatics with the linearized Poisson-Boltzmann equation.
Fogolari, F; Zuccato, P; Esposito, G; Viglino, P
1999-01-01
Electrostatics plays a key role in many biological processes. The Poisson-Boltzmann equation (PBE) and its linearized form (LPBE) allow prediction of electrostatic effects for biomolecular systems. The discrepancies between the solutions of the PBE and those of the LPBE are well known for systems with a simple geometry, but much less for biomolecular systems. Results for high charge density systems show that there are limitations to the applicability of the LPBE at low ionic strength and, to a lesser extent, at higher ionic strength. For systems with a simple geometry, the onset of nonlinear effects has been shown to be governed by the ratio of the electric field over the Debye screening constant. This ratio is used in the present work to correct the LPBE results to reproduce fairly accurately those obtained from the PBE for systems with a simple geometry. Since the correction does not involve any geometrical parameter, it can be easily applied to real biomolecular systems. The error on the potential for the LPBE (compared to the PBE) spans few kT/q for the systems studied here and is greatly reduced by the correction. This allows for a more accurate evaluation of the electrostatic free energy of the systems. PMID:9876118
Gamba, Irene M.; Haack, Jeffrey R.
2014-08-01
We present the formulation of a conservative spectral method for the Boltzmann collision operator with anisotropic scattering cross-sections. The method is an extension of the conservative spectral method of Gamba and Tharkabhushanam [17,18], which uses the weak form of the collision operator to represent the collisional term as a weighted convolution in Fourier space. The method is tested by computing the collision operator with a suitably cut-off angular cross section and comparing the results with the solution of the Landau equation. We analytically study the convergence rate of the Fourier transformed Boltzmann collision operator in the grazing collisions limit to the Fourier transformed Landau collision operator under the assumption of some regularity and decay conditions of the solution to the Boltzmann equation. Our results show that the angular singularity which corresponds to the Rutherford scattering cross section is the critical singularity for which a grazing collision limit exists for the Boltzmann operator. Additionally, we numerically study the differences between homogeneous solutions of the Boltzmann equation with the Rutherford scattering cross section and an artificial cross section, which give convergence to solutions of the Landau equation at different asymptotic rates. We numerically show the rate of the approximation as well as the consequences for the rate of entropy decay for homogeneous solutions of the Boltzmann equation and Landau equation.
The lattice Boltzmann model for the second-order Benjamin-Ono equations
NASA Astrophysics Data System (ADS)
Lai, Huilin; Ma, Changfeng
2010-04-01
In this paper, in order to extend the lattice Boltzmann method to deal with more complicated nonlinear equations, we propose a 1D lattice Boltzmann scheme with an amending function for the second-order (1 + 1)-dimensional Benjamin-Ono equation. With the Taylor expansion and the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The equilibrium distribution function and the amending function are obtained. Numerical simulations are carried out for the 'good' Boussinesq equation and the 'bad' one to validate the proposed model. It is found that the numerical results agree well with the analytical solutions. The present model can be used to solve more kinds of nonlinear partial differential equations.
Progress in the understanding of the fluctuating lattice Boltzmann equation
NASA Astrophysics Data System (ADS)
Dünweg, Burkhard; Schiller, Ulf D.; Ladd, Anthony J. C.
2009-04-01
We give a brief account of the development of methods to include thermal fluctuations into lattice Boltzmann algorithms. Emphasis is put on our recent work [B. Dünweg, U.D. Schiller, A.J.C. Ladd, Phys. Rev. E 76 (2007) 036704] which provides a clear understanding in terms of statistical mechanics.
On a derivation of the Boltzmann equation in Quantum Field Theory
NASA Astrophysics Data System (ADS)
Leiler, Gregor
The Boltzmann equation (BE) is a commonly used tool for the study of non-equilibrium many particle systems. It has been introduced in 1872 by Ludwig Boltzmann and has been widely generalized throughout the years. Today it is commonly used in physical applications, from the study of ordinary fluids to problems in particle Cosmology where Quantum Field Theoretical techniques are essential. Despite its numerous experimental successes, the conceptual basis of the BE is not entirely clear. For instance, it is well known that it is not a fundamental equation of physics like, say, the Heisenberg equation (HE). A natural question then arises whether it is possible to derive the BE from physical first principles, i.e. the Heisenberg equation in Quantum Field Theory. In this work we attempted to answer this question and succeeded in deriving the BE from the HE, thus further clarifying its conceptual status. In particular, the results we have obtained are as follows. Firstly, we establish the non-perturbative validity of what we call the "pre-Boltzmann equation". The crucial point here is that this latter equation is equivalent to the Heisenberg equation. Secondly, we proceed to consider various limits of the pre-Boltzmann equation, namly the "low density" and the "weak coupling" limits, to obtain two equations that can be considered as generalizations of the BE. These limits are always taken together with the "long time" limit, which allows us to interpret the BE as an appropriate long time limit of the HE. The generalization we obtain consists in additional "correction" terms to the usual Boltzmann collision factor, and can be associated to multiple particle scattering. Unlike the pre-Boltzmann equation, these latter results are only valid pertubatively. Finally, we briefly consider the possibility to extend these results beyond said limits and outline some important aspects in this case.
Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow.
Guo, Zhaoli; Zheng, Chuguang; Shi, Baochang
2008-03-01
The standard lattice Boltzmann equation (LBE) is inadequate for simulating gas flows with a large Knudsen number. In this paper we propose a generalized lattice Boltzmann equation with effective relaxation times based on a recently developed generalized Navier-Stokes constitution [Guo, Europhys Lett. 80, 24001 (2007)] for nonequilibrium flows. A kinetic boundary condition corresponding to a generalized second-order slip scheme is also designed for the model. The LBE model and the boundary condition are analyzed for a unidirectional flow, and it is found that in order to obtain the generalized Navier-Stokes equations, the relaxation times must be properly chosen and are related to the boundary condition. Numerical results show that the proposed method is able to capture the Knudsen layer phenomenon and can yield improved predictions in comparison with the standard lattice Boltzmann equation.
A discrete velocity direction model for the Boltzmann equation and applications to micro gas flows
NASA Astrophysics Data System (ADS)
Zhang, Zhenyu; Xu, Jianzhong; Qi, Zhiguo; Xi, Guang
2008-05-01
A discrete velocity direction model for the Boltzmann equation is proposed in this paper, which provides an alternative technique to the rarefied gas flows. In this model, the directions of molecular velocities are discrete, which are restricted in eight fixed directions, while the molecular speed rate is still continuous. By this approximation, the Boltzmann equation in the six-dimensional phase space is replaced by eight differential-integral equations in three-dimensional space. Thus, the computational cost is reduced greatly by reduction of three dimensions. The number of discrete velocities is not fixed in the present model because the speed rate can be truncated arbitrarily. This is distinguished from the conventional discrete velocity models (DVM). To test this technique, it was applied to the Couette flow and Poiseuille flow. The computed results agree well with those by the linearized Boltzmann equation and the DSMC method.
An asymptotic preserving Monte Carlo method for the multispecies Boltzmann equation
NASA Astrophysics Data System (ADS)
Zhang, Bin; Liu, Hong; Jin, Shi
2016-01-01
An asymptotic preserving (AP) scheme is efficient in solving multiscale kinetic equations with a wide range of the Knudsen number. In this paper, we generalize the asymptotic preserving Monte Carlo method (AP-DSMC) developed in [25] to the multispecies Boltzmann equation. This method is based on the successive penalty method [26] originated from the BGK-penalization-based AP scheme developed in [7]. For the multispecies Boltzmann equation, the penalizing Maxwellian should use the unified Maxwellian as suggested in [12]. We give the details of AP-DSMC for multispecies Boltzmann equation, show its AP property, and verify through several numerical examples that the scheme can allow time step much larger than the mean free time, thus making it much more efficient for flows with possibly small Knudsen numbers than the classical DSMC.
Lattice Boltzmann method for solving the bioheat equation.
Zhang, Haifeng
2008-02-01
In this work, we develop the lattice Boltzmann method (LBM) as a potential solver for the bioheat problems. The accuracy of the present LBM algorithm is validated through comparison with the analytical solution and the finite element simulation. The results show that the LBM can give a precise prediction of the temperature distribution, and it is efficient to deal with the space- and time-dependent heat source, which are often encountered in the treatment planning of tumor hyperthermia.
A New Regularization Mechanism for the Boltzmann Equation Without Cut-Off
NASA Astrophysics Data System (ADS)
Silvestre, Luis
2016-11-01
We apply recent results on regularity for general integro-differential equations to derive a priori estimates in Hölder spaces for the space homogeneous Boltzmann equation in the non cut-off case. We also show an a priori estimate in {L^∞} which applies in the space inhomogeneous case as well, provided that the macroscopic quantities remain bounded.
A note on a Discrete Boltzmann Equation with multiple collisions
NASA Astrophysics Data System (ADS)
Oliveira, Filipe; Soares, Ana Jacinta
2008-05-01
We compute a non-trivial explicit solution for the one-dimensional plane 6-velocity discrete Boltzmann model with multiple collisions introduced in [E. Longo, R. Monaco, On the discrete kinetic theory with multiple collisions: Plane six-velocity and unsteady Couette flow, in: Muntz, et al. (Eds.), The Proceedings of Rarefied Gas Dynamics, in: AIAA Publ., vol. 118, 1989, pp. 118-130] which asymptotically connects two particular equilibrium states. We prove that such a solution exists provided that a suitable condition on the differential elastic cross sections holds.
The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential
NASA Astrophysics Data System (ADS)
Liu, Shuangqian; Yang, Xiongfeng
2016-08-01
Boundary effects are central to the dynamics of the dilute particles governed by the Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for the Boltzmann equation with a soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate {L^∞} analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in {L^∞} space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori {L^∞} estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the {L^2-L^∞} theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted {L^∞} theory so that we could deal with the greater difficulties stemming from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove that the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in {L^∞} space with the aid of the L 2 theory and a bootstrap argument. These methods, in the latter case, can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.
An H Theorem for Boltzmann's Equation for the Yard-Sale Model of Asset Exchange
NASA Astrophysics Data System (ADS)
Boghosian, Bruce M.; Johnson, Merek; Marcq, Jeremy A.
2015-12-01
In recent work (Boghosian, Phys Rev E 89:042804-042825, 2014; Boghosian, Int J Mod Phys 25:1441008-1441015, 2014), Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy—complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an H function of both the Boltzmann and Fokker-Planck equations for the model.
Arnold, J.; Kosson, D.S.; Garrabrants, A.; Meeussen, J.C.L.; Sloot, H.A. van der
2013-02-15
A robust numerical solution of the nonlinear Poisson-Boltzmann equation for asymmetric polyelectrolyte solutions in discrete pore geometries is presented. Comparisons to the linearized approximation of the Poisson-Boltzmann equation reveal that the assumptions leading to linearization may not be appropriate for the electrochemical regime in many cementitious materials. Implications of the electric double layer on both partitioning of species and on diffusive release are discussed. The influence of the electric double layer on anion diffusion relative to cation diffusion is examined.
NASA Astrophysics Data System (ADS)
Watanabe, Hirofumi; Okiyama, Yoshio; Nakano, Tatsuya; Tanaka, Shigenori
2010-11-01
We developed FMO-PB method, which incorporates solvation effects into the Fragment Molecular Orbital calculation with the Poisson-Boltzmann equation. This method retains good accuracy in energy calculations with reduced computational time. We calculated the solvation free energies for polyalanines, Alpha-1 peptide, tryptophan cage, and complex of estrogen receptor and 17 β-estradiol to show the applicability of this method for practical systems. From the calculated results, it has been confirmed that the FMO-PB method is useful for large biomolecules in solution. We also discussed the electric charges which are used in solving the Poisson-Boltzmann equation.
High-accuracy deterministic solution of the Boltzmann equation for the shock wave structure
NASA Astrophysics Data System (ADS)
Malkov, E. A.; Bondar, Ye. A.; Kokhanchik, A. A.; Poleshkin, S. O.; Ivanov, M. S.
2015-07-01
A new deterministic method of solving the Boltzmann equation has been proposed. The method has been employed in numerical studies of the plane shock wave structure in a hard sphere gas. Results for Mach numbers and have been compared with predictions of the direct simulation Monte Carlo (DSMC) method, which has been used to obtain the reference solution. Particular attention in estimating the solution accuracy has been paid to a fine structural effect: the presence of a total temperature peak exceeding the temperature value further downstream. The results of solving the Boltzmann equation for the shock wave structure are in excellent agreement with the DSMC predictions.
Equations of motion of test particles for solving the spin-dependent Boltzmann-Vlasov equation
NASA Astrophysics Data System (ADS)
Xia, Yin; Xu, Jun; Li, Bao-An; Shen, Wen-Qing
2016-08-01
A consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-dependent Boltzmann-Vlasov equation is presented. The resulting EOMs in phase space are similar to the canonical equations in Hamiltonian dynamics, and the EOM of spin is the same as that in the Heisenburg picture of quantum mechanics. Considering further the quantum nature of spin and choosing the direction of total angular momentum in heavy-ion reactions as a reference of measuring nucleon spin, the EOMs of spin-up and spin-down nucleons are given separately. The key elements affecting the spin dynamics in heavy-ion collisions are identified. The resulting EOMs provide a solid foundation for using the test-particle approach in studying spin dynamics in heavy-ion collisions at intermediate energies. Future comparisons of model simulations with experimental data will help to constrain the poorly known in-medium nucleon spin-orbit coupling relevant for understanding properties of rare isotopes and their astrophysical impacts.
Numerical investigations of low-density nozzle flow by solving the Boltzmann equation
NASA Technical Reports Server (NTRS)
Deng, Zheng-Tao; Liaw, Goang-Shin; Chou, Lynn Chen
1995-01-01
A two-dimensional finite-difference code to solve the BGK-Boltzmann equation has been developed. The solution procedure consists of three steps: (1) transforming the BGK-Boltzmann equation into two simultaneous partial differential equations by taking moments of the distribution function with respect to the molecular velocity u(sub z), with weighting factors 1 and u(sub z)(sup 2); (2) solving the transformed equations in the physical space based on the time-marching technique and the four-stage Runge-Kutta time integration, for a given discrete-ordinate. The Roe's second-order upwind difference scheme is used to discretize the convective terms and the collision terms are treated as source terms; and (3) using the newly calculated distribution functions at each point in the physical space to calculate the macroscopic flow parameters by the modified Gaussian quadrature formula. Repeating steps 2 and 3, the time-marching procedure stops when the convergent criteria is reached. A low-density nozzle flow field has been calculated by this newly developed code. The BGK Boltzmann solution and experimental data show excellent agreement. It demonstrated that numerical solutions of the BGK-Boltzmann equation are ready to be experimentally validated.
Solutions of Boltzmann`s Equation for Mono-energetic Neutrons in an Infinite Homogeneous Medium
DOE R&D Accomplishments Database
Wigner, E. P.
1943-11-30
Boltzman's equation is solved for the case of monoenergetic neutrons created by a plane or point source in an infinite medium which has spherically symmetric scattering. The customary solution of the diffusion equation appears to be multiplied by a constant factor which is smaller than 1. In addition to this term the total neutron density contains another term which is important in the neighborhood of the source. It varies as 1/r{sup 2} in the neighborhood of a point source. (auth)
A Revisiting of the -Stability Theory of the Boltzmann Equation Near Global Maxwellians
NASA Astrophysics Data System (ADS)
Ha, Seung-Yeal; Xiao, Qinghua
2015-07-01
We study the -stability theory of the Boltzmann equation near a global Maxwellian. When an initial datum is a perturbation of a global Maxwellian, we show that the -distance between two classical solutions can be controlled by the initial data in a Lipschitz manner, which illustrates the Lipschitz continuity of the solution operator for the Boltzmann equation in -topology. Our local-in-time -stability results cover cutoff very soft potentials as well as non-cutoff hard and soft potentials. These cases were not treated in the previous work (Ha et al. in Arch Ration Mech Anal 197:657-688, 2010). Thus, our results together with the results in Ha et al. (2010) complete the -stability theory for the Boltzmann equation near a global Maxwellian. For this -stability estimate, we use the coercivity estimate of the linearized collision operator, the smallness of perturbation in a mixed Lebesgue norm, and Strichartz-type estimates of perturbation. We also show that for all classical solutions available in the literature, the Lipschitz constant can be chosen as independent of time to obtain the uniform -stability of the Boltzmann equation.
The charge conserving Poisson-Boltzmann equations: Existence, uniqueness, and maximum principle
Lee, Chiun-Chang
2014-05-15
The present article is concerned with the charge conserving Poisson-Boltzmann (CCPB) equation in high-dimensional bounded smooth domains. The CCPB equation is a Poisson-Boltzmann type of equation with nonlocal coefficients. First, under the Robin boundary condition, we get the existence of weak solutions to this equation. The main approach is variational, based on minimization of a logarithm-type energy functional. To deal with the regularity of weak solutions, we establish a maximum modulus estimate for the standard Poisson-Boltzmann (PB) equation to show that weak solutions of the CCPB equation are essentially bounded. Then the classical solutions follow from the elliptic regularity theorem. Second, a maximum principle for the CCPB equation is established. In particular, we show that in the case of global electroneutrality, the solution achieves both its maximum and minimum values at the boundary. However, in the case of global non-electroneutrality, the solution may attain its maximum value at an interior point. In addition, under certain conditions on the boundary, we show that the global non-electroneutrality implies pointwise non-electroneutrality.
Radiative or neutron transport modeling using a lattice Boltzmann equation framework
NASA Astrophysics Data System (ADS)
Bindra, H.; Patil, D. V.
2012-07-01
In this paper, the lattice Boltzmann equation (LBE)-based framework is used to obtain the solution for the linear radiative or neutron transport equation. The LBE framework is devised for the integrodifferential forms of these equations which arise due to the inclusion of the scattering terms. The interparticle collisions are neglected, hence omitting the nonlinear collision term. Furthermore, typical representative examples for one-dimensional or two-dimensional geometries and inclusion or exclusion of the scattering term (isotropic and anisotropic) in the Boltzmann transport equation are illustrated to prove the validity of the method. It has been shown that the solution from the LBE methodology is equivalent to the well-known Pn and Sn methods. This suggests that the LBE can potentially provide a more convenient and easy approach to solve the physical problems of neutron and radiation transport.
Radiative or neutron transport modeling using a lattice Boltzmann equation framework.
Bindra, H; Patil, D V
2012-07-01
In this paper, the lattice Boltzmann equation (LBE)-based framework is used to obtain the solution for the linear radiative or neutron transport equation. The LBE framework is devised for the integrodifferential forms of these equations which arise due to the inclusion of the scattering terms. The interparticle collisions are neglected, hence omitting the nonlinear collision term. Furthermore, typical representative examples for one-dimensional or two-dimensional geometries and inclusion or exclusion of the scattering term (isotropic and anisotropic) in the Boltzmann transport equation are illustrated to prove the validity of the method. It has been shown that the solution from the LBE methodology is equivalent to the well-known P(n) and S(n) methods. This suggests that the LBE can potentially provide a more convenient and easy approach to solve the physical problems of neutron and radiation transport.
Rapid scatter estimation for CBCT using the Boltzmann transport equation
NASA Astrophysics Data System (ADS)
Sun, Mingshan; Maslowski, Alex; Davis, Ian; Wareing, Todd; Failla, Gregory; Star-Lack, Josh
2014-03-01
Scatter in cone-beam computed tomography (CBCT) is a significant problem that degrades image contrast, uniformity and CT number accuracy. One means of estimating and correcting for detected scatter is through an iterative deconvolution process known as scatter kernel superposition (SKS). While the SKS approach is efficient, clinically significant errors on the order 2-4% (20-40 HU) still remain. We have previously shown that the kernel method can be improved by perturbing the kernel parameters based on reference data provided by limited Monte Carlo simulations of a first-pass reconstruction. In this work, we replace the Monte Carlo modeling with a deterministic Boltzmann solver (AcurosCTS) to generate the reference scatter data in a dramatically reduced time. In addition, the algorithm is improved so that instead of adjusting kernel parameters, we directly perturb the SKS scatter estimates. Studies were conducted on simulated data and on a large pelvis phantom scanned on a tabletop system. The new method reduced average reconstruction errors (relative to a reference scan) from 2.5% to 1.8%, and significantly improved visualization of low contrast objects. In total, 24 projections were simulated with an AcurosCTS execution time of 22 sec/projection using an 8-core computer. We have ported AcurosCTS to the GPU, and current run-times are approximately 4 sec/projection using two GPU's running in parallel.
Convergence of Solutions to the Boltzmann Equation in the Incompressible Euler Limit
NASA Astrophysics Data System (ADS)
SAINT-RAYMOND, LAURE
We consider here the problem of deriving rigorously, for well-prepared initial data and without any additional assumption, dissipative or smooth solutions of the incompressible Euler equations from renormalized solutions of the Boltzmann equation. This completes the partial results obtained by Golse [B. Perthame and L. Desvillettes eds., Series in Applied Mathematics 4 (2000), Gauthier-Villars, Paris] and Lions & Masmoudi [Arch. Rational Mech. Anal. 158 (2001), 195-211].
Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method
NASA Technical Reports Server (NTRS)
Chen, Hudong; Chen, Shiyi; Matthaeus, William H.
1992-01-01
A lattice Boltzmann model is presented which gives the complete Navier-Stokes equation and may provide an efficient parallel numerical method for solving various fluid problems. The model uses the single-time relaxation approximation and a particular Maxwell-type distribution. The model eliminates exactly (1) the non-Galilean invariance caused by a density-dependent coefficient in the convection term and (2) a velocity-dependent equation of state.
The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium
NASA Astrophysics Data System (ADS)
Briant, Marc; Daus, Esther S.
2016-07-01
We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in {L^1_vL^∞_x(m)} , where {m˜ (1+ |v|^k)} is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an {L^2-L^E&infty}; theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the {L^1_vL^∞_x} framework is dealt with for any {k > k_0} , recovering the optimal physical threshold of finite energy {k_0=2} in the particular case of a multi-species hard spheres mixture with the same masses.
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. PMID:24229110
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.
General approach to constructing models of the Boltzmann equation
NASA Astrophysics Data System (ADS)
Gorban, Alexander N.; Karlin, Iliya V.
1994-05-01
The problem of thermodynamic parameterization of an arbitrary approximation of reduced description is solved. On the base of this solution a new class of model kinetic equations is constructed that gives a model extension of the chosen approximation to a kinetic model. Model equations describe two processes: rapid relaxation to the chosen approximation along the planes of rapid motions, and the slow motion caused by the chosen approximation. The H-theorem is proved for these models. It is shown, that the rapid process always leads to entropy growth, and also a neighborhood of the approximation is determined inside which the slow process satisfies the H-theorem. Kinetic models for Grad moment approximations and for the Tamm-Mott-Smith approximation are constructed explicitly. In particular, the problem of concordance of the ES-model with the H-theorem is solved.
Fully-Lagrangian and Lattice-Boltzmann Methods for Solving Systems of Conservation Equations
NASA Astrophysics Data System (ADS)
Ancona, M. G.
1994-11-01
A class of "fully-Lagrangian" methods for solving systems of conservation equations is defined. The key step in formulating these methods is the definition of a new set of field variables for which Lagrangian discretization is trivial. Recently popular lattice-Boltzmann simulation schemes for solving such systems are shown to be a useful sub-class of these fully-Lagrangian methods in which (a) the conservation laws are satisfied at each grid point, (b) the Lagrangian variables are expanded perturbatively, and (c) discretization error is used to represent physics. Such schemes are typically derived using methods of kinetic theory. Our numerical analysis approach shows that the conventional physical derivation, while certainly valid and fruitful, is not essential, that it often confuses physics and numerics and that it can be unnecessarily constraining. For example, we show that lattice-Boltzmann-like methods can be non-perturbative and can be made higher-order, implicit and/or with non-uniform grids. Furthermore, our approach provides new perspective on the relationship between lattice-Boltzmann methods and finite-difference techniques. Among other things, we show that the lattice-Boltzmann schemes are only conditionally consistent and in some cases are identical to the well-known Dufort-Frankel method. Through this connection, the lattice-Boltzmann method provides a rational basis for understanding Dufort-Frankel and gives a pathway for its generalization. At the same time, that Dufort Frankel is no longer much used suggests that the lattice-Boltzmann approach might also share this fate.
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.
NASA Astrophysics Data System (ADS)
Feng, Yue
Plasma is currently a hot topic and it has many significant applications due to its composition of both positively and negatively charged particles. The energy distribution function is important in plasma science since it characterizes the ability of the plasma to affect chemical reactions, affect physical outcomes, and drive various applications. The Boltzmann Transport Equation is an important kinetic equation that provides an accurate basis for characterizing the distribution function---both in energy and space. This dissertation research proposes a multi-term approximation to solve the Boltzmann Transport Equation by treating the relaxation process using an expansion of the electron distribution function in Legendre polynomials. The elastic and 29 inelastic cross sections for electron collisions with nitrogen molecules (N2) and singly ionized nitrogen molecules ( N+2 ) have been used in this application of the Boltzmann Transport Equation. Different numerical methods have been considered to compare the results. The numerical methods discussed in this thesis are the implicit time-independent method, the time-dependent Euler method, the time-dependent Runge-Kutta method, and finally the implicit time-dependent relaxation method by generating the 4-way grid with a matrix solver. The results show that the implicit time-dependent relaxation method is the most accurate and stable method for obtaining reliable results. The results were observed to match with the published experimental data rather well.
Global solutions in the critical Besov space for the non-cutoff Boltzmann equation
NASA Astrophysics Data System (ADS)
Morimoto, Yoshinori; Sakamoto, Shota
2016-10-01
The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin-Lerner space. In order to analyze the collisional term of the equation, a Chemin-Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an a priori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn-Banach extension theorem, the desired solution is obtained.
NASA Astrophysics Data System (ADS)
Ketenoğlu, D.; Ünal, B.
2012-08-01
In this study the Green function solution of the Boltzmann transport equation on semiconducting thin film with irregular walls has been applied for the first time. The effects of electron scattering caused by these irregularities on the electrical conductivity have been investigated. First of all by using coordinate transformations, the irregularities on the walls have been transferred into the volume and in this way the both surfaces have been brought into flat forms. By taking two models, Gaussian and exponential, for random potential energy term contained in the transformed Hamiltonian as the perturbation, the resistivity results have been calculated and compared with the ones obtained from the methods widely known in the literature. The Boltzmann transport equation has been solved in relaxation time approximation for the irregular walled system in the case of no magnetic field.
The solution of the relaxation problem for the Boltzmann equation by the integral iteration method
NASA Technical Reports Server (NTRS)
Limar, Y. F.
1972-01-01
The Boltzmann equation is considered in terms of the problem of relaxation of some initial distribution function which depends only on velocities, to Maxwell's distribution function. The Boltzmann equation is given for the relaxation problem in which the distribution function f(t, u, v) is time dependent and is also dependent on two other variables u and v (the velocities of rigid spherical molecules). An iteration process is discussed in which the velocity space u, v is subdivided into squares, the distribution function in each square being approximated by the second-order surface from the values of the distribution function at nine points. The set of all of these points forms a network of u, v values at the nodes of which the distribution function can be found.
Evaluation of Various Types of Wall Boundary Conditions for the Boltzmann Equation
NASA Astrophysics Data System (ADS)
Wilson, Christopher D.; Agarwal, Ramesh K.; Tcheremissine, Felix G.
2011-05-01
This paper presents the evaluation of several solid wall boundary conditions when used in the numerical solution of the Boltzmann equation using the finite-difference/finite-volume methods. Five solid wall boundary conditions are considered: (a) adsorption, (b) specular reflection, (c) diffuse reflection, (d) Maxwellian reflection, and (e) adsorptive Maxwellian reflection. The boundary conditions are applied on a two-dimensional discretized velocity space mesh. Methods for applying the same boundary conditions on a three-dimensional velocity space grid are also presented. The boundary conditions are implemented for the numerical solution of the hypersonic rarefied flow over a flat plate using a three-dimensional generalized Boltzmann equation (GBE) solver. The derivatives that contribute to heat transfer and skin friction at the solid boundary are calculated and compared. Recommendations for further evaluation of the boundary conditions are made.
Numerical solution of the Boltzmann equation for the collective modes of trapped Fermi gases
Lepers, Thomas; Davesne, Dany; Chiacchiera, Silvia; Urban, Michael
2010-08-15
We numerically solve the Boltzmann equation for trapped fermions in the normal phase by using the test-particle method. After discussing a couple of tests in order to estimate the reliability of the method, we apply it to the description of collective modes in a spherical harmonic trap. The numerical results are compared with those obtained previously by taking moments of the Boltzmann equation. We find that the general shape of the response function is very similar in both methods, but the relaxation time obtained from the simulation is significantly longer than that predicted by the method of moments. It is shown that the result of the method of moments can be corrected by including fourth-order moments in addition to the usual second-order ones and that this method agrees very well with our numerical simulations.
NASA Astrophysics Data System (ADS)
Xie, Dexuan; Jiang, Yi
2016-10-01
The nonlocal dielectric approach has been studied for more than forty years but only limited to water solvent until the recent work of Xie et al. (2013) [20]. As the development of this recent work, in this paper, a nonlocal modified Poisson-Boltzmann equation (NMPBE) is proposed to incorporate nonlocal dielectric effects into the classic Poisson-Boltzmann equation (PBE) for protein in ionic solvent. The focus of this paper is to present an efficient finite element algorithm and a related software package for solving NMPBE. Numerical results are reported to validate this new software package and demonstrate its high performance for protein molecules. They also show the potential of NMPBE as a better predictor of electrostatic solvation and binding free energies than PBE.
Coulomb collisions in the Boltzmann equation for electrons in low-temperature gas discharge plasmas
NASA Astrophysics Data System (ADS)
Hagelaar, G. J. M.
2016-02-01
This paper investigates the effects of electron-electron and electron-ion Coulomb collisions on the electron distribution function and transport coefficients obtained from the Boltzmann equation for simple dc gas discharge conditions. Expressions are provided for the full Coulomb collision terms acting on both the isotropic and anisotropic parts of the electron distribution function, which are then incorporated in the freeware Boltzmann equation solver BOLSIG+. Different Coulomb collision effects are demonstrated and discussed on the basis of BOLSIG+ results for argon gas. It is shown that the anisotropic part of the electron-electron collision term, neglected in previous work, can in certain cases have a large effect on the electron mobility and is essential when describing the transition towards the Coulomb-collision dominated regime characterized by Spitzer transport coefficients. Finally, a brief overview is presented of the discharge conditions for which different Coulomb collision effects occur in different gases.
Dilaton and off-shell (non-critical string) effects in Boltzmann equation for species abundances
NASA Astrophysics Data System (ADS)
Lahanas, Ab; Mavromatos, Ne; Nanopoulos, Dv
In this work we derive the modifications to the Boltzmann equation governing the cosmic evolution of relic abundances induced by dilaton dissipative-source and non-critical-string terms in dilaton-driven non-equilibrium string Cosmologies. We also discuss briefly the most important phenomenological consequences, including modifications of the constraints on the available parameter space of cosmologically appealing particle physics models, imposed by recent precision data of astrophysical measurements.
Recent applications of the Boltzmann master equation to heavy ion precompound decay phenomena
Blann, M.; Remington, B.A.
1988-06-01
The Boltzmann master equation (BME) is described and used as a tool to interpret preequilibrium neutron emission from heavy ion collisions gated on evaporation residue or fission fragments. The same approach is used to interpret neutron spectra gated on deep inelastic and quasi-elastic heavy ion collisions. Less successful applications of BME to proton inclusive data with 40 MeV/u incident /sup 12/C ions are presented, and improvements required in the exciton injection term are discussed.
Two-dimensional boltzmann transport equation approach to simulation of local ion implantation
NASA Astrophysics Data System (ADS)
Komarov, F. F.; Mozolevski, I. E.; Rogach, V. P.
1995-05-01
A new theoretical model and software tool is proposed for simulation of two-dimensional local ion implantation in a target of arbitrary geometry. The program uses an algorithm of numerical solution of the boundary value problem for Boltzmann transport equation in two dimensions and permits to calculate the angular and energy distribution function of the particles moving in a multilayered multicomponent target. The program is essentially time saving and can be implemented on an IBM PC AT standard configuration computer.
Bouchard, Hugo; Bielajew, Alex
2015-07-01
To establish a theoretical framework for generalizing Monte Carlo transport algorithms by adding external electromagnetic fields to the Boltzmann radiation transport equation in a rigorous and consistent fashion. Using first principles, the Boltzmann radiation transport equation is modified by adding a term describing the variation of the particle distribution due to the Lorentz force. The implications of this new equation are evaluated by investigating the validity of Fano's theorem. Additionally, Lewis' approach to multiple scattering theory in infinite homogeneous media is redefined to account for the presence of external electromagnetic fields. The equation is modified and yields a description consistent with the deterministic laws of motion as well as probabilistic methods of solution. The time-independent Boltzmann radiation transport equation is generalized to account for the electromagnetic forces in an additional operator similar to the interaction term. Fano's and Lewis' approaches are stated in this new equation. Fano's theorem is found not to apply in the presence of electromagnetic fields. Lewis' theory for electron multiple scattering and moments, accounting for the coupling between the Lorentz force and multiple elastic scattering, is found. However, further investigation is required to develop useful algorithms for Monte Carlo and deterministic transport methods. To test the accuracy of Monte Carlo transport algorithms in the presence of electromagnetic fields, the Fano cavity test, as currently defined, cannot be applied. Therefore, new tests must be designed for this specific application. A multiple scattering theory that accurately couples the Lorentz force with elastic scattering could improve Monte Carlo efficiency. The present study proposes a new theoretical framework to develop such algorithms.
Parallel FE Approximation of the Even/Odd Parity Form of the Linear Boltzmann Equation
Drumm, Clifton R.; Lorenz, Jens
1999-07-21
A novel solution method has been developed to solve the linear Boltzmann equation on an unstructured triangular mesh. Instead of tackling the first-order form of the equation, this approach is based on the even/odd-parity form in conjunction with the conventional mdtigroup discrete-ordinates approximation. The finite element method is used to treat the spatial dependence. The solution method is unique in that the space-direction dependence is solved simultaneously, eliminating the need for the conventional inner iterations, and the method is well suited for massively parallel computers.
From microscopic theory to Boltzmann kinetic equation: Application to vortex dynamics
NASA Astrophysics Data System (ADS)
Blatter, G.; Geshkenbein, V. B.; Kopnin, N. B.
1999-06-01
We show how to lift the problem of calculating the force acting on a topological defect in a superfluid from the microscopic to the semiclassical level: Starting from the microscopic kinetic equations for a clean superconductor, we derive a Boltzmann equation for the quasiparticle distribution function in and around the defect. The velocity q˙ and force p˙ appearing in this Boltzmann equation are given through the Hamiltonian equations q˙=∂pEn(p,q) and p˙=-∂qEn(p,q), where En(p,q) denotes the (nth branch in the) spectrum of the quasiparticles in the vicinity of the defect. Second, we reformulate the microscopic expression for the force acting on the defect in terms of the total momentum transfer of the quasiparticles from the heat bath to the vortex core. We illustrate our result with an application to vortices in s-wave superconductors, where we derive the vortex equation of motion and identify the Magnus, Hall, and dissipative forces.
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.
NASA Astrophysics Data System (ADS)
Takane, Yositake; Hayashi, Masahiko; Ebisawa, Hiromichi
2016-08-01
The time-dependent Ginzburg-Landau equation and the Boltzmann transport equation for charge-density-wave (CDW) conductors are derived from a microscopic one-dimensional model by applying the Keldysh Green's function approach under a quasiclassical approximation. The effects of an external electric field and impurity pinning of the CDW are fully taken into account without relying on a phenomenological argument. These equations simultaneously describe the spatiotemporal dynamics of both the CDW and quasiparticles; thus, they serve as a starting point to develop a general framework to analyze various nonequilibrium phenomena, such as current conversion between the CDW condensate and quasiparticles, in realistic CDW conductors. It is shown that, in typical situations, the equations correctly describe the nonlinear behavior of electric conductivity in a simpler manner.
A conservative multi-group approach to the Boltzmann equations for reactive gas mixtures
NASA Astrophysics Data System (ADS)
Bisi, M.; Rossani, A.; Spiga, G.
2015-11-01
Starting from a simple kinetic model for a quaternary mixture of gases undergoing a bimolecular chemical reaction, multi-group integro-differential equations are derived for the particle distribution functions of all species. The procedure takes advantage of a suitable probabilistic formulation, based on the underlying collision frequencies and transition probabilities, of the relevant reactive kinetic equations of Boltzmann type. Owing to an appropriate choice of a sufficiently large number of weight functions, it is shown that the proposed multi-group equations are able to fulfil exactly, at any order of approximation, the correct conservation laws that must be inherited from the original kinetic equations, where speed was a continuous variable. Future developments are also discussed.
NASA Astrophysics Data System (ADS)
Yano, Ryosuke; Suzuki, Kojiro; Kuroda, Hisayasu
2007-01-01
The direct description of chemical reactions by the Boltzmann equation still involves some difficulties in the kinetic theory. In this paper, we describe diatomic molecular dissociation due to transitions of vibrational quantum states resulting from inelastic collisions. These can be described by the Wang Chang-Uhlenbeck (WCU) equation. To avoid direct evaluation of the strong nonlinear collision kernel of the WCU equation, we used a kinetic equation. For accurate description of the dissociation process, we describe improvements we made to the conventional inelastic collision model (the so-called Morse model). Combining this inelastic collision model with the gas mixture model by Oguchi, we formulated a model for representing diatomic molecular dissociations. We validated this model by simulating a hypersonic shock layer with diatomic molecular dissociation.
NASA Astrophysics Data System (ADS)
Zhou, Jian Guo; Liu, Haifei
2013-08-01
The bed slope in the shallow-water equations reflects the bed topography. It is not a flow variable and cannot be determined in the solution to the flow equations. An immovable nonflat bed affects a flow as a force term, but the flow has no effect on it. Only when the bed term is correctly represented in a numerical method can it generate an accurate solution. In the enhanced lattice Boltzmann method for the shallow-water equations (eLABSWE), using an individual Chapman-Enskog analysis, it is found that such a correct representation can be achieved by retaining Cα=2λα, in which Cα is the coefficient for bed elevation in the lattice Boltzmann equation and λα is that for the water depth in the local equilibrium distribution function. The finding has been validated through simulations of a water at rest in a dish-shaped lake, a wind-induced shallow flow in the same lake, and a steady flow over a two-dimensional bed hump.
Zhou, Jian Guo; Liu, Haifei
2013-08-01
The bed slope in the shallow-water equations reflects the bed topography. It is not a flow variable and cannot be determined in the solution to the flow equations. An immovable nonflat bed affects a flow as a force term, but the flow has no effect on it. Only when the bed term is correctly represented in a numerical method can it generate an accurate solution. In the enhanced lattice Boltzmann method for the shallow-water equations (eLABSWE), using an individual Chapman-Enskog analysis, it is found that such a correct representation can be achieved by retaining C(α)=2λ(α), in which C(α) is the coefficient for bed elevation in the lattice Boltzmann equation and λ(α) is that for the water depth in the local equilibrium distribution function. The finding has been validated through simulations of a water at rest in a dish-shaped lake, a wind-induced shallow flow in the same lake, and a steady flow over a two-dimensional bed hump.
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).
NASA Astrophysics Data System (ADS)
Ayissi, Raoul Domingo; Noutchegueme, Norbert
2015-01-01
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
Ayissi, Raoul Domingo Noutchegueme, Norbert
2015-01-15
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
Linearized Boltzmann equation: A preliminary exploration of its range of applicability
NASA Astrophysics Data System (ADS)
Ghiroldi, Gian Pietro; Gibelli, Livio; Dagna, Paride; Invernizzi, Alice
2012-11-01
We investigate the scaling behavior of non-linear solutions of the two dimensional driven cavity flow with respect to the Mach number. Accurate numerical solutions of the hard-sphere Boltzmann equation have been obtained by means of a novel deterministic method which combines a finite volume discretization of the free-streaming term with a Gauss-Hermite evaluation of the collision integral. The results are of interest for DSMC applications to low speed flows, since they reveal the strategy to rescale solutions obtained at relatively high Mach numbers down to smaller values.
NASA Technical Reports Server (NTRS)
Yoshikawa, K. K.
1979-01-01
The direct simulation Monte Carlo method is applied to solve the Boltzmann equation for collisions between internally excited diatomic gases in highly nonequilibrium states. The semiclassical transition probability is 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 are impractical. The validity of the local Maxwellian assumption and relaxation time, rotational-translational energy transition, and a velocity analysis of the inelastic collision are discussed in detail.
Numerical solution of a three-dimensional cubic cavity flow by using the Boltzmann equation
NASA Technical Reports Server (NTRS)
Hwang, Danny P.
1992-01-01
A three-dimensional cubic cavity flow has been analyzed for diatomic gases by using the Boltzmann equation with the Bhatnagar-Gross-Krook (B-G-K) model. The method of discrete ordinate was applied, and the diffuse reflection boundary condition was assumed. The results, which show a consistent trend toward the Navier-Stokes solution as the Knudson number is reduced, give us confidence to apply the method to a three-dimensional geometry for practical predictions of rarefied-flow characteristics. The CPU time and the main memory required for a three-dimensional geometry using this method seem reasonable.
On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates
NASA Astrophysics Data System (ADS)
Lu, Xuguang; Mouhot, Clément
The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.
Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation.
Khurana, Saheba; Thachuk, Mark
2016-03-14
A numerical method using B-splines is used to solve the linear Boltzmann equation describing the energy relaxation of massive tracer particles moving through a dilute bath gas. The smooth and rough hard sphere and Maxwell molecule models are used with a variety of mass ratios and initial energies to test the capability of the numerical method. Massive tracers are initialized with energies typically found in energy loss experiments in mass spectrometry using biomolecules. The method is also used to examine the applicability of known expressions for the kinetic energy decay from the Fokker-Planck equation for the Rayleigh gas, where we find that results are generally good provided that the initial energy is properly bounded. Otherwise, the energy decay is not constant and a more complex behaviour occurs. The validity of analytical expressions for drag coefficients for spherical particles under specular and diffuse scattering is also tested. We find such expressions are generally good for hard spheres but cannot account, as expected, for the softer repulsive walls of the Maxwell (and real) molecules. Overall, the numerical method performed well even when tracers more than 400 times as massive as the bath were initialized with energies very far from equilibrium. This is a range of applicability beyond many of the standard methods for solving the Boltzmann equation. PMID:26979675
Transient oscillations in a macroscopic effective theory of the Boltzmann equation
NASA Astrophysics Data System (ADS)
Bazow, Dennis; Martinez, Mauricio; Heinz, Ulrich
2016-02-01
A new transient effective theory of the relativistic Boltzmann equation is derived for locally momentum-anisotropic systems. In the expansion of the distribution function around a local "quasi-equilibrium" state, a nonhydrodynamic dynamical degree of freedom is introduced at leading order that breaks local momentum isotropy. By replacing the deviation of the distribution function from this quasi-equilibrium state in terms of moments of the leading-order distribution and applying a systematic power-counting scheme that orders the nonhydrodynamic modes by their microscopic time scales, a closed set of equations for the dynamical degrees of freedom is obtained. Truncating this set at the level of the slowest nonhydroynamic mode, we find that it exhibits transient oscillatory behavior—a phenomenon previously found only in strongly coupled theories, where it appears to be generic. In weakly coupled systems described by the Boltzmann equation, these transient oscillations depend on the breaking of local momentum isotropy being treated nonperturbatively at leading order in the expansion of the distribution function.
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.
Coupling lattice Boltzmann and continuum equations for flow and reactive transport in porous media.
Coon, Ethan; Porter, Mark L.; Kang, Qinjun; Moulton, John D.; Lichtner, Peter C.
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's 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.
NASA Astrophysics Data System (ADS)
Cobos, Agustín C.; Poma, Ana L.; Alvarez, Guillermo D.; Sanz, Darío E.
2016-10-01
We introduce an alternative method to calculate the steady state solution of the angular photon flux after a numerical evolution of the time-dependent Boltzmann transport equation (BTE). After a proper discretization the transport equation was converted into an ordinary system of differential equations that can be iterated as a weighted Richardson algorithm. As a different approach, in this work the time variable regulates the iteration process and convergence criteria is based on physical parameters. Positivity and convergence was assessed from first principles and a modified Courant-Friedrichs-Lewy condition was devised to guarantee convergence. The Penelope Monte Carlo method was used to test the convergence and accuracy of our approach for different phase space discretizations. Benchmarking was performed by calculation of total fluence and photon spectra in different one-dimensional geometries irradiated with 60Co and 6 MV photon beams and radiological applications were devised.
Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations
NASA Astrophysics Data System (ADS)
Wang, H. L.; Chai, Z. H.; Shi, B. C.; Liang, H.
2016-09-01
In this paper, a comparative study of the lattice Boltzmann (LB) models for the Allen-Cahn (A-C) and Cahn-Hilliard (C-H) equations is conducted. To this end, a new LB model for the A-C equation is first proposed, where the equilibrium distribution function and the source term distribution function are delicately designed to recover the A-C equation correctly. The gradient term in this model can be computed by the nonequilibrium part of the distribution function such that the collision process can be implemented locally. Then a detailed numerical study on several classical problems is performed to give a comparison between the present model for the A-C equation and the recently developed LB model [H. Liang et al., Phys. Rev. E 89, 053320 (2014), 10.1103/PhysRevE.89.053320] for the C-H equation in terms of tracking the interface of two-phase flow. The results show that the present LB model for the A-C equation is more accurate and more stable, and also has a second-order convergence rate in space, while the convergence rate of the previous LB model for the C-H equation is only about 1.5.
Nanoscale roughness effect on Maxwell-like boundary conditions for the Boltzmann equation
NASA Astrophysics Data System (ADS)
Brull, S.; Charrier, P.; Mieussens, L.
2016-08-01
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 implemented 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.
One-dimensional compressible gas dynamics calculations using the Boltzmann equation
Reitz, R.D.
1981-07-01
One-dimensional inviscid gas dynamics computations are made using a new method to solve the Boltzmann equation. The numerical method is explicit and is based on concepts from the kinetic theory of gases. The gas density, velocity and temperature are computed by integrating numerically the molecular velocity distribution function. This in turn is computed from the Boltzmann equation using an operator splitting approach. The basic algorithm is shown to be efficient and unconditionally stable. The method is tested for a single component diatomic ideal gas on initial-boundary value problems. These include the Riemann shock-tube problem and shock wave reflection from a stationary wall for a range of incident Mach numbers up to M = 10. The results show that the method can offer significant advantages over standard finite difference methods for certain problems. Shock waves are resolved well with minimal oscillations in the solution, and accurate results are obtained with Courant numbers an order of magnitude larger than the usual stability limit. The method performs best in regions of the flow which are close to thermodynamic equilibrium and is first order accurate in regions which are far from equilibrium, as would be predicted from kinetic theory arguments.
Direct solution of the Boltzmann equation for a binary mixture on GPUs
NASA Astrophysics Data System (ADS)
Frezzotti, Aldo; Pietro Ghiroldi, Gian; Gibelli, Livio
2011-05-01
We show how to accelerate the numerical solution of the Boltzmann equation for a binary gas mixture by using Graphics Processing Units (GPUs). In order to fully exploit the computational power of the GPU, we adopt a semi-regular method of solution which combines a finite difference discretization of the free-streaming term with a Monte Carlo evaluation of the collision integral. The efficiency of the code is demonstrated by studying the propagation of plane harmonic waves of small amplitude in a binary gas mixture of hard spheres for a wide range of Knudsen numbers and wave frequencies. The GPU-based code is about two order of magnitudes faster than the CPU version thus proving that GPUs can substantially speedup the numerical solution of kinetic equations.
NASA Astrophysics Data System (ADS)
Chiloyan, Vazrik; Zeng, Lingping; Huberman, Samuel; Maznev, Alexei A.; Nelson, Keith A.; Chen, Gang
2016-07-01
The phonon Boltzmann transport equation (BTE) is widely utilized to study non-diffusive thermal transport. We find a solution of the BTE in the thin film transient thermal grating (TTG) experimental geometry by using a recently developed variational approach with a trial solution supplied by the Fourier heat conduction equation. We obtain an analytical expression for the thermal decay rate that shows excellent agreement with Monte Carlo simulations. We also obtain a closed form expression for the effective thermal conductivity that demonstrates the full material property and heat transfer geometry dependence, and recovers the limits of the one-dimensional TTG expression for very thick films and the Fuchs-Sondheimer expression for very large grating spacings. The results demonstrate the utility of the variational technique for analyzing non-diffusive phonon-mediated heat transport for nanostructures in multi-dimensional transport geometries, and will assist the probing of the mean free path distribution of materials via transient grating experiments.
Boltzmann Equation Analysis Of Electron Swarms For Non Thermal Flue Gas Discharge Modeling
NASA Astrophysics Data System (ADS)
Yousfi, M.
1997-10-01
The aim of this presentation is to give an overview on the electron swarm development in the flue gas mixture discharges involving N2, O2, H2O and CO2. The corresponding electron basic data needed for the non thermal plasma device for pollution control are given in typical flue gases from Boltzmann equation solution including the dominant collision processes (elastic, inelastic and super-elastic). These data are first the electron-molecule collision cross sections for each gas of the mixture and then the transport and reaction coefficients of electron swarms in the gas mixture. The strong coupling between this electron swarm model with the different models used for the non thermal plasma device of our interest are emphasized. This concerns the electron Boltzamnn equation coupled with the charged particle (or electrical) model, the gas dynamics and also the chemical kinetics models. Some illustrative results of this coupling are then given.
A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation
Larsen, Edward
2013-06-17
The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation.
A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
Colmenares, José; Galizia, Antonella; Ortiz, Jesús; Clematis, Andrea; Rocchia, Walter
2014-01-01
The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs. PMID:25013789
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
Boltzmann equation and Monte Carlo studies of electron transport in resistive plate chambers
NASA Astrophysics Data System (ADS)
Bošnjaković, D.; Petrović, Z. Lj; White, R. D.; Dujko, S.
2014-10-01
A multi term theory for solving the Boltzmann equation and Monte Carlo simulation technique are used to investigate electron transport in Resistive Plate Chambers (RPCs) that are used for timing and triggering purposes in many high energy physics experiments at CERN and elsewhere. Using cross sections for electron scattering in C2H2F4, iso-C4H10 and SF6 as an input in our Boltzmann and Monte Carlo codes, we have calculated data for electron transport as a function of reduced electric field E/N in various C2H2F4/iso-C4H10/SF6 gas mixtures used in RPCs in the ALICE, CMS and ATLAS experiments. Emphasis is placed upon the explicit and implicit effects of non-conservative collisions (e.g. electron attachment and/or ionization) on the drift and diffusion. Among many interesting and atypical phenomena induced by the explicit effects of non-conservative collisions, we note the existence of negative differential conductivity (NDC) in the bulk drift velocity component with no indication of any NDC for the flux component in the ALICE timing RPC system. We systematically study the origin and mechanisms for such phenomena as well as the possible physical implications which arise from their explicit inclusion into models of RPCs. Spatially-resolved electron transport properties are calculated using a Monte Carlo simulation technique in order to understand these phenomena.
Obliger, Amaël; Duvail, Magali; Jardat, Marie; Coelho, Daniel; Békri, Samir; Rotenberg, Benjamin
2013-07-01
We report the calculation of all the transfer coefficients which couple the solvent and ionic fluxes through a charged pore under the effect of pressure, electrostatic potential, and concentration gradients. We use a combination of analytical calculations at the Poisson-Nernst-Planck and Navier-Stokes levels of description and mesoscopic lattice simulations based on kinetic theory. In the absence of added salt, i.e., when the only ions present in the fluid are the counterions compensating the charge of the surface, exact analytical expressions for the fluxes in cylindrical pores allow us to validate a new lattice-Boltzmann electrokinetics (LBE) scheme which accounts for the osmotic contribution to the transport of all species. The influence of simulation parameters on the numerical accuracy is thoroughly investigated. In the presence of an added salt, we assess the range of validity of approximate expressions of the fluxes computed from the linearized Poisson-Boltzmann equation by a systematic comparison with LBE simulations. PMID:23944561
Lattice Boltzmann methods for some 2-D nonlinear diffusion equations:Computational results
Elton, B.H.; Rodrigue, G.H. . Dept. of Applied Science Lawrence Livermore National Lab., CA ); Levermore, C.D. . Dept. of Mathematics)
1990-01-01
In this paper we examine two lattice Boltzmann methods (that are a derivative of lattice gas methods) for computing solutions to two two-dimensional nonlinear diffusion equations of the form {partial derivative}/{partial derivative}t u = v ({partial derivative}/{partial derivative}x D(u){partial derivative}/{partial derivative}x u + {partial derivative}/{partial derivative}y D(u){partial derivative}/{partial derivative}y u), where u = u({rvec x},t), {rvec x} {element of} R{sup 2}, v is a constant, and D(u) is a nonlinear term that arises from a Chapman-Enskog asymptotic expansion. In particular, we provide computational evidence supporting recent results showing that the methods are second order convergent (in the L{sub 1}-norm), conservative, conditionally monotone finite difference methods. Solutions computed via the lattice Boltzmann methods are compared with those computed by other explicit, second order, conservative, monotone finite difference methods. Results are reported for both the L{sub 1}- and L{sub {infinity}}-norms.
Yong, Wen-An; Zhao, Weifeng; Luo, Li-Shi
2016-03-01
We propose using the Maxwell iteration to derive the hydrodynamic equations from the lattice Boltzmann equation (LBE) with an external forcing term. The proposed methodology differs from existing approaches in several aspects. First, it need not explicitly introduce multiple-timescales or the Knudsen number, both of which are required in the Chapman-Enskog analysis. Second, it need not use the Hilbert expansion of the hydrodynamic variables, which is necessary in the asymptotic analysis of the LBE. The proposed methodology assumes the acoustic scaling (or the convective scaling) δ(t)∼δ(x), thus δ(t) is the only expansion parameter in the analysis of the LBE system, and it leads to the Navier-Stokes equations in compressible form. The forcing density derived in this work can reproduce existing forcing schemes by adjusting appropriate parameters. The proposed methodology also analyzes the numerical accuracy of the LBE. In particular, it shows the Mach number Ma should scale as O(δ(t)(1/3)) to maintain the truncation errors due to Ma and δ(t) in balance when δ(t)→0, so that the LBE can converge to the expected hydrodynamic equations effectively and efficiently. PMID:27078487
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.
NASA Astrophysics Data System (ADS)
Hu, Jingwei; Wang, Li
2015-01-01
We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for electron mass and energy as mean free path goes to zero. As opposed to the classical drift-diffusion limit where the stiff collisions are all in one scale, new difficulties arise in the two-scale stiff collision terms because the simple BGK penalization [15] fails to drive the solution to the correct limit. We propose to set up a spatially dependent threshold on the penalization of the stiffer collision operator such that the evolution of the solution resembles a Hilbert expansion at the continuous level. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of our scheme.
Hu, Jingwei; Wang, Li
2015-01-15
We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for electron mass and energy as mean free path goes to zero. As opposed to the classical drift-diffusion limit where the stiff collisions are all in one scale, new difficulties arise in the two-scale stiff collision terms because the simple BGK penalization [15] fails to drive the solution to the correct limit. We propose to set up a spatially dependent threshold on the penalization of the stiffer collision operator such that the evolution of the solution resembles a Hilbert expansion at the continuous level. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of our scheme.
Guo, Zhaoli; Shi, Baochang; Zhao, T S; Zheng, Chuguang
2007-11-01
The lattice Boltzmann equation (LBE) has shown its promise in the simulation of microscale gas flows. One of the critical issues with this advanced method is to specify suitable slip boundary conditions to ensure simulation accuracy. In this paper we study two widely used kinetic boundary conditions in the LBE: the combination of the bounce-back and specular-reflection scheme and the discrete Maxwell's scheme. We show that (i) both schemes are virtually equivalent in principle, and (ii) there exist discrete effects in both schemes. A strategy is then proposed to adjust the parameters in the two kinetic boundary conditions such that an accurate slip boundary condition can be implemented. The numerical results demonstrate that the corrected boundary conditions are robust and reliable.
NASA Astrophysics Data System (ADS)
Wang, Shyh-Wei; Guo, Shuang-Fa
1998-07-01
A stepwise Boltzmann transport equation (BTE) simulation using non-uniform energy grid momentum matrix and exact nuclear scattering cross-section is successfully parallelized to simulate the ion implantation of multi-component targets. Assuming that the interactions of ion with different target atoms are independent, the scattering of ions with different components can be calculated concurrently by different processors. It is developed on CONVEX SPP-1000 and the software environment of parallel virtual machine (PVM) with a master-slave paradigm. A speedup of 3.3 has been obtained for the simulation of As ions implanted into AZ1350 (C6.2H6O1N0.15S0.06) which is composed of five components. In addition, our new scheme gives better agreement with the experimental results for heavy ion implantation than the conventional method using a uniform energy grid and approximated scattering function.
Fillion-Gourdeau, F; Herrmann, H J; Mendoza, M; Palpacelli, S; Succi, S
2013-10-18
We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and nonrelativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wave function) propagates in space-time as a classical statistical process (Boltzmann distribution).
Baryon asymmetry of the Universe without Boltzmann or Kadanoff-Baym equations
NASA Astrophysics Data System (ADS)
Gagnon, Jean-Sébastien; Shaposhnikov, Mikhail
2011-03-01
We present a formalism that allows the computation of the baryon asymmetry of the Universe, from the first principles of statistical physics and quantum field theory, that is applicable to certain types of physics beyond the standard model (such as the neutrino minimal standard model) and does not require the solution of Boltzmann or Kadanoff-Baym equations. The formalism works if a thermal bath of standard model particles is very weakly coupled to a new sector (sterile neutrinos in the neutrino minimal standard model case) that is out-of-equilibrium. The key point that allows a computation without kinetic equations is that the number of sterile neutrinos produced during the relevant cosmological period remains small. In such a case, it is possible to expand the formal solution of the von Neumann equation perturbatively and obtain a master formula for the lepton asymmetry expressed in terms of nonequilibrium Wightman functions. The master formula neatly separates CP-violating contributions from finite temperature correlation functions and satisfies all three Sakharov conditions. These correlation functions can then be evaluated perturbatively; the validity of the perturbative expansion depends on the parameters of the model considered. Here, we choose a toy model (containing only two active and two sterile neutrinos) to illustrate the use of the formalism, but it could be applied to other models.
Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation.
Ren, Feng; Song, Baowei; Sukop, Michael C; Hu, Haibao
2016-08-01
The primary and key task of binary fluid flow modeling is to track the interface with good accuracy, which is usually challenging due to the sharp-interface limit and numerical dispersion. This article concentrates on further development of the conservative Allen-Cahn equation (ACE) [Geier et al., Phys. Rev. E 91, 063309 (2015)10.1103/PhysRevE.91.063309] under the framework of the lattice Boltzmann method (LBM), with incorporation of the incompressible hydrodynamic equations [Liang et al., Phys. Rev. E 89, 053320 (2014)10.1103/PhysRevE.89.053320]. Utilizing a modified equilibrium distribution function and an additional source term, this model is capable of correctly recovering the conservative ACE through the Chapman-Enskog analysis. We also simulate four phase-tracking benchmark cases, including one three-dimensional case; all show good accuracy as well as low numerical dispersion. By coupling the incompressible hydrodynamic equations, we also simulate layered Poiseuille flow and the Rayleigh-Taylor instability, illustrating satisfying performance in dealing with complex flow problems, e.g., high viscosity ratio, high density ratio, and high Reynolds number situations. The present work provides a reliable and efficient solution for binary flow modeling. PMID:27627416
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.
Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation
NASA Astrophysics Data System (ADS)
Ren, Feng; Song, Baowei; Sukop, Michael C.; Hu, Haibao
2016-08-01
The primary and key task of binary fluid flow modeling is to track the interface with good accuracy, which is usually challenging due to the sharp-interface limit and numerical dispersion. This article concentrates on further development of the conservative Allen-Cahn equation (ACE) [Geier et al., Phys. Rev. E 91, 063309 (2015), 10.1103/PhysRevE.91.063309] under the framework of the lattice Boltzmann method (LBM), with incorporation of the incompressible hydrodynamic equations [Liang et al., Phys. Rev. E 89, 053320 (2014), 10.1103/PhysRevE.89.053320]. Utilizing a modified equilibrium distribution function and an additional source term, this model is capable of correctly recovering the conservative ACE through the Chapman-Enskog analysis. We also simulate four phase-tracking benchmark cases, including one three-dimensional case; all show good accuracy as well as low numerical dispersion. By coupling the incompressible hydrodynamic equations, we also simulate layered Poiseuille flow and the Rayleigh-Taylor instability, illustrating satisfying performance in dealing with complex flow problems, e.g., high viscosity ratio, high density ratio, and high Reynolds number situations. The present work provides a reliable and efficient solution for binary flow modeling.
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.
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.
Ion strength limit of computed excess functions based on the linearized Poisson-Boltzmann equation.
Fraenkel, Dan
2015-12-01
The linearized Poisson-Boltzmann (L-PB) equation is examined for its κ-range of validity (κ, Debye reciprocal length). This is done for the Debye-Hückel (DH) theory, i.e., using a single ion size, and for the SiS treatment (D. Fraenkel, Mol. Phys. 2010, 108, 1435), which extends the DH theory to the case of ion-size dissimilarity (therefore dubbed DH-SiS). The linearization of the PB equation has been claimed responsible for the DH theory's failure to fit with experiment at > 0.1 m; but DH-SiS fits with data of the mean ionic activity coefficient, γ± (molal), against m, even at m > 1 (κ > 0.33 Å(-1) ). The SiS expressions combine the overall extra-electrostatic potential energy of the smaller ion, as central ion-Ψa>b (κ), with that of the larger ion, as central ion-Ψb>a (κ); a and b are, respectively, the counterion and co-ion distances of closest approach. Ψa>b and Ψb>a are derived from the L-PB equation, which appears to conflict with their being effective up to moderate electrolyte concentrations (≈1 m). However, the L-PB equation can be valid up to κ ≥ 1.3 Å(-1) if one abandons the 1/κ criterion for its effectiveness and, instead, use, as criterion, the mean-field electrostatic interaction potential of the central ion with its ion cloud, at a radial distance dividing the cloud charge into two equal parts. The DH theory's failure is, thus, not because of using the L-PB equation; the lethal approximation is assigning a single size to the positive and negative ions. PMID:26493019
Improved Multiple-Coarsening Methods for Sn Discretizations of the Boltzmann Equation
Lee, Barry
2010-06-01
In a recent series of articles, the author presented a multiple-coarsening multigrid method for solving $S_n$ discretizations of the Boltzmann transport equation. This algorithm is applied to an integral equation for the scalar flux or moments. Although this algorithm is very efficient over parameter regimes that describe realistic neutron/photon transport applications, improved methods that can reduce the computational cost are presented in this paper. These improved methods are derived through a careful examination of the frequencies, particularly the near-nullspace, of the integral equation. In the earlier articles, the near-nullspace components were shown to be smooth in angle in the sense that the angular fluxes generated by these components are smooth in angle. In this paper, we present a spatial description of these near-nullspace components. Using the angular description of the earlier papers together with the spatial description reveals the intrinsic space-angle dependence of the integral equation's frequencies. This space-angle dependence is used to determine the appropriate space-angle grids to represent and efficiently attenuate the near-nullspace error components on. It will be shown that these components can have multiple spatial scales. By using only the appropriate space-angle grids that can represent these spatial scales in the original multiple-coarsening algorithm, an improved algorithm is obtained. Moreover, particularly for anisotropic scattering, recognizing the strong angle dependence of the angular fluxes generated by the high frequencies of the integral equation, another improved multiple-coarsening scheme is derived. Restricting this scheme to the appropriate space-angle grids produces a very efficient method.
Ion strength limit of computed excess functions based on the linearized Poisson-Boltzmann equation.
Fraenkel, Dan
2015-12-01
The linearized Poisson-Boltzmann (L-PB) equation is examined for its κ-range of validity (κ, Debye reciprocal length). This is done for the Debye-Hückel (DH) theory, i.e., using a single ion size, and for the SiS treatment (D. Fraenkel, Mol. Phys. 2010, 108, 1435), which extends the DH theory to the case of ion-size dissimilarity (therefore dubbed DH-SiS). The linearization of the PB equation has been claimed responsible for the DH theory's failure to fit with experiment at > 0.1 m; but DH-SiS fits with data of the mean ionic activity coefficient, γ± (molal), against m, even at m > 1 (κ > 0.33 Å(-1) ). The SiS expressions combine the overall extra-electrostatic potential energy of the smaller ion, as central ion-Ψa>b (κ), with that of the larger ion, as central ion-Ψb>a (κ); a and b are, respectively, the counterion and co-ion distances of closest approach. Ψa>b and Ψb>a are derived from the L-PB equation, which appears to conflict with their being effective up to moderate electrolyte concentrations (≈1 m). However, the L-PB equation can be valid up to κ ≥ 1.3 Å(-1) if one abandons the 1/κ criterion for its effectiveness and, instead, use, as criterion, the mean-field electrostatic interaction potential of the central ion with its ion cloud, at a radial distance dividing the cloud charge into two equal parts. The DH theory's failure is, thus, not because of using the L-PB equation; the lethal approximation is assigning a single size to the positive and negative ions.
Improved Multiple-Coarsening Methods for Sn Discretizations of the Boltzmann Equation
Lee, B
2008-12-01
In a recent series of articles, the author presented a multiple-coarsening multigrid method for solving S{sub n} discretizations of the Boltzmann transport equation. This algorithm is applied to an integral equation for the scalar flux or moments. Although this algorithm is very efficient over parameter regimes that describe realistic neutron/photon transport applications, improved methods that can reduce the computational cost are presented in this paper. These improved methods are derived through a careful examination of the frequencies, particularly the near-nullspace, of the integral equation. In the earlier articles, the near-nullspace components were shown to be smooth in angle in the sense that the angular fluxes generated by these components are smooth in angle. In this paper, we present a spatial description of these near-nullspace components. Using the angular description of the earlier papers together with the spatial description reveals the intrinsic space-angle dependence of the integral equation's frequencies. This space-angle dependence is used to determine the appropriate space-angle grids to represent and efficiently attenuate the near-nullspace error components on. It will be shown that these components can have multiple spatial scales. By using only the appropriate space-angle grids that can represent these spatial scales in the original multiple-coarsening algorithm, an improved algorithm is obtained. Moreover, particularly for anisotropic scattering, recognizing the strong angle dependence of the angular fluxes generated by the high frequencies of the integral equation, another improved multiple-coarsening scheme is derived. Restricting this scheme to the appropriate space-angle grids produces a very efficient method.
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
Leptogenesis with heavy neutrino flavours: from density matrix to Boltzmann equations
Blanchet, Steve; Bari, Pasquale Di; Jones, David A.; Marzola, Luca E-mail: pdb1d08@soton.ac.uk E-mail: daj1g08@soton.ac.uk
2013-01-01
Leptogenesis with heavy neutrino flavours is discussed within a density matrix formalism. We write the density matrix equation, describing the generation of the matter-antimatter asymmetry, for an arbitrary choice of the right-handed (RH) neutrino masses. For hierarchical RH neutrino masses lying in the fully flavoured regimes, this reduces to multiple-stage Boltzmann equations. In this case we recover and extend results previously derived within a quantum state collapse description. We confirm the generic existence of phantom terms. However, taking into account the effect of gauge interactions, we show that they are washed out at the production with a wash-out rate that is halved compared to that one acting on the total asymmetry. In the N{sub 1}-dominated scenario they cancel without contributing to the final baryon asymmetry. In other scenarios they do not in general and they have to be taken into account. We also confirm that there is a (orthogonal) component in the asymmetry produced by the heavier RH neutrinos which completely escapes the washout from the lighter RH neutrinos and show that phantom terms additionally contribute to it. The other (parallel) component is washed out with the usual exponential factor, even for weak washout. Finally, as an illustration, we study the two RH neutrino model in the light of the above findings, showing that phantom terms can contribute to the final asymmetry also in this case.
A simple Boltzmann transport equation for ballistic to diffusive transient heat transport
Maassen, Jesse Lundstrom, Mark
2015-04-07
Developing simplified, but accurate, theoretical approaches to treat heat transport on all length and time scales is needed to further enable scientific insight and technology innovation. Using a simplified form of the Boltzmann transport equation (BTE), originally developed for electron transport, we demonstrate how ballistic phonon effects and finite-velocity propagation are easily and naturally captured. We show how this approach compares well to the phonon BTE, and readily handles a full phonon dispersion and energy-dependent mean-free-path. This study of transient heat transport shows (i) how fundamental temperature jumps at the contacts depend simply on the ballistic thermal resistance, (ii) that phonon transport at early times approach the ballistic limit in samples of any length, and (iii) perceived reductions in heat conduction, when ballistic effects are present, originate from reductions in temperature gradient. Importantly, this framework can be recast exactly as the Cattaneo and hyperbolic heat equations, and we discuss how the key to capturing ballistic heat effects is to use the correct physical boundary conditions.
Boltzmann-equation simulations of radio-frequency-driven, low-temperature plasmas
Drallos, P.J.; Riley, M.E.
1995-01-01
We present a method for the numerical solution of the Boltzmann equation (BE) describing plasma electrons. We apply the method to a capacitively-coupled, radio-frequency-driven He discharge in parallel-plate (quasi-1D) geometry which contains time scales for physical processes spanning six orders of magnitude. Our BE solution procedure uses the method of characteristics for the Vlasov operator with interpolation in phase space at early time, allowing storage of the distribution function on a fixed phase-space grid. By alternating this BE method with a fluid description of the electrons, or with a novel time-cycle-average equation method, we compute the periodic steady state of a He plasma by time evolution from startup conditions. We find that the results compare favorably with measured current-voltage, plasma density, and ``cited state densities in the ``GEC`` Reference Cell. Our atomic He model includes five levels (some are summed composites), 15 electronic transitions, radiation trapping, and metastable-metastable collisions.
Leptogenesis with heavy neutrino flavours: from density matrix to Boltzmann equations
NASA Astrophysics Data System (ADS)
Blanchet, Steve; Di Bari, Pasquale; Jones, David A.; Marzola, Luca
2013-01-01
Leptogenesis with heavy neutrino flavours is discussed within a density matrix formalism. We write the density matrix equation, describing the generation of the matter-antimatter asymmetry, for an arbitrary choice of the right-handed (RH) neutrino masses. For hierarchical RH neutrino masses lying in the fully flavoured regimes, this reduces to multiple-stage Boltzmann equations. In this case we recover and extend results previously derived within a quantum state collapse description. We confirm the generic existence of phantom terms. However, taking into account the effect of gauge interactions, we show that they are washed out at the production with a wash-out rate that is halved compared to that one acting on the total asymmetry. In the N1-dominated scenario they cancel without contributing to the final baryon asymmetry. In other scenarios they do not in general and they have to be taken into account. We also confirm that there is a (orthogonal) component in the asymmetry produced by the heavier RH neutrinos which completely escapes the washout from the lighter RH neutrinos and show that phantom terms additionally contribute to it. The other (parallel) component is washed out with the usual exponential factor, even for weak washout. Finally, as an illustration, we study the two RH neutrino model in the light of the above findings, showing that phantom terms can contribute to the final asymmetry also in this case.
Catt, B; Snyder, M
2014-06-15
Purpose: To investigate the use of the linear Boltzmann transport equation as a dose calculation tool which can account for interface effects, while still having faster computation times than Monte Carlo methods. In particular, we introduce a forward scattering approximation, in hopes of improving calculation time without a significant hindrance to accuracy. Methods: Two coupled Boltzmann transport equations were constructed, one representing the fluence of photons within the medium, and the other, the fluence of electrons. We neglect the scattering term within the electron transport equation, resulting in an extreme forward scattering approximation to reduce computational complexity. These equations were then solved using a numerical technique for solving partial differential equations, known as a finite difference scheme, where the fluence at each discrete point in space is calculated based on the fluence at the previous point in the particle's path. Using this scheme, it is possible to develop a solution to the Boltzmann transport equations by beginning with boundary conditions and iterating across the entire medium. The fluence of electrons can then be used to find the dose at any point within the medium. Results: Comparisons with Monte Carlo simulations indicate that even simplistic techniques for solving the linear Boltzmann transport equation yield expected interface effects, which many popular dose calculation algorithms are not capable of predicting. Implementation of a forward scattering approximation does not appear to drastically reduce the accuracy of this algorithm. Conclusion: Optimized implementations of this algorithm have been shown to be very accurate when compared with Monte Carlo simulations, even in build up regions where many models fail. Use of a forward scattering approximation could potentially give a reasonably accurate dose distribution in a shorter amount of time for situations where a completely accurate dose distribution is not
A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations
NASA Astrophysics Data System (ADS)
Liu, Chang; Xu, Kun; Sun, Quanhua; Cai, Qingdong
2016-06-01
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, the 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
A first-order system least-squares finite element method for the Poisson-Boltzmann equation.
Bond, Stephen D; Chaudhry, Jehanzeb Hameed; Cyr, Eric C; Olson, Luke N
2010-06-01
The Poisson-Boltzmann equation is an important tool in modeling solvent in biomolecular systems. In this article, we focus on numerical approximations to the electrostatic potential expressed in the regularized linear Poisson-Boltzmann equation. We expose the flux directly through a first-order system form of the equation. Using this formulation, we propose a system that yields a tractable least-squares finite element formulation and establish theory to support this approach. The least-squares finite element approximation naturally provides an a posteriori error estimator and we present numerical evidence in support of the method. The computational results highlight optimality in the case of adaptive mesh refinement for a variety of molecular configurations. In particular, we show promising performance for the Born ion, Fasciculin 1, methanol, and a dipole, which highlights robustness of our approach.
ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSON-BOLTZMANN EQUATION.
Holst, Michael; McCammon, James Andrew; Yu, Zeyun; Zhou, Youngcheng; Zhu, Yunrong
2012-01-01
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
ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSON-BOLTZMANN EQUATION
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
Shrinkage of bubbles and drops in the lattice Boltzmann equation method for nonideal gases.
Zheng, Lin; Lee, Taehun; Guo, Zhaoli; Rumschitzki, David
2014-03-01
One characteristic of multiphase lattice Boltzmann equation (LBE) methods is that the interfacial region has a finite (i.e., noninfinitesimal) thickness known as a diffuse interface. In simulations of, e.g., bubble or drop dynamics, for problems involving nonideal gases, one frequently observes that the diffuse interface method produces a spontaneous, nonphysical shrinkage of the bubble or drop radius. In this paper, we analyze in detail a single-fluid two-phase model and use a LBE model for nonideal gases in order to explain this fundamental problem. For simplicity, we only investigate the static bubble or droplet problem. We find that the method indeed produces a density shift, bubble or droplet shrinkage, as well as a critical radius below which the bubble or droplet eventually vanishes. Assuming that the ratio between the interface thickness D and the initial bubble or droplet radius r0 is small, we analytically show the existence of this density shift, bubble or droplet radius shrinkage, and critical bubble or droplet survival radius. Numerical results confirm our analysis. We also consider droplets on a solid surface with different curvatures, contact angles, and initial droplet volumes. Numerical results show that the curvature, contact angle, and the initial droplet volume have an effect on this spontaneous shrinkage process, consistent with the survival criterion.
NASA Astrophysics Data System (ADS)
Grubert, G. K.; Loffhagen, D.
2014-01-01
Kinetic studies of the electrons in spatially inhomogeneous, bounded plasmas have been performed by means of two different numerical techniques: the solution of the space-dependent electron Boltzmann equation (BE) using a multiterm approximation of the Legendre polynomial expansion of the electron velocity distribution function and Monte Carlo (MC) simulations. Appropriate conditions at the boundaries of gas discharge plasmas have been deduced, which are adequate for the direct comparison of electron BE and MC simulation results. In particular, extended boundary conditions at the electron emitting cathode are represented and discussed. The investigations are performed for dc discharges in oxygen at conditions typical of abnormal glow discharges. The analysis of the results shows extreme alterations of the properties of the electrons in the discharge region, where pronounced groups of electrons are found, which are generated by electron impact dissociation processes and ionization processes when assuming equal energy sharing in ionizing collisions and isotropic scattering in inelastic electron collisions. The influence of these assumptions on the results obtained is discussed. The very good agreement between the results of the BE and MC calculations verifies the consistence of the derived extended boundary conditions at the cathode and of both the kinetic approaches.
Zheng, Lin; Guo, Zhaoli; Shi, Baochang
2012-07-01
The lattice Boltzmann equation (LBE) method has been shown to be a promising tool for microscale gas flows. However, few works focus on the microtube flows, and there still are some fundamental problems for the LBE to such flows. In this paper, a recently proposed axisymmetric LBE with three kinetic boundary conditions, i.e., the combination of bounceback and specular reflection scheme, the combination of the Maxwell and specular-reflection scheme, and the combination of the Maxwell and bounceback scheme, have been investigated in detail. By analyzing the micro-Hagen-Poiseuille flow, we observed the discrete boundary condition effect and provided a revised boundary scheme to overcome such effect near the slip flow regime. Some numerical tests for the micro-Hagen-Poiseuille have been carried out to validate the analysis, and the numerical results of the revised boundary schemes agree well with the analytic solutions which confirmed our theoretical analysis. In addition, we also applied the revised combination of the Maxwell and bounceback scheme to microtube flow with sudden expansion and contraction, the numerical results of the pressure distribution and normalized slip velocity agree well with the theoretical ones. PMID:23005568
High Order Finite Volume Nonlinear Schemes for the Boltzmann Transport Equation
Bihari, B L; Brown, P N
2005-03-29
The authors apply the nonlinear WENO (Weighted Essentially Nonoscillatory) scheme to the spatial discretization of the Boltzmann Transport Equation modeling linear particle transport. The method is a finite volume scheme which ensures not only conservation, but also provides for a more natural handling of boundary conditions, material properties and source terms, as well as an easier parallel implementation and post processing. It is nonlinear in the sense that the stencil depends on the solution at each time step or iteration level. By biasing the gradient calculation towards the stencil with smaller derivatives, the scheme eliminates the Gibb's phenomenon with oscillations of size O(1) and reduces them to O(h{sup r}), where h is the mesh size and r is the order of accuracy. The current implementation is three-dimensional, generalized for unequally spaced meshes, fully parallelized, and up to fifth order accurate (WENO5) in space. For unsteady problems, the resulting nonlinear spatial discretization yields a set of ODE's in time, which in turn is solved via high order implicit time-stepping with error control. For the steady-state case, they need to solve the non-linear system, typically by Newton-Krylov iterations. There are several numerical examples presented to demonstrate the accuracy, non-oscillatory nature and efficiency of these high order methods, in comparison with other fixed-stencil schemes.
Computations of ion diffusion coefficients from the Boltzmann-Fokker-Planck equation
NASA Technical Reports Server (NTRS)
Roussel-Dupre, R.
1981-01-01
The Boltzmann-Fokker-Planck equation is solved with the Chapman-Enskog method of analysis for the velocity distribution functions of helium, carbon, nitrogen, and oxygen. The analysis is a perturbation scheme based on the assumption of a collision-dominated gas, and the calculations are carried out to first order. The elements considered are treated as trace constituents in an electron-proton gas. From the resulting distribution functions, diffusion coefficients are computed which are found to be 20-30% less than those obtained by Chapman and Burgers. In addition, it is shown that the return current of cold electrons needed to maintain quasi-neutrality in a plasma with a temperature gradient contributes a term in the thermal diffusion coefficient omitted erroneously in previous works. This added term resolves the longstanding controversy over the discrepancy between the coefficients of Chapman and Burgers, which are seen to be completely equivalent in the light of this analysis. The viscosity coefficient for an electron-proton gas is also computed and found to be 7% less than that obtained by Braginskii.
Hard-thermal-loop corrections in leptogenesis II: solving the Boltzmann equations
Kießig, Clemens P.; Plümacher, Michael E-mail: pluemi@mpp.mpg.de
2012-09-01
We investigate hard-thermal-loop (HTL) corrections to the final lepton asymmetry in leptogenesis. To this end we solve the Boltzmann equations with HTL-corrected rates and CP asymmetries, which we calculated in paper I of this series. We pay special attention to the influence of the two leptonic quasiparticles that arise at non-zero temperature. We include only decays and inverse decays and allow for the lepton modes to be either decoupled from each other, or to be in chemical equilibrium by some strong interaction, simulating the interaction with gauge bosons. In two additional cases, we approximate the full HTL lepton propagators with zero-temperature propagators, where we replace the zero-temperature mass by the thermal mass of the leptons m{sub l}(T) or the asymptotic mass (2){sup 1/2} m{sub l}(T). We compare the final lepton asymmetries of the four thermal cases and the zero-temperature case for zero, thermal and dominant initial neutrino abundance. The final lepton asymmetries of the thermal cases differ considerably from the vacuum case and from each other in the weak washout regime for zero initial neutrino abundance and in the intermediate regime for dominant initial neutrino abundance. In the strong washout regime, the final lepton asymmetry can be enhanced by a factor of two in the case of strongly interacting lepton modes.
On anisotropy function in crystal growth simulations using Lattice Boltzmann equation
NASA Astrophysics Data System (ADS)
Younsi, Amina; Cartalade, Alain
2016-11-01
In this paper, we present the ability of the Lattice Boltzmann (LB) equation, usually applied to simulate fluid flows, to simulate various shapes of crystals. Crystal growth is modeled with a phase-field model for a pure substance, numerically solved with a LB method in 2D and 3D. This study focuses on the anisotropy function that is responsible for the anisotropic surface tension between the solid phase and the liquid phase. The anisotropy function involves the unit normal vectors of the interface, defined by gradients of phase-field. Those gradients have to be consistent with the underlying lattice of the LB method in order to avoid unwanted effects of numerical anisotropy. Isotropy of the solution is obtained when the directional derivatives method, specific for each lattice, is applied for computing the gradient terms. With the central finite differences method, the phase-field does not match with its rotation and the solution is not any more isotropic. Next, the method is applied to simulate simultaneous growth of several crystals, each of them being defined by its own anisotropy function. Finally, various shapes of 3D crystals are simulated with standard and nonstandard anisotropy functions which favor growth in <100>-, <110>- and <111>-directions.
Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation
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
Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions
NASA Astrophysics Data System (ADS)
Hu, Zexi; Huang, Juntao; Yong, Wen-An
2016-04-01
In this work, we propose an interfacial scheme accompanying the lattice Boltzmann method for convection-diffusion equations with general interfacial conditions, including conjugate conditions with or without jumps in heat and mass transfer, continuity of macroscopic variables and normal fluxes in ion diffusion in porous media with different porosity, and the Kapitza resistance in heat transfer. The construction of this scheme is based on our boundary schemes [Huang and Yong, J. Comput. Phys. 300, 70 (2015), 10.1016/j.jcp.2015.07.045] for Robin boundary conditions on straight or curved boundaries. It gives second-order accuracy for straight interfaces and first-order accuracy for curved ones. In addition, the new scheme inherits the advantage of the boundary schemes in which only the current lattice nodes are involved. Such an interfacial scheme is highly desirable for problems with complex geometries or in porous media. The interfacial scheme is numerically validated with several examples. The results show the utility of the constructed scheme and very well support our theoretical predications.
Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation.
Yang, Jaw-Yen; Yan, Chin-Yuan; Huang, Juan-Chen; Li, Zhihui
2014-08-01
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
Boltzmann equation analysis of electron-molecule collision cross sections in water vapor and ammonia
NASA Astrophysics Data System (ADS)
Yousfi, M.; Benabdessadok, M. D.
1996-12-01
Sets of electron-molecule collision cross sections for H2O and NH3 have been determined from a classical technique of electron swarm parameter unfolding. This deconvolution method is based on a simplex algorithm using a powerful multiterm Boltzmann equation analysis established in the framework of the classical hydrodynamic approximation. It is well adapted for the simulation of the different classes of swarm experiments (i.e., time resolved, time of flight, and steady state experiments). The sets of collision cross sections that exist in the literature are reviewed and analyzed. Fitted sets of cross sections are determined for H2O and NH3 which exhibit features characteristic of polar molecules such as high rotational excitation collision cross sections. The hydrodynamic swarm parameters (i.e., drift velocity, longitudinal and transverse diffusion coefficients, ionization and attachment coefficients) calculated from the fitted sets are in excellent agreement with the measured ones. These sets are finally used to calculate the transport and reaction coefficients needed for discharge modeling in two cases of typical gas mixtures for which experimental swarm data are very sparse or nonexistent (i.e., flue gas mixtures and gas mixtures for rf plasma surface treatment).
NASA Astrophysics Data System (ADS)
Romano, Giuseppe; Esfarjani, Keivan; Strubbe, David A.; Broido, David; Kolpak, Alexie M.
2016-01-01
Nanostructured materials exhibit low thermal conductivity because of the additional scattering due to phonon-boundary interactions. As these interactions are highly sensitive to the mean free path (MFP) of phonons, MFP distributions in nanostructures can be dramatically distorted relative to bulk. Here we calculate the MFP distribution in periodic nanoporous Si for different temperatures, using the recently developed MFP-dependent Boltzmann transport equation. After analyzing the relative contribution of each phonon branch to thermal transport in nanoporous Si, we find that at room temperature optical phonons contribute 17 % to heat transport, compared to 5 % in bulk Si. Interestingly, we observe a constant thermal conductivity over the range 200 K
White, R D; Ness, K F; Robson, R E; Li, B
1999-08-01
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 psi. The hierarchy resulting from a spherical harmonic decomposition of the Boltzmann equation in the hydrodynamic regime [Ness, Phys. Rev. A 47, 327 (1993)] is solved numerically by representing the speed dependence of the phase-space distribution function in terms of an expansion in Sonine polynomials about a weighted sum of Maxwellian distributions at different temperatures. Results are given for charged-particle swarms in certain model gases over a range of psi and field strengths. The variation of the transport coefficients with psi is addressed using physical arguments. The errors associated with the two-term approximation and inadequacies of Legendre polynomial expansions are highlighted.
Hashimoto, K; Kanki, K; Tanaka, S; Petrosky, T
2016-02-01
Irreversible processes of weakly coupled one-dimensional quantum perfect Lorentz gas are studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouville-von Neumann operator. Without any phenomenological operations, such as a coarse-graining of space-time, or a truncation of the higher order correlation, we obtained irreversible processes in a purely dynamical basis in all space and time scale including the microscopic atomic interaction range that is much smaller than the mean-free length. Based on this solution, a limitation of the usual phenomenological Boltzmann equation, as well as an extension of the Boltzmann equation to entire space-time scale, is discussed.
Hashimoto, K; Kanki, K; Tanaka, S; Petrosky, T
2016-02-01
Irreversible processes of weakly coupled one-dimensional quantum perfect Lorentz gas are studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouville-von Neumann operator. Without any phenomenological operations, such as a coarse-graining of space-time, or a truncation of the higher order correlation, we obtained irreversible processes in a purely dynamical basis in all space and time scale including the microscopic atomic interaction range that is much smaller than the mean-free length. Based on this solution, a limitation of the usual phenomenological Boltzmann equation, as well as an extension of the Boltzmann equation to entire space-time scale, is discussed. PMID:26986313
NASA Astrophysics Data System (ADS)
Bernhoff, N.
2012-11-01
Half-space problems for the Boltzmann equation are of great importance in the study of the asymptotic behavior of the solutions of boundary value problems of the Boltzmann equation for small Knudsen numbers. Half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-type equations in a neighborhood of the boundary. Here we consider a half-space problem of condensation for a pure vapor in the presence of a non-condensable gas by using discrete velocity models (DVMs) of the Boltzmann equation. The Boltzmann equation can be approximated by DVMs up to any order, and these DVMs can be applied for numerical methods, but also for mathematical studies to bring deeper understanding and new ideas. For one-dimensional half-space problems, the discrete Boltzmann equation (the general DVM) reduces to a system of ODEs. We obtain that the number of parameters to be specified in the boundary conditions depends on whether the condensing vapor flow is subsonic or supersonic. This behavior has earlier been found numerically. We want to stress that our results are valid for any finite number of velocities. This is an extension of known results for single-component gases (and for binary mixtures of two vapors) to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian, with a flow velocity towards the condensed phase, at infinity, while the non-condensable gas tends to zero at infinity. Steady condensation of the vapor takes place at the condensed phase, which is held at a constant temperature. We assume that the vapor is completely absorbed, that the non-condensable gas is diffusively reflected at the condensed phase, and that vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution at the condensed phase, needed for the existence of a unique solution of the
Morel, J.E.; Lorence, L.J. Jr.; Kensek, R.P.; Halbleib, J.A.; Sloan, D.P.
1996-11-01
A hybrid multigroup/continuous-energy Monte Carlo algorithm is developed for solving the Boltzmann-Fokker-Planck equation. This algorithm differs significantly from previous charged-particle Monte Carlo algorithms. Most importantly, it can be used to perform both forward and adjoint transport calculations, using the same basic multigroup cross-section data. The new algorithm is fully described, computationally tested, and compared with a standard condensed history algorithm for coupled electron-photon transport calculations.
NASA Astrophysics Data System (ADS)
Fiorentini, Mattia; Bonini, Nicola
2016-08-01
We present a first-principles computational approach to calculate thermoelectric transport coefficients via the exact solution of the linearized Boltzmann transport equation, also including the effect of nonequilibrium phonon populations induced by a temperature gradient. We use density functional theory and density functional perturbation theory for an accurate description of the electronic and vibrational properties of a system, including electron-phonon interactions; carriers' scattering rates are computed using standard perturbation theory. We exploit Wannier interpolation (both for electronic bands and electron-phonon matrix elements) for an efficient sampling of the Brillouin zone, and the solution of the Boltzmann equation is achieved via a fast and stable conjugate gradient scheme. We discuss the application of this approach to n -doped silicon. In particular, we discuss a number of thermoelectric properties such as the thermal and electrical conductivities of electrons, the Lorenz number and the Seebeck coefficient, including the phonon drag effect, in a range of temperatures and carrier concentrations. This approach gives results in good agreement with experimental data and provides a detailed characterization of the nature and the relative importance of the individual scattering mechanisms. Moreover, the access to the exact solution of the Boltzmann equation for a realistic system provides a direct way to assess the accuracy of different flavors of relaxation time approximation, as well as of models that are popular in the thermoelectric community to estimate transport coefficients.
Li, Zhihui; Ma, Qiang; Wu, Junlin; Jiang, Xinyu; Zhang, Hanxin
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 ordinate 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.
NASA Astrophysics Data System (ADS)
Li, Zhihui; Wu, Junlin; Ma, Qiang; Jiang, Xinyu; Zhang, Hanxin
2014-12-01
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 ordinate 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.
Chakavorty, Arghya; Li, Lin; Alexov, Emil
2016-10-30
Macromolecular interactions are essential for understanding numerous biological processes and are typically characterized by the binding free energy. Important component of the binding free energy is the electrostatics, which is frequently modeled via the solutions of the Poisson-Boltzmann Equations (PBE). However, numerous works have shown that the electrostatic component (ΔΔGelec ) of binding free energy is very sensitive to the parameters used and modeling protocol. This prompted some researchers to question the robustness of PBE in predicting ΔΔGelec . We argue that the sensitivity of the absolute ΔΔGelec calculated with PBE using different input parameters and definitions does not indicate PBE deficiency, rather this is what should be expected. We show how the apparent sensitivity should be interpreted in terms of the underlying changes in several numerous and physical parameters. We demonstrate that PBE approach is robust within each considered force field (CHARMM-27, AMBER-94, and OPLS-AA) once the corresponding structures are energy minimized. This observation holds despite of using two different molecular surface definitions, pointing again that PBE delivers consistent results within particular force field. The fact that PBE delivered ΔΔGelec values may differ if calculated with different modeling protocols is not a deficiency of PBE, but natural results of the differences of the force field parameters and potential functions for energy minimization. In addition, while the absolute ΔΔGelec values calculated with different force field differ, their ordering remains practically the same allowing for consistent ranking despite of the force field used. © 2016 Wiley Periodicals, Inc.
NASA Astrophysics Data System (ADS)
Wang, Mingliang; Wong, Chung F.; Liu, Jianhong; Zhang, Peixin
2007-07-01
We have successfully coupled the Kohn-Sham with Poisson-Boltzmann equations to predict the solvation free energy, where the Kohn-Sham equations were solved by implementing the flexible pseudo atomic orbitals as in S IESTA package. It was found that the calculated solvation free energy is in good agreement with experimental results for small neutral molecules, and its standard error is 1.33 kcal/mol, the correlation coefficient is 0.97. Due to its high efficiency and accuracy, the proposed model can be a promising tool for computing solvation free energies in computer aided drug design in future.
A lattice-Boltzmann scheme of the Navier-Stokes equations on a 3D cuboid lattice
NASA Astrophysics Data System (ADS)
Min, Haoda; Peng, Cheng; Wang, Lian-Ping
2015-11-01
The standard lattice-Boltzmann method (LBM) for fluid flow simulation is based on a square (in 2D) or cubic (in 3D) lattice grids. Recently, two new lattice Boltzmann schemes have been developed on a 2D rectangular grid using the MRT (multiple-relaxation-time) collision model, by adding a free parameter in the definition of moments or by extending the equilibrium moments. Here we developed a lattice Boltzmann model on 3D cuboid lattice, namely, a lattice grid with different grid lengths in different spatial directions. We designed our MRT-LBM model by matching the moment equations from the Chapman-Enskog expansion with the Navier-Stokes equations. The model guarantees correct hydrodynamics. A second-order term is added to the equilibrium moments in order to restore the isotropy of viscosity on a cuboid lattice. The form and the coefficients of the extended equilibrium moments are determined through an inverse design process. An additional benefit of the model is that the viscosity can be adjusted independent of the stress-moment relaxation parameter, thus improving the numerical stability of the model. The resulting cuboid MRT-LBM model is then validated through benchmark simulations using laminar channel flow, turbulent channel flow, and the 3D Taylor-Green vortex flow.
Lloyd, S. A. M.; Ansbacher, W.
2013-01-15
Purpose: Acuros external beam (Acuros XB) is a novel dose calculation algorithm implemented through the ECLIPSE treatment planning system. The algorithm finds a deterministic solution to the linear Boltzmann transport equation, the same equation commonly solved stochastically by Monte Carlo methods. This work is an evaluation of Acuros XB, by comparison with Monte Carlo, for dose calculation applications involving high-density materials. Existing non-Monte Carlo clinical dose calculation algorithms, such as the analytic anisotropic algorithm (AAA), do not accurately model dose perturbations due to increased electron scatter within high-density volumes. Methods: Acuros XB, AAA, and EGSnrc based Monte Carlo are used to calculate dose distributions from 18 MV and 6 MV photon beams delivered to a cubic water phantom containing a rectangular high density (4.0-8.0 g/cm{sup 3}) volume at its center. The algorithms are also used to recalculate a clinical prostate treatment plan involving a unilateral hip prosthesis, originally evaluated using AAA. These results are compared graphically and numerically using gamma-index analysis. Radio-chromic film measurements are presented to augment Monte Carlo and Acuros XB dose perturbation data. Results: Using a 2% and 1 mm gamma-analysis, between 91.3% and 96.8% of Acuros XB dose voxels containing greater than 50% the normalized dose were in agreement with Monte Carlo data for virtual phantoms involving 18 MV and 6 MV photons, stainless steel and titanium alloy implants and for on-axis and oblique field delivery. A similar gamma-analysis of AAA against Monte Carlo data showed between 80.8% and 87.3% agreement. Comparing Acuros XB and AAA evaluations of a clinical prostate patient plan involving a unilateral hip prosthesis, Acuros XB showed good overall agreement with Monte Carlo while AAA underestimated dose on the upstream medial surface of the prosthesis due to electron scatter from the high-density material. Film measurements
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
Premnath, Kannan N; Pattison, Martin J; Banerjee, Sanjoy
2009-02-01
In this paper, we present a framework based on the generalized lattice Boltzmann equation (GLBE) using multiple relaxation times with forcing term for eddy capturing simulation of wall-bounded turbulent flows. Due to its flexibility in using disparate relaxation times, the GLBE is well suited to maintaining numerical stability on coarser grids and in obtaining improved solution fidelity of near-wall turbulent fluctuations. The subgrid scale (SGS) turbulence effects are represented by the standard Smagorinsky eddy viscosity model, which is modified by using the van Driest wall-damping function to account for reduction of turbulent length scales near walls. In order to be able to simulate a wider class of problems, we introduce forcing terms, which can represent the effects of general nonuniform forms of forces, in the natural moment space of the GLBE. Expressions for the strain rate tensor used in the SGS model are derived in terms of the nonequilibrium moments of the GLBE to include such forcing terms, which comprise a generalization of those presented in a recent work [Yu, Comput. Fluids 35, 957 (2006)]. Variable resolutions are introduced into this extended GLBE framework through a conservative multiblock approach. The approach, whose optimized implementation is also discussed, is assessed for two canonical flow problems bounded by walls, viz., fully developed turbulent channel flow at a shear or friction Reynolds number (Re) of 183.6 based on the channel half-width and three-dimensional (3D) shear-driven flows in a cubical cavity at a Re of 12 000 based on the side length of the cavity. Comparisons of detailed computed near-wall turbulent flow structure, given in terms of various turbulence statistics, with available data, including those from direct numerical simulations (DNS) and experiments showed good agreement. The GLBE approach also exhibited markedly better stability characteristics and avoided spurious near-wall turbulent fluctuations on coarser grids
Premnath, Kannan N; Pattison, Martin J; Banerjee, Sanjoy
2009-02-01
In this paper, we present a framework based on the generalized lattice Boltzmann equation (GLBE) using multiple relaxation times with forcing term for eddy capturing simulation of wall-bounded turbulent flows. Due to its flexibility in using disparate relaxation times, the GLBE is well suited to maintaining numerical stability on coarser grids and in obtaining improved solution fidelity of near-wall turbulent fluctuations. The subgrid scale (SGS) turbulence effects are represented by the standard Smagorinsky eddy viscosity model, which is modified by using the van Driest wall-damping function to account for reduction of turbulent length scales near walls. In order to be able to simulate a wider class of problems, we introduce forcing terms, which can represent the effects of general nonuniform forms of forces, in the natural moment space of the GLBE. Expressions for the strain rate tensor used in the SGS model are derived in terms of the nonequilibrium moments of the GLBE to include such forcing terms, which comprise a generalization of those presented in a recent work [Yu, Comput. Fluids 35, 957 (2006)]. Variable resolutions are introduced into this extended GLBE framework through a conservative multiblock approach. The approach, whose optimized implementation is also discussed, is assessed for two canonical flow problems bounded by walls, viz., fully developed turbulent channel flow at a shear or friction Reynolds number (Re) of 183.6 based on the channel half-width and three-dimensional (3D) shear-driven flows in a cubical cavity at a Re of 12 000 based on the side length of the cavity. Comparisons of detailed computed near-wall turbulent flow structure, given in terms of various turbulence statistics, with available data, including those from direct numerical simulations (DNS) and experiments showed good agreement. The GLBE approach also exhibited markedly better stability characteristics and avoided spurious near-wall turbulent fluctuations on coarser grids
Conjugate heat and mass transfer in the lattice Boltzmann equation method.
Li, Like; Chen, Chen; Mei, Renwei; Klausner, James F
2014-04-01
An interface treatment for conjugate heat and mass transfer in the lattice Boltzmann equation method is proposed based on our previously proposed second-order accurate Dirichlet and Neumann boundary schemes. The continuity of temperature (concentration) and its flux at the interface for heat (mass) transfer is intrinsically satisfied without iterative computations, and the interfacial temperature (concentration) and their fluxes are conveniently obtained from the microscopic distribution functions without finite-difference calculations. The present treatment takes into account the local geometry of the interface so that it can be directly applied to curved interface problems such as conjugate heat and mass transfer in porous media. For straight interfaces or curved interfaces with no tangential gradient, the coupling between the interfacial fluxes along the discrete lattice velocity directions is eliminated and thus the proposed interface schemes can be greatly simplified. Several numerical tests are conducted to verify the applicability and accuracy of the proposed conjugate interface treatment, including (i) steady convection-diffusion in a channel containing two different fluids, (ii) unsteady convection-diffusion in the channel, (iii) steady heat conduction inside a circular domain with two different solid materials, and (iv) unsteady mass transfer from a spherical droplet in an extensional creeping flow. The accuracy and order of convergence of the simulated interior temperature (concentration) field, the interfacial temperature (concentration), and heat (mass) flux are examined in detail and compared with those obtained from the "half-lattice division" treatment in the literature. The present analysis and numerical results show that the half-lattice division scheme is second-order accurate only when the interface is fixed at the center of the lattice links, while the present treatment preserves second-order accuracy for arbitrary link fractions. For curved
Conjugate heat and mass transfer in the lattice Boltzmann equation method
Li, LK; Chen, C; Mei, RW; Klausner, JF
2014-04-22
An interface treatment for conjugate heat and mass transfer in the lattice Boltzmann equation method is proposed based on our previously proposed second-order accurate Dirichlet and Neumann boundary schemes. The continuity of temperature (concentration) and its flux at the interface for heat (mass) transfer is intrinsically satisfied without iterative computations, and the interfacial temperature (concentration) and their fluxes are conveniently obtained from the microscopic distribution functions without finite-difference calculations. The present treatment takes into account the local geometry of the interface so that it can be directly applied to curved interface problems such as conjugate heat and mass transfer in porous media. For straight interfaces or curved interfaces with no tangential gradient, the coupling between the interfacial fluxes along the discrete lattice velocity directions is eliminated and thus the proposed interface schemes can be greatly simplified. Several numerical tests are conducted to verify the applicability and accuracy of the proposed conjugate interface treatment, including (i) steady convection-diffusion in a channel containing two different fluids, (ii) unsteady convection-diffusion in the channel, (iii) steady heat conduction inside a circular domain with two different solid materials, and (iv) unsteady mass transfer from a spherical droplet in an extensional creeping flow. The accuracy and order of convergence of the simulated interior temperature (concentration) field, the interfacial temperature (concentration), and heat (mass) flux are examined in detail and compared with those obtained from the "half-lattice division" treatment in the literature. The present analysis and numerical results show that the half-lattice division scheme is second-order accurate only when the interface is fixed at the center of the lattice links, while the present treatment preserves second-order accuracy for arbitrary link fractions. For curved
NASA Astrophysics Data System (ADS)
Rukolaine, Sergey A.
2016-05-01
In classical kinetic models a particle free path distribution is exponential, but this is more likely to be an exception than a rule. In this paper we derive a generalized linear Boltzmann equation (GLBE) for a general free path distribution in the framework of Alt's model. In the case that the free path distribution has at least first and second finite moments we construct an asymptotic solution to the initial value problem for the GLBE for small mean free paths. In the special case of the one-speed transport problem the asymptotic solution results in a diffusion approximation to the GLBE.
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
Iyer, Ramakrishnan; Mukhopadhyay, Ayan
2010-04-15
The AdS/CFT correspondence defines a sector with universal strongly coupled dynamics in the field theory as the dual of pure gravity in AdS described by Einstein's equation with a negative cosmological constant. We explain here, from the field-theoretic viewpoint how the dynamics in this sector gets determined by the expectation value of the energy-momentum tensor alone. We first show that the Boltzmann equation has very special solutions which could be functionally completely determined in terms of the energy-momentum tensor alone. We call these solutions conservative solutions. We indicate why conservative solutions should also exist when we refine this kinetic description to go closer to the exact microscopic theory or even move away from the regime of weak coupling so that no kinetic description could be employed. We argue that these conservative solutions form the universal sector dual to pure gravity at strong coupling and large N. Based on this observation, we propose a regularity condition on the energy-momentum tensor so that the dual solution in pure gravity has a smooth future horizon. We also study if irreversibility emerges only at long time scales of observation, unlike the case of the Boltzmann equation.
Igor D. Kaganovich; Oleg Polomarov
2003-05-19
In low-pressure discharges, when the electron mean free path is larger or comparable with the discharge length, the electron dynamics is essentially non-local. Moreover, the electron energy distribution function (EEDF) deviates considerably from a Maxwellian. Therefore, an accurate kinetic description of the low-pressure discharges requires knowledge of the non-local conductivity operator and calculation of the non-Maxwellian EEDF. The previous treatments made use of simplifying assumptions: a uniform density profile and a Maxwellian EEDF. In the present study a self-consistent system of equations for the kinetic description of nonlocal, non-uniform, nearly collisionless plasmas of low-pressure discharges is derived. It consists of the nonlocal conductivity operator and the averaged kinetic equation for calculation of the non-Maxwellian EEDF. The importance of accounting for the non-uniform plasma density profile on both the current density profile and the EEDF is demonstrated.
Magneto-Hydro-Dynamic Waves In The Collisionless Space Plasma
NASA Astrophysics Data System (ADS)
Dzhalilov, N. S.; Kuznetsov, V. D.; Staude, J.
2007-12-01
The instability of magneto-hydro-dynamic (MHD) waves in an anisotropic, collisionless, rarefied hot plasma is studied. Anisotropy properties of such a plasma are caused by a strong magnetic field, when the thermal gas pressures across and along the field become unequal. Moreover, there appears an anisotropy of the thermal fluxes. The study of the anisotropy features of the plasma are motivated by observed solar coronal data. The 16 moments equations derived from the Boltzmann-Vlasov kinetic equation are used. These equations strongly differ from the usual isotropic MHD case. For linear disturbances the wave equations in homogenous anisotropic plasma are deduced. The general dispersion relation for the incompressible wave modes is derived, solved and analyzed. It is shown that a wide wave spectrum with stable and unstable behavior is possible, in contrast to the usual isotropic MHD case. The dependence of the instability on magnetic field, pressure anisotropy, and heat fluxes is investigated. The general instability condition is obtained. The results can be applied to the theory of solar and stellar coronal heating, to wind models and in other modeling, where the collisionless approximation is valid.
Numerical modeling of the splitting of magnetic droplets by multiphase lattice Boltzmann equation
NASA Astrophysics Data System (ADS)
Clime, L.; Brassard, D.; Veres, T.
2009-04-01
A multiphase lattice Boltzmann numerical model driven by an isothermal interaction potential is applied for the splitting of magnetic droplets in electrowetting-on-dielectric devices. A hydrophilic magnetic plug is considered inside the liquid droplet and successive uniform force fields are applied in order to split this droplet. The numerical results are compared with experiments on water droplets containing plugs of superparamagnetic beads and good agreement is obtained.
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 AdS2 Ⓧ S2. 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 does not matchmore » 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
NASA Astrophysics Data System (ADS)
Silva, Goncalo; Semiao, Viriato
2011-03-01
When the lattice Boltzmann (LB) method is used to solve hydrodynamic problems containing a body force term varying in space and/or time, its modelling at the mesoscopic scale must be verified in terms of consistency in order to avoid the appearance of non-hydrodynamic error terms at the macroscopic scale. In the present work it is shown that the modelling of spatially varying steady body force terms in the LB equation must be different from the time-dependent case, when a steady-state flow solution is sought. For that, the Chapman-Enskog analysis is used to derive the LB body force model for the LB BGK equations in a steady-state flow problem. The theoretical findings are supported by numerical tests performed on two different 2D steady-state laminar flows driven by spatially varying body forces with known analytical solutions.
NASA Astrophysics Data System (ADS)
Lahiri, T.; Pal Majumder, T.; Ghosh, N. K.
2014-07-01
Commercialization of ferroelectric liquid crystal displays (FLCDs) suffers from mechanical and electro-convective instabilities. Impurity ions play a pivotal role in the latter case, and therefore we developed a mean-field type model to understand the complex role of space charges, particularly ions in a ferroelectric liquid crystal. Considering an effective ion-chirality relation, we obtained a modified Poisson-Boltzmann equation for ions dissolved into a chiral solvent like the ferroelectric smectic phase. A nonuniform director profile induced by the mean electrostatic potential of the ions is then calculated by solving an Euler-Lagrange equation for a helically twisted smectic state. A combination of effects resulting from molecular chirality and an electrostatically driven twist created by the ions seems to produce this nonuniform fluctuation in the director orientation. Finally, both theoretical and experimental points of view are presented on the prediction of this mean-field model.
NASA Astrophysics Data System (ADS)
Shizgal, Bernie
2016-03-01
The paper by Burini et al. [7] presents an interesting use of the Boltzmann equation of kinetic theory to model real learning processes. The authors provide a comprehensive discussion of the basic concepts involved in their modelling work. The Boltzmann equation as used by physicists and chemists to model a variety of transport processes in many diverse fields is based on the notion of the binary collisions between identifiable particles in the defined system [9]. The particles exchange energy on collision and the distribution function, which depends on the three velocity components and the three spatial coordinates, varies with time. The classical or quantum collision dynamics between particles play a central role in the definition of the kernels in the integral operators that define the Boltzmann equation [8].
NASA Technical Reports Server (NTRS)
Yoshikawa, K. K.
1978-01-01
Theoretical results pertaining to internally excited translational-rotational energy relaxation in a spatially uniform diatomic gas far removed from solid boundaries are obtained by solving the Boltzmann equation by means of the Monte Carlo direct simulation method. The analysis is based on calculations involving three different types of initial conditions: equilibrium, nonequilibrium-equipartition (i.e., equipartition is satisfied, but the distributions are perturbed), and nonequilibrium-nonequipartition (i.e., both equipartition and the distributions are perturbed). Results of monatomic-gas simulations are also included to facilitate comparisons with the coupled translational-rotational relaxation simulations, and some simulations for a normal shock-wave structure are briefly examined. The results show that: (1) single-step transitions are the significant mechanisms of intermodal energy transfer; (2) translational-rotational transitions are coupled most efficiently for low-lying states of rotationally excited molecules and least efficiently for highly rotationally excited molecules; and (3) relaxation occurs via a successive set of distributions that are not Maxwell-Boltzmann (nonlocal Maxwellian).
NASA Astrophysics Data System (ADS)
Halliday, I.; Lishchuk, S. V.; Spencer, T. J.; Pontrelli, G.; Evans, P. C.
2016-08-01
We present a method for applying a class of velocity-dependent forces within a multicomponent lattice Boltzmann equation simulation that is designed to recover continuum regime incompressible hydrodynamics. This method is applied to the problem, in two dimensions, of constraining to uniformity the tangential velocity of a vesicle membrane implemented within a recent multicomponent lattice Boltzmann simulation method, which avoids the use of Lagrangian boundary tracers. The constraint of uniform tangential velocity is carried by an additional contribution to an immersed boundary force, which we derive here from physical arguments. The result of this enhanced immersed boundary force is to apply a physically appropriate boundary condition at the interface between separated lattice fluids, defined as that region over which the phase-field varies most rapidly. Data from this enhanced vesicle boundary method are in agreement with other data obtained using related methods [e.g., T. Krüger, S. Frijters, F. Günther, B. Kaoui, and J. Harting, Eur. Phys. J. 222, 177 (2013), 10.1140/epjst/e2013-01834-y] and underscore the importance of a correct vesicle membrane condition.
Halliday, I; Lishchuk, S V; Spencer, T J; Pontrelli, G; Evans, P C
2016-08-01
We present a method for applying a class of velocity-dependent forces within a multicomponent lattice Boltzmann equation simulation that is designed to recover continuum regime incompressible hydrodynamics. This method is applied to the problem, in two dimensions, of constraining to uniformity the tangential velocity of a vesicle membrane implemented within a recent multicomponent lattice Boltzmann simulation method, which avoids the use of Lagrangian boundary tracers. The constraint of uniform tangential velocity is carried by an additional contribution to an immersed boundary force, which we derive here from physical arguments. The result of this enhanced immersed boundary force is to apply a physically appropriate boundary condition at the interface between separated lattice fluids, defined as that region over which the phase-field varies most rapidly. Data from this enhanced vesicle boundary method are in agreement with other data obtained using related methods [e.g., T. Krüger, S. Frijters, F. Günther, B. Kaoui, and J. Harting, Eur. Phys. J. 222, 177 (2013)10.1140/epjst/e2013-01834-y] and underscore the importance of a correct vesicle membrane condition. PMID:27627411
NASA Astrophysics Data System (ADS)
Donko, Zoltan; Dyatko, Nikolay
2016-06-01
The Negative Differential Conductivity and Transient Negative Mobility effects in xenon gas are analyzed by a first-principles particle simulation technique and via an approximate solution of the Boltzmann transport equation (BE). The particle simulation method is devoid of the approximations that are traditionally adopted in the BE solutions in which: (i) the distribution function is searched for in a two-term form; (ii) the Coulomb part of the collision integral for the anisotropic part of the distribution function is neglected; (iii) Coulomb collisions are treated as binary events; and (iv) the range of the electron-electron interaction is limited to a cutoff distance. The results obtained from the two methods are, for both effects, in good qualitative agreement, small differences are attributed to the approximations listed above.
NASA Technical Reports Server (NTRS)
Nathenson, M.; Baganoff, D.; Yen, S. M.
1974-01-01
Data obtained from a numerical solution of the Boltzmann equation for shock-wave structure are used to test the accuracy of accepted approximate expressions for the two moments of the collision integral Delta (Q) for general intermolecular potentials in systems with a large translational nonequilibrium. The accuracy of the numerical scheme is established by comparison of the numerical results with exact expressions in the case of Maxwell molecules. They are then used in the case of hard-sphere molecules, which are the furthest-removed inverse power potential from the Maxwell molecule; and the accuracy of the approximate expressions in this domain is gauged. A number of approximate solutions are judged in this manner, and the general advantages of the numerical approach in itself are considered.
Path length differencing and energy conservation of the S[sub N] Boltzmann/Spencer-Lewis equation
Filippone, W.L.; Monahan, S.P. )
1993-02-01
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 negative flux fixup destroys energy balance when applied to path legend, a modified fixup scheme that does not is presented.
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.
Li, Yunqi; Zhao, Qin; Huang, Qingrong
2014-01-30
A combination of turbidimetric titration, a sigmoidal Boltzmann equation approach and Monte Carlo simulation has been used to study the complex coacervation in serum albumin and pectin mixtures. The effects of the mass ratio of protein to polysaccharide on the critical pH values, the probability of complex coacervation and the electrostatic interaction from charge patches in serum albumin were investigated. Turbidimetric titration results showed an optimum pH for complex coacervation (pHm), which corresponded to the maximum turbidity in the protein/polysaccharide mixture. The pHm monotonically decreased as the ratio decreased, and could be fitted using the sigmoidal Boltzmann equation. It suggests that pHm could be a good ordering parameter to characterize the phase behavior associated with protein/polysaccharide complex coacervation. Qualitative understanding of pHm by taking into account the minimization of electrostatic interaction, as well as quantitative matching of pHm according to the concept of charge neutralization were both achieved. Our results suggest that the serum albumin/pectin complexes were ultimately neutralized by the partial charges originated from the titratable residues in protein and polysaccharide chains at pHm. The Monte Carlo simulation provided consistent phase boundaries for complex coacervation in the same system, and the intermolecular association strength was determined to be several kBT below the given ionic strength. The strongest binding site in the protein is convergent to the largest positive charge patch if pure electrostatic interaction was considered. Further inclusion of contribution from excluded volume resulted in the binding site distribution over five different positive charge patches at different protein/polysaccharide ratios and pH values. PMID:24299810
NASA Astrophysics Data System (ADS)
Khisamutdinov, A. I.; Velker, N. N.
2014-05-01
The talk examines a system of pairwise interaction particles, which models a rarefied gas in accordance with the nonlinear Boltzmann equation, the master equations of Markov evolution of this system and corresponding numerical Monte Carlo methods. Selection of some optimal method for simulation of rarefied gas dynamics depends on the spatial size of the gas flow domain. For problems with the Knudsen number Kn of order unity "imitation", or "continuous time", Monte Carlo methods ([2]) are quite adequate and competitive. However if Kn <= 0.1 (the large sizes), excessive punctuality, namely, the need to see all the pairs of particles in the latter, leads to a significant increase in computational cost(complexity). We are interested in to construct the optimal methods for Boltzmann equation problems with large enough spatial sizes of the flow. Speaking of the optimal, we mean that we are talking about algorithms for parallel computation to be implemented on high-performance multi-processor computers. The characteristic property of large systems is the weak dependence of sub-parts of each other at a sufficiently small time intervals. This property is taken into account in the approximate methods using various splittings of operator of corresponding master equations. In the paper, we develop the approximate method based on the splitting of the operator of master equations system "over groups of particles" ([7]). The essence of the method is that the system of particles is divided into spatial subparts which are modeled independently for small intervals of time, using the precise"imitation" method. The type of splitting used is different from other well-known type "over collisions and displacements", which is an attribute of the known Direct simulation Monte Carlo methods. The second attribute of the last ones is the grid of the "interaction cells", which is completely absent in the imitation methods. The main of talk is parallelization of the imitation algorithms with
NASA Astrophysics Data System (ADS)
Wang, Lian-Ping
2010-03-01
Particle-laden turbulent flow is of importance to many engineering applications and natural phenomena, such as aerosol and pollutant transport, interaction of cloud droplets, spay combustion, and chemical processes. In general, the dynamics of dispersed phase and that of the carrier fluid phase are closely coupled. Most previous studies utilize the point particle approach to study the effects of particles on the carrier turbulence, under the assumptions that the particle size is significantly smaller than the smallest turbulence length scale and the particle volume fraction is low. The present study focuses on the motion and hydrodynamic interactions of finite-size freely moving particles in a turbulent background flow. To simulate carrier fluid turbulence, a mesoscopic lattice Boltzmann approach is applied with the multiple relaxation-time collision model, which yields a more robust viscous flow simulation method than the single-relaxation collision model. The no-slip boundary condition on the moving surface of each particle is implemented using an interpolated bounce-back scheme. The refill problem resulting from the moving boundary is handled by a non-equilibrium correction method to reduce the unphysical force fluctuations acting on the particles. The short-range lubrication force not resolved by the simulation is represented by a physical model involving particle relative location and velocity. For the carrier fluid phase, computational results are discussed in terms of the change of energy spectrum compared with the particle-free turbulence, the time evolution of the turbulent kinetic energy and the dissipation rate. For the dispersed phase, the focus will be on the particle-pair statistics such as the relative velocity and radial distribution function as well as particle-particle collision rate. The effects of varying particle size, volume fraction, and particle-to-fluid density ratio will be examined. The results will be compared to those from the previous
Pattison, Martin J; Premnath, Kannan N; Banerjee, Sanjoy
2009-02-01
Turbulent flow in a straight square duct driven by a pressure gradient exhibits remarkable flow structures such as the presence of mean streamwise vorticity or secondary flows. These secondary circulations take the form of two counter-rotating vortices near each corner of the duct. Even though their magnitudes are small compared with primary streamwise motions, they have a significant influence on flow and scalar transport and are challenging to accurately predict using computational approaches. In this paper, we employ a recently developed formulation of the generalized lattice Boltzmann equation (GLBE) with forcing term to perform large eddy simulation of fully developed turbulent flow in a square duct at a shear Reynolds number based on duct width equal to 300. Subgrid scale effects are represented by the Smagorinsky eddy viscosity model, which is modified by the van Driest damping function in the near-wall regions, in this GLBE approach, which is based on multiple relaxation times. It was found that the GLBE is able to correctly reproduce the existence of mean secondary motions and the computed detailed structure of first- and second-order statistics of main and secondary motions are in good agreement with prior direct numerical simulations based on the solution of the Navier-Stokes equations and experimental data.
NASA Astrophysics Data System (ADS)
Goffin, Mark A.; Baker, Christopher M. J.; Buchan, Andrew G.; Pain, Christopher C.; Eaton, Matthew D.; Smith, Paul N.
2013-06-01
This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, k, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for k with directional dependence. General error estimators are derived for any given functional of the flux and applied to k to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The k goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.
Ringe, Stefan; Oberhofer, Harald; Hille, Christoph; Matera, Sebastian; Reuter, Karsten
2016-08-01
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. PMID:27323006
Goffin, Mark A.; Baker, Christopher M.J.; Buchan, Andrew G.; Pain, Christopher C.; Eaton, Matthew D.; Smith, Paul N.
2013-06-01
This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, k{sub eff}, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for k{sub eff} with directional dependence. General error estimators are derived for any given functional of the flux and applied to k{sub eff} to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The k{sub eff} goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.
Chai, Zhenhua; Zhao, T S
2014-07-01
In this paper, we propose a local nonequilibrium scheme for computing the flux of the convection-diffusion equation with a source term in the framework of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Both the Chapman-Enskog analysis and the numerical results show that, at the diffusive scaling, the present nonequilibrium scheme has a second-order convergence rate in space. A comparison between the nonequilibrium scheme and the conventional second-order central-difference scheme indicates that, although both schemes have a second-order convergence rate in space, the present nonequilibrium scheme is more accurate than the central-difference scheme. In addition, the flux computation rendered by the present scheme also preserves the parallel computation feature of the LBM, making the scheme more efficient than conventional finite-difference schemes in the study of large-scale problems. Finally, a comparison between the single-relaxation-time model and the MRT model is also conducted, and the results show that the MRT model is more accurate than the single-relaxation-time model, both in solving the convection-diffusion equation and in computing the flux.
Ringe, Stefan; Oberhofer, Harald; Hille, Christoph; Matera, Sebastian; Reuter, Karsten
2016-08-01
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.
NASA Technical Reports Server (NTRS)
Khabibrakhmanov, I. KH.; Galeev, A. A.; Galinskii, V. L.
1993-01-01
Consideration is given to a collisionless parallel shock based on solitary-type solutions of the modified derivative nonlinear Schroedinger equation (MDNLS) for parallel Alfven waves. The standard derivative nonlinear Schroedinger equation is generalized in order to include the possible anisotropy of the plasma distribution and higher-order Korteweg-de Vies-type dispersion. Stationary solutions of MDNLS are discussed. The anisotropic nature of 'adiabatic' reflections leads to the asymmetric particle distribution in the upstream as well as in the downstream regions of the shock. As a result, nonzero heat flux appears near the front of the shock. It is shown that this causes the stochastic behavior of the nonlinear waves, which can significantly contribute to the shock thermalization.
Fluid dynamic description of flocking via the Povzner-Boltzmann equation
NASA Astrophysics Data System (ADS)
Fornasier, Massimo; Haskovec, Jan; Toscani, Giuseppe
2011-01-01
We introduce and discuss the possible dynamics of groups of indistinguishable agents, which are interacting according to their relative positions, with the aim of deriving hydrodynamic equations. These models are developed to mimic the collective motion of groups of species such as bird flocks, fish schools, herds of quadrupeds or bacteria colonies. Our starting model for these interactions is the Povzner equation [21], which describes a dilute gas in which binary collisions of elastic spheres depend on their relative positions. Following the Cucker and Smale model [9], we will consider binary interactions between agents that are dissipative collisions in which the coefficient of restitution depends on their relative distance. Under the assumption of weak dissipation, it is shown that the Povzner equation is modified through a correction in the form of a nonlinear friction type operator. Using this correction, we formally obtain from the Povzner equation in a direct way a fluid dynamic description of a system of agents with weak dissipative interactions, with a coefficient of restitution that depends on their relative distance.
St Aubin, J. Keyvanloo, A.; Fallone, B. G.; Vassiliev, O.
2015-02-15
Purpose: Accurate radiotherapy dose calculation algorithms are essential to any successful radiotherapy program, considering the high level of dose conformity and modulation in many of today’s treatment plans. As technology continues to progress, such as is the case with novel MRI-guided radiotherapy systems, the necessity for dose calculation algorithms to accurately predict delivered dose in increasingly challenging scenarios is vital. To this end, a novel deterministic solution has been developed to the first order linear Boltzmann transport equation which accurately calculates x-ray based radiotherapy doses in the presence of magnetic fields. Methods: The deterministic formalism discussed here with the inclusion of magnetic fields is outlined mathematically using a discrete ordinates angular discretization in an attempt to leverage existing deterministic codes. It is compared against the EGSnrc Monte Carlo code, utilizing the emf-macros addition which calculates the effects of electromagnetic fields. This comparison is performed in an inhomogeneous phantom that was designed to present a challenging calculation for deterministic calculations in 0, 0.6, and 3 T magnetic fields oriented parallel and perpendicular to the radiation beam. The accuracy of the formalism discussed here against Monte Carlo was evaluated with a gamma comparison using a standard 2%/2 mm and a more stringent 1%/1 mm criterion for a standard reference 10 × 10 cm{sup 2} field as well as a smaller 2 × 2 cm{sup 2} field. Results: Greater than 99.8% (94.8%) of all points analyzed passed a 2%/2 mm (1%/1 mm) gamma criterion for all magnetic field strengths and orientations investigated. All dosimetric changes resulting from the inclusion of magnetic fields were accurately calculated using the deterministic formalism. However, despite the algorithm’s high degree of accuracy, it is noticed that this formalism was not unconditionally stable using a discrete ordinate angular discretization
Shestakov, A I; Milovich, J L; Noy, A
2002-03-01
The nonlinear Poisson-Boltzmann (PB) equation is solved using Newton-Krylov iterations coupled with pseudo-transient continuation. The PB potential is used to compute the electrostatic energy and evaluate the force on a user-specified contour. The PB solver is embedded in a existing, 3D, massively parallel, unstructured-grid, finite element code. Either Dirichlet or mixed boundary conditions are allowed. The latter specifies surface charges, approximates far-field conditions, or linearizes conditions "regulating" the surface charge. Stability and robustness are proved using results for backward Euler differencing of diffusion equations. Potentials and energies of charged spheres and plates are computed and results compared to analysis. An approximation to the potential of the nonlinear, spherical charge is derived by combining two analytic formulae. The potential and force due to a conical probe interacting with a flat plate are computed for two types of boundary conditions: constant potential and constant charge. The second case is compared with direct force measurements by chemical force microscopy. The problem is highly nonlinear-surface potentials of the linear and nonlinear PB equations differ by over an order of magnitude. Comparison of the simulated and experimentally measured forces shows that approximately half of the surface carboxylic acid groups, of density 1/(0.2 nm2), ionize in the electrolyte implying surface charges of 0.4 C/m2, surface potentials of 0.27 V, and a force of 0.6 nN when the probe and plate are 8.7 nm apart. PMID:16290441
NASA Astrophysics Data System (ADS)
Huang, Juan-Chen; Hsieh, Tse-Yang; Yang, Jaw-Yen
2015-06-01
A robust and efficient numerical method in phase space combining the conservative discrete ordinate method and high order spatial discretization is proposed for the semiclassical Boltzmann equation with Bhatnagar-Gross-Krook (BGK) relaxation time approximation. A general Maxwell type partially diffuse and partially specular reflection solid wall boundary condition for modeling the gas-surface interactions for semiclassical rarefied gas flows is devised. The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term. The high order weighted essentially non-oscillatory scheme is then implemented to evolve the solution in physical space and time. The conservation property of the BGK collision integral is ensured to be fulfilled at the discrete quadrature level approximation. Both specular and diffuse reflection boundary conditions are implemented. Extensive computations of one- and two-dimensional semiclassical rarefied gas dynamical flows covering wide range of flow regimes for three statistics are presented to illustrate the method and boundary conditions treatment. The effect of accommodation coefficient on the rarefied flow field is also examined.
NASA Astrophysics Data System (ADS)
Illg, Christian; Haag, Michael; Teeny, Nicolas; Wirth, Jens; Fähnle, Manfred
2016-03-01
Scatterings of electrons at quasiparticles or photons are very important for many topics in solid-state physics, e.g., spintronics, magnonics or photonics, and therefore a correct numerical treatment of these scatterings is very important. For a quantum-mechanical description of these scatterings, Fermi's golden rule is used to calculate the transition rate from an initial state to a final state in a first-order time-dependent perturbation theory. One can calculate the total transition rate from all initial states to all final states with Boltzmann rate equations involving Brillouin zone integrations. The numerical treatment of these integrations on a finite grid is often done via a replacement of the Dirac delta distribution by a Gaussian. The Dirac delta distribution appears in Fermi's golden rule where it describes the energy conservation among the interacting particles. Since the Dirac delta distribution is a not a function it is not clear from a mathematical point of view that this procedure is justified. We show with physical and mathematical arguments that this numerical procedure is in general correct, and we comment on critical points.
Hua, Chengyun; Minnich, Austin J.
2015-05-07
Cross-plane heat transport in thin films with thicknesses comparable to the phonon mean free paths is of both fundamental and practical interest for applications such as light-emitting diodes and quantum well lasers. However, physical insight is difficult to obtain for the cross-plane geometry due to the challenge of solving the Boltzmann equation in a finite domain. Here, we present a semi-analytical series expansion method to solve the transient, frequency-dependent Boltzmann transport equation that is valid from the diffusive to ballistic transport regimes and rigorously includes the frequency-dependence of phonon properties. Further, our method is more than three orders of magnitude faster than prior numerical methods and provides a simple analytical expression for the thermal conductivity as a function of film thickness. Our result enables a straightforward physical understanding of cross-plane heat conduction in thin films.
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.
Kópházi, József Lathouwers, Danny
2015-09-15
In this paper a new method for the discretization of the radiation transport equation is presented, based on a discontinuous Galerkin method in space and angle that allows for local refinement in angle where any spatial element can support its own angular discretization. To cope with the discontinuous spatial nature of the solution, a generalized Riemann procedure is required to distinguish between incoming and outgoing contributions of the numerical fluxes. A new consistent framework is introduced that is based on the solution of a generalized eigenvalue problem. The resulting numerical fluxes for the various possible cases where neighboring elements have an equal, higher or lower level of refinement in angle are derived based on tensor algebra and the resulting expressions have a very clear physical interpretation. The choice of discontinuous trial functions not only has the advantage of easing local refinement, it also facilitates the use of efficient sweep-based solvers due to decoupling of unknowns on a large scale thereby approaching the efficiency of discrete ordinates methods with local angular resolution. The approach is illustrated by a series of numerical experiments. Results show high orders of convergence for the scalar flux on angular refinement. The generalized Riemann upwinding procedure leads to stable and consistent solutions. Further the sweep-based solver performs well when used as a preconditioner for a Krylov method.
Huang, Cheng-Hsuan; Cheng, Wen-Li; He, Yan-Ying; Lee, Eric
2010-08-12
Electrophoresis of a soft particle along the centerline of a cylindrical pore is investigated theoretically in this study. The soft particle consists of an inner hard sphere covered by a concentric porous layer with fixed charge uniformly distributed in it. The polarization effect, the deformation of ion clouds surrounding the particle due to convection flow, is taken into account properly by adopting the full nonlinear Poisson-Boltzmann equation. The study reveals that recent investigation in the literature without consideration of the polarization effect could severely overestimate the particle mobility up to nearly two times if the fixed charge in the porous layer is high. The boundary effect in terms of the reduction of particle mobility is very significant when the double layer is thick and diminishes as it gets very thin. The effect of the highly charged cylindrical wall is analyzed, in particular, among other factors of electrokinetic interest. The presence of the cylindrical wall retards the particle motion in general, as compared with an isolated particle. With the generation of an electroosmotic flow, however, the charged wall can either enhance the particle motion or deter it, depending on the surface potential on the wall and the double-layer thickness. The thinner the double layer, the more significant the influence of the osmotic flow on the particle motion in general. The direction of particle motion may even change twice as the reciprocal of the double-layer thickness increases when both the wall and the particle are highly charged. This is due to the competition between the electric driving force of the charged particle and the hydrodynamic retarding force from the background electroosmotic flow. This has direct impact in practical applications of nanofluidics when a weak electric field is applied. Conducting operations near these critical double-layer thicknesses should be avoided in practice.
Fluctuating multicomponent lattice Boltzmann model
NASA Astrophysics Data System (ADS)
Belardinelli, D.; Sbragaglia, M.; Biferale, L.; Gross, M.; Varnik, F.
2015-02-01
Current implementations of fluctuating lattice Boltzmann equations (FLBEs) describe single component fluids. In this paper, a model based on the continuum kinetic Boltzmann equation for describing multicomponent fluids is extended to incorporate the effects of thermal fluctuations. The thus obtained fluctuating Boltzmann equation is first linearized to apply the theory of linear fluctuations, and expressions for the noise covariances are determined by invoking the fluctuation-dissipation theorem directly at the kinetic level. Crucial for our analysis is the projection of the Boltzmann equation onto the orthonormal Hermite basis. By integrating in space and time the fluctuating Boltzmann equation with a discrete number of velocities, the FLBE is obtained for both ideal and nonideal multicomponent fluids. Numerical simulations are specialized to the case where mean-field interactions are introduced on the lattice, indicating a proper thermalization of the system.
Mikell, Justin K.; Klopp, Ann H.; Gonzalez, Graciela M.N.; Kisling, Kelly D.; Price, Michael J.; Berner, Paula A.; Eifel, Patricia J.; Mourtada, Firas
2012-07-01
Purpose: To investigate the dosimetric impact of the heterogeneity dose calculation Acuros (Transpire Inc., Gig Harbor, WA), a grid-based Boltzmann equation solver (GBBS), for brachytherapy in a cohort of cervical cancer patients. Methods and Materials: The impact of heterogeneities was retrospectively assessed in treatment plans for 26 patients who had previously received {sup 192}Ir intracavitary brachytherapy for cervical cancer with computed tomography (CT)/magnetic resonance-compatible tandems and unshielded colpostats. The GBBS models sources, patient boundaries, applicators, and tissue heterogeneities. Multiple GBBS calculations were performed with and without solid model applicator, with and without overriding the patient contour to 1 g/cm{sup 3} muscle, and with and without overriding contrast materials to muscle or 2.25 g/cm{sup 3} bone. Impact of source and boundary modeling, applicator, tissue heterogeneities, and sensitivity of CT-to-material mapping of contrast were derived from the multiple calculations. American Association of Physicists in Medicine Task Group 43 (TG-43) guidelines and the GBBS were compared for the following clinical dosimetric parameters: Manchester points A and B, International Commission on Radiation Units and Measurements (ICRU) report 38 rectal and bladder points, three and nine o'clock, and {sub D2cm3} to the bladder, rectum, and sigmoid. Results: Points A and B, D{sub 2} cm{sup 3} bladder, ICRU bladder, and three and nine o'clock were within 5% of TG-43 for all GBBS calculations. The source and boundary and applicator account for most of the differences between the GBBS and TG-43 guidelines. The D{sub 2cm3} rectum (n = 3), D{sub 2cm3} sigmoid (n = 1), and ICRU rectum (n = 6) had differences of >5% from TG-43 for the worst case incorrect mapping of contrast to bone. Clinical dosimetric parameters were within 5% of TG-43 when rectal and balloon contrast were mapped to bone and radiopaque packing was not overridden. Conclusions
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 does 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.
NASA Technical Reports Server (NTRS)
Bernhardt, P. A.
1979-01-01
The paper is motivated by a need for describing high-altitude gas releases from rockets or the Space Shuttle. In the tenuous upper atmosphere, the injected gases expand from a collisionless to a collision-dominated state. The analysis presented extends the efforts of Baum (1973-1974) to include the effects of a nonuniform background atmosphere and a velocity-dependent collision frequency. The unsteady expansion of gas releases is analyzed using gas kinetic theory. The Boltzmann equation with the Krook collision integral is solved numerically. At late times (after many collisions with the background atmosphere) the solution is identical to the one given by diffusion in a nonuniform atmosphere (Bernhardt, 1979). The theoretical model predicts elongation and heating of the vapor trails due to collisions.
Cui, Shuqi; Hong, Ning; Shi, Baochang; Chai, Zhenhua
2016-04-01
In this paper, we will focus on the multiple-relaxation-time (MRT) lattice Boltzmann model for two-dimensional convection-diffusion equations (CDEs), and analyze the discrete effect on the halfway bounce-back (HBB) boundary condition (or sometimes called bounce-back boundary condition) of the MRT model where three different discrete velocity models are considered. We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in the x or y direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a second-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the single-relaxation-time model or the Bhatnagar-Gross-Krook model unless the relaxation parameter related to the diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that the numerical results are consistent with our theoretical analysis. Finally, we would also like to point out the present analysis can be extended to other boundary conditions of lattice Boltzmann models for CDEs.
Cui, Shuqi; Hong, Ning; Shi, Baochang; Chai, Zhenhua
2016-04-01
In this paper, we will focus on the multiple-relaxation-time (MRT) lattice Boltzmann model for two-dimensional convection-diffusion equations (CDEs), and analyze the discrete effect on the halfway bounce-back (HBB) boundary condition (or sometimes called bounce-back boundary condition) of the MRT model where three different discrete velocity models are considered. We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in the x or y direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a second-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the single-relaxation-time model or the Bhatnagar-Gross-Krook model unless the relaxation parameter related to the diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that the numerical results are consistent with our theoretical analysis. Finally, we would also like to point out the present analysis can be extended to other boundary conditions of lattice Boltzmann models for CDEs. PMID:27176432
Dielectric and permeability effects in collisionless plasmas. [in collisionless plasmas
NASA Technical Reports Server (NTRS)
Cole, K. D.
1984-01-01
Using the unabridged Maxwell equations (including vectors D, E and H) new effects in collisionless plasmas are uncovered. In a steady state, it is found that spatially varying energy density of the electric field (E perpendicular) orthogonal to B produces electric current leading, under certain conditions, to the relationship P perpendicular + B(2)/8 pi-epsilon E perpendicular(2)/8 pi = constant, where epsilon is the dielectric constant of the plasma for fields orthogonal to B. In steady state quasi-two-dimensional flows in plasmas, a general relationship between the components of electric field parallel and perpendicular to B is found. These effects are significant in geophysical and astrophysical plasmas. The general conditions for a steady state in collisionless plasma are deduced. With time variations in a plasma, slow compared to ion-gyroperiod, there is a general current, (j-asterisk), which includes the well-known polarization current, given by J-asterisk = d/dt (E x M) + (P x B) x B B(-2) where M and P are the magnetization and polarization vectors respectively.
Honey, D.A.
1989-12-01
The collisional Boltzmann equation was solved numerically to obtain excitation rates for use in a CO{sub 2} laser design program. The program was written in Microsoft QuickBasic for use on the IBM Personal Computer or equivalent. Program validation involved comparisons of computed transport coefficients with experimental data and previous theoretical work. Four different numerical algorithms were evaluated in terms of accuracy and efficiency. L-U decomposition was identified as the preferred approach. The calculated transport coefficients were found to agree with empirical data within one to five percent. The program was integrated into a CO{sub 2} laser design program. Studies were then performed to evaluate the effects on predicted laser output power and energy density as parameters affecting electron kinetics were changed. Plotting routines were written for both programs.
Stellar dynamics around a massive black hole - I. Secular collisionless theory
NASA Astrophysics Data System (ADS)
Sridhar, S.; Touma, Jihad R.
2016-06-01
We present a theory in three parts, of the secular dynamics of a (Keplerian) stellar system of mass M orbiting a black hole of mass M• ≫ M. Here we describe the collisionless dynamics; Papers II and III are on the (collisional) theory of resonant relaxation. The mass ratio, ε = M/M• ≪ 1, is a natural small parameter implying a separation of time-scales between the short Kepler orbital periods and the longer orbital precessional periods. The collisionless Boltzmann equation (CBE) for the stellar distribution function (DF) is averaged over the fast Kepler orbital phase using the method of multiple scales. The orbit-averaged system is described by a secular DF, F, in a reduced phase space. F obeys a secular CBE that includes stellar self-gravity, general relativistic corrections up to 1.5 post-Newtonian order, and external sources varying over secular times. Secular dynamics, even with general time dependence, conserves the semimajor axis of every star. This additional integral of motion promotes extra regularity of the stellar orbits, and enables the construction of equilibria, F0, through a secular Jeans theorem. A linearized secular CBE determines the response and stability of F0. Spherical, non-rotating equilibria may support long-lived, warp-like distortions. We also prove that an axisymmetric, zero-thickness, flat disc is secularly stable to all in-plane perturbations, when its DF, F0, is a monotonic function of the angular momentum at fixed energy.
Reversible collisionless magnetic reconnection
Ishizawa, A.; Watanabe, T.-H.
2013-10-15
Reversible magnetic reconnection is demonstrated for the first time by means of gyrokinetic numerical simulations of a collisionless magnetized plasma. Growth of a current-driven instability in a sheared magnetic field is accompanied by magnetic reconnection due to electron inertia effects. Following the instability growth, the collisionless reconnection is accelerated with development of a cross-shaped structure of current density, and then all field lines are reconnected. The fully reconnected state is followed by the secondary reconnection resulting in a weakly turbulent state. A time-reversed simulation starting from the turbulent state manifests that the collisionless reconnection process proceeds inversely leading to the initial state. During the reversed reconnection, the kinetic energy is reconverted into the original magnetic field energy. In order to understand the stability of reversed process, an external perturbation is added to the fully reconnected state, and it is found that the accelerated reconnection is reversible when the deviation of the E × B streamlines due to the perturbation is comparable with or smaller than a current layer width.
NASA Astrophysics Data System (ADS)
Evesque, Pierre
2013-06-01
A 1d Boltzmann equation is introduced to describe the speed distribution function in granular gas system with local collision dissipation. It leads to introduce a new term, equivalent to an acceleration This term was always assumed to be 0, but it is not zero in general, even when the system is steady (i.e. when local mean flow equals 0). This shows that the flow (+ boundary) exerts a force on any extra steady particle (or plane) that drives it to the center. This result is analyzed, compared and interpreted using the Lagrangian & Eulerian view points of the mechanics; it demonstrates that classic view point of hydrodynamics does not hold anymore. The paper investigates different cases and gives experimental evidences of the features: it explains while local speed distribution f(v,r) of granular gas in a box subjected to vibration is non symmetric in the direction of vibration, while the system is stationary (mean local speed equals 0). Papers giving local experimental or simulated distributions are quoted, where two local pressures P± = Σv>0,orv<0 (mv2) in +Ox and -Ox direction are different. It implies also introducing two local temperatures T± in the ±Ox vibration direction. These points are confirmed using 2d and 3d granular gas simulation. It should apply likely to get deeper understanding of different effects as the "granular Leidenfrost effect", the stoppage of vibrated-hourglass, some turbulent flow, and the granular-Maxwell-demon.
Kan, Monica W. K.; Yu, Peter K. N.; Leung, Lucullus H. T.
2013-01-01
Deterministic linear Boltzmann transport equation (D-LBTE) solvers have recently been developed, and one of the latest available software codes, Acuros XB, has been implemented in a commercial treatment planning system for radiotherapy photon beam dose calculation. One of the major limitations of most commercially available model-based algorithms for photon dose calculation is the ability to account for the effect of electron transport. This induces some errors in patient dose calculations, especially near heterogeneous interfaces between low and high density media such as tissue/lung interfaces. D-LBTE solvers have a high potential of producing accurate dose distributions in and near heterogeneous media in the human body. Extensive previous investigations have proved that D-LBTE solvers were able to produce comparable dose calculation accuracy as Monte Carlo methods with a reasonable speed good enough for clinical use. The current paper reviews the dosimetric evaluations of D-LBTE solvers for external beam photon radiotherapy. This content summarizes and discusses dosimetric validations for D-LBTE solvers in both homogeneous and heterogeneous media under different circumstances and also the clinical impact on various diseases due to the conversion of dose calculation from a conventional convolution/superposition algorithm to a recently released D-LBTE solver. PMID:24066294
Ness, K F; Robson, R E; Brunger, M J; White, R D
2012-01-14
This paper revisits the issues surrounding computation of electron transport properties in water vapour as a function of E/n(0) (the ratio of the applied electric field to the water vapour number density) up to 1200 Td. We solve the Boltzmann equation using an improved version of the code of Ness and Robson [Phys. Rev. A 38, 1446 (1988)], facilitating the calculation of transport coefficients to a considerably higher degree of accuracy. This allows a correspondingly more discriminating test of the various electron-water vapour cross section sets proposed by a number of authors, which has become an important issue as such sets are now being applied to study electron driven processes in atmospheric phenomena [P. Thorn, L. Campbell, and M. Brunger, PMC Physics B 2, 1 (2009)] and in modeling charged particle tracks in matter [A. Munoz, F. Blanco, G. Garcia, P. A. Thorn, M. J. Brunger, J. P. Sullivan, and S. J. Buckman, Int. J. Mass Spectrom. 277, 175 (2008)]. PMID:22260590
Talon, Laurent; Bauer, Daniela
2013-12-01
Simulating flow of a Bingham fluid in porous media still remains a challenging task as the yield stress may significantly alter the numerical stability and precision. We present a Lattice-Boltzmann TRT scheme that allows the resolution of this type of flow in stochastically reconstructed porous media. LB methods have an intrinsic error associated to the boundary conditions. Depending on the schemes this error might be directly linked to the effective viscosity. As for non-Newtonian fluids viscosity varies in space the error becomes inhomogeneous and very important. In contrast to that, the TRT scheme does not present this deficiency and is therefore adequate to be used for simulations of non-Newtonian fluid flow. We simulated Bingham fluid flow in porous media and determined a generalized Darcy equation depending on the yield stress, the effective viscosity, the pressure drop and a characteristic length of the porous medium. By evaluating the flow in the porous structure, we distinguished three different scaling regimes. Regime I corresponds to the situation where fluid is flowing in only one channel. Here, the relation between flow rate and pressure drop is given by the non-Newtonian Poiseuille law. During Regime II an increase in pressure triggers the opening of new paths and the relation between flow rate and the difference in pressure to the critical yield pressure becomes quadratic: [Formula: see text]. Finally, Regime III corresponds to the situation where all the fluid is flowing. In this case, [Formula: see text]. PMID:24326905
Coupled electron and ion nonlinear oscillations in a collisionless plasma
Karimov, A. R.
2013-05-15
Dynamics of coupled electrostatic electron and ion nonlinear oscillations in a collisionless plasma is studied with reference to a kinetic description. Proceeding from the exact solution of Vlasov-Maxwell equations written as a function of linear functions in the electron and ion velocities, we arrive at the two coupled nonlinear equations which describe the evolution of the system.
Expansion techniques for collisionless stellar dynamical simulations
NASA Astrophysics Data System (ADS)
Meiron, Yohai
2016-02-01
We present ETICS, a collisionless N-body code based on two kinds of series expansions of the Poisson equation, implemented for graphics processing units (GPUs). The code is publicly available and can be used as a standalone program or as a library (an AMUSE plugin is included). One of the two expansion methods available is the self-consistent field (SCF) method, which is a Fourier-like expansion of the density field in some basis set; the other is the multipole expansion (MEX) method, which is a Taylor-like expansion of the Green's function. MEX, which has been advocated in the past, has not gained as much popularity as SCF. Both are particle-field methods and optimized for collisionless galactic dynamics, but while SCF is a ``pure'' expansion, MEX is an expansion in just the angular part; thus, MEX is capable of capturing radial structure easily, while SCF needs a large number of radial terms.
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.
Nonlinear Collisionless Magnetic Reconnection
Grasso, D.; Tassi, E.; Borgogno, D.; Pegoraro, F.
2008-10-15
We review some recent results that have been obtained in the investigation of collisionless reconnection in two and three dimensional magnetic configurations with a strong guide field in regimes of interest for laboratory plasmas. First, we adopt a two-field plasma model where two distinct regimes, laminar and turbulent, can be identified. Then, we show that these regimes may combine when we consider a more general four-field model, where perturbation of the magnetic and velocity fields are allowed also along the ignorable coordinate.
NASA Astrophysics Data System (ADS)
Dyatko, Nikolay; Donkó, Zoltán
2015-08-01
At low reduced electric fields the electron energy distribution function in heavy noble gases can take two distinct shapes. This ‘bistability effect’—in which electron-electron (Coulomb) collisions play an essential role—is analyzed here for Xe with a Boltzmann equation approach and with a first principles particle simulation method. The solution of the Boltzmann equation adopts the usual approximations of (i) searching for the distribution function in the form of two terms (‘two-term approximation’), (ii) neglecting the Coulomb part of the collision integral for the anisotropic part of the distribution function, (iii) treating Coulomb collisions as binary events, and (iv) truncating the range of the electron-electron interaction beyond a characteristic distance. The particle-based simulation method avoids these approximations: the many-body interactions within the electron gas with a true (un-truncated) Coulomb potential are described by a molecular dynamics algorithm, while the collisions between electrons and the background gas atoms are treated with Monte Carlo simulation. We find a good general agreement between the results of the two techniques, which confirms, to a certain extent, the approximations used in the solution of the Boltzmann equation. The differences observed between the results are believed to originate from these approximations and from the presence of statistical noise in the particle simulations.
Prinja, A.K.
1995-08-01
We have developed and successfully implemented a two-dimensional bilinear discontinuous in space and time, used in conjunction with the S{sub N} angular approximation, to numerically solve the time dependent, one-dimensional, one-speed, slab geometry, (ion) transport equation. Numerical results and comparison with analytical solutions have shown that the bilinear-discontinuous (BLD) scheme is third-order accurate in the space ad time dimensions independently. Comparison of the BLD results with diamond-difference methods indicate that the BLD method is both quantitavely and qualitatively superior to the DD scheme. We note that the form of the transport operator is such that these conclusions carry over to energy dependent problems that include the constant-slowing-down-approximation term, and to multiple space dimensions or combinations thereof. An optimized marching or inversion scheme or a parallel algorithm should be investigated to determine if the increased accuracy can compensate for the extra overhead required for a BLD solution, and then could be compared to other discretization methods such as nodal or characteristic schemes.
Lattice Boltzmann modeling of phonon transport
NASA Astrophysics Data System (ADS)
Guo, Yangyu; Wang, Moran
2016-06-01
A novel lattice Boltzmann scheme is proposed for phonon transport based on the phonon Boltzmann equation. Through the Chapman-Enskog expansion, the phonon lattice Boltzmann equation under the gray relaxation time approximation recovers the classical Fourier's law in the diffusive limit. The numerical parameters in the lattice Boltzmann model are therefore rigorously correlated to the bulk material properties. The new scheme does not only eliminate the fictitious phonon speed in the diagonal direction of a square lattice system in the previous lattice Boltzmann models, but also displays very robust performances in predicting both temperature and heat flux distributions consistent with analytical solutions for diverse numerical cases, including steady-state and transient, macroscale and microscale, one-dimensional and multi-dimensional phonon heat transport. This method may provide a powerful numerical tool for deep studies of nonlinear and nonlocal heat transports in nanosystems.
Nonlinear Gyroviscous Force in a Collisionless Plasma
Belova, E.V.
2001-05-23
Nonlinear gyroviscous forces in a collisionless plasma with temperature variations are calculated from the gyrofluid moments of the gyrokinetic Vlasov equation. The low-frequency gyrokinetic ordering and electrostatic perturbations are assumed, and an additional finite Larmor radius (FLR) expansion is performed. This approach leads naturally to an expression for the gyroviscous force in terms of the gyrocenter distribution function, thus including all resonant effects, and represents a systematic FLR expansion in a general form (no assumption of any closure is made). The expression for the gyroviscous force is also calculated in terms of the particle-fluid moments by making the transformation from the gyrocenter to particle coordinates. The calculated expression represents a modification of the Braginskii gyroviscosity for a collisionless plasma with nonuniform temperature. It is compared with previous calculations based on the traditional fluid approach. As a byproduct of the gyroviscosity calculations, we derive a set of nonlinear reduced gyrofluid (and a corresponding set of particle-fluid) moment equations with FLR corrections, which exhibit a generalized form of the ''gyroviscous cancellation.''
NASA Astrophysics Data System (ADS)
Schaap, M. G.
2013-12-01
This DOE-funded study is a collaboration between Oregon State University (led by Dr. Dorthe Wildenschild) and the University of Arizona to investigate pore-scale aspects of capillary trapping to enhance the efficiency of geological CO2 sequestration in deep saline aquifers where super-critical conditions prevail. Compared to most current reservoir-scale studies, our research takes several steps back in scale to observe and model trapping at the pore-scale using a combination of computed micro-tomography imaging (performed by OSU) and multi-phase/multi-component lattice Boltzmann (LB) simulations (carried out by UA). The main objective is to quantify how pore-scale mechanisms translate into continuum scale properties that can subsequently support improved modelling of sequestration at large spatio-temporal scales. For the purposes of this project it is important to correctly simulate the physical conditions under which super-critical CO2 will be present after injection into the host rock. In practice this means that the LB model should be able to handle the pressures (P), densities (ρ), temperatures (T) that prevail in deep geological media. A logical way of dealing with is is to combine a single-component LB model with and Equation of State (EOS) that describes the physical interrelations among P, ρ and T (Yuan and Scheafer, 2006). Previously, we showed that the Peng-Robinson (PR) EOS provides an excellent fit to super-critical conditions for the pure CO2 system. However, simulating pure-CO2 systems is not sufficient as the super-critical CO2 will co-exist (and interact) with brine present in the saline aquifers. In effect this means that we need to simulate multi-component systems: one phase being the super-critical CO2, the other phase being a brine of varying salinity. Previously, we have used used a Shan-Chen-type model (Shan Chen, 1993, 1994) as modified by Martys and Chen (1996) for simplified capillari pressure dominated air-water systems in porous media
A collisionless shock wave experiment
Winske, D.; Jones, M.E.; Sgro, A.G.; Thomas, V.A.
1995-04-01
Collisionless shock waves are a very important heating mechanism for plasmas and are commonly found in space and astrophysical environments. Collisionless shocks were studied in the laboratory more than 20 years ago, and more recently in space via in situ satellite measurements. The authors propose a new laboratory shock wave experiment to address unresolved issues related to the differences in the partition of plasma heating between electrons and ions in space and laboratory plasmas, which can have important implications for a number of physical systems.
Accurate deterministic solutions for the classic Boltzmann shock profile
NASA Astrophysics Data System (ADS)
Yue, Yubei
The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.
Generalized Boltzmann formalism for oscillating neutrinos
Strack, P.; Burrows, A.
2005-05-01
In the standard approaches to neutrino transport in the simulation of core-collapse supernovas, one will often start from the classical Boltzmann equation for the neutrino's spatial, temporal, and spectral evolution. For each neutrino species, and its antiparticle, the classical density in phase space, or the associated specific intensity, will be calculated as a function of time. The neutrino radiation is coupled to matter by source and sink terms on the 'right-hand side' of the transport equation and together with the equations of hydrodynamics this set of coupled partial differential equations for classical densities describes, in principle, the evolution of core collapse and explosion. However, with the possibility of neutrino oscillations between species, a purely quantum-physical effect, how to generalize this set of Boltzmann equations for classical quantities to reflect oscillation physics has not been clear. To date, the formalisms developed have retained the character of quantum operator physics involving complex quantities and have not been suitable for easy incorporation into standard supernova codes. In this paper, we derive generalized Boltzmann equations for quasiclassical, real-valued phase-space densities that retain all the standard oscillation phenomenology, including the matter-enhanced resonant flavor conversion (Mikheev-Smirnov-Wolfenstein effect), neutrino self-interactions, and the interplay between decohering matter coupling and flavor oscillations. With this formalism, any code(s) that can now handle the solution of the classical Boltzmann or transport equation can easily be generalized to include neutrino oscillations in a quantum-physically consistent fashion.
MHD instabilities of collisionless space plasma with heat fluxes
NASA Astrophysics Data System (ADS)
Kuznetsov, V. D.; Dzhalilov, N. S.
2014-12-01
Properties of instabilities in a collisionless plasma are considered based on 16-moment MHD equations with allowance for differences between the heat fluxes along the magnetic field due to longitudinal and transverse thermal ion motions. It is shown that the increments and thresholds appreciably depend on these two heat fluxes for all compressible instabilities arising in the MHD approach (second compressible fire-hose, mirror, and thermal instabilities).
Intercomponent momentum transport and electrical conductivity of collisionless plasma
NASA Technical Reports Server (NTRS)
Wilhelm, H. E.
1973-01-01
Based on the Lenard-Balescu equation, the interaction integral for the intercomponent momentum transfer in a two-component, collisionless plasma is evaluated in closed form. The distribution functions of the electrons and ions are represented in the form of nonisothermal, displaced Maxwellians corresponding to the 5-moment approximation. As an application, the transport of electrical current in an electric field is discussed for infrasonic up to sonic electron-ion drift velocities.
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.
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-Langevin theory of Coulomb drag
NASA Astrophysics Data System (ADS)
Chen, W.; Andreev, A. V.; Levchenko, A.
2015-06-01
We develop a Boltzmann-Langevin description of the Coulomb drag effect in clean double-layer systems with large interlayer separation d as compared to the average interelectron distance λF. Coulomb drag arises from density fluctuations with spatial scales of order d . At low temperatures, their characteristic frequencies exceed the intralayer equilibration rate of the electron liquid, and Coulomb drag may be treated in the collisionless approximation. As temperature is raised, the electron mean free path becomes short due to electron-electron scattering. This leads to local equilibration of electron liquid, and consequently drag is determined by hydrodynamic density modes. Our theory applies to both the collisionless and the hydrodynamic regimes, and it enables us to describe the crossover between them. We find that drag resistivity exhibits a nonmonotonic temperature dependence with multiple crossovers at distinct energy scales. At the lowest temperatures, Coulomb drag is dominated by the particle-hole continuum, whereas at higher temperatures of the collision-dominated regime it is governed by the plasmon modes. We observe that fast intralayer equilibration mediated by electron-electron collisions ultimately renders a stronger drag effect.
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.
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.
Crystallographic Lattice Boltzmann Method
NASA Astrophysics Data System (ADS)
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-06-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows.
Crystallographic Lattice Boltzmann Method.
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-01-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows. PMID:27251098
Crystallographic Lattice Boltzmann Method
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-01-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows. PMID:27251098
Is collisionless heating in capacitively coupled plasmas really collisionless?
NASA Astrophysics Data System (ADS)
Lafleur, T.; Chabert, P.
2015-08-01
By performing a combination of test-particle and particle-in-cell simulations, we investigate electron heating in single frequency capacitively coupled plasmas (CCPs). In agreement with previous theoretical considerations highlighted in Kaganovich et al (1996 Appl. Phys. Lett. 69 3818), we show that the level of true collisionless/stochastic heating in typical CCPs is significantly smaller than that due to collisional interactions; even at very low pressures and wide gap lengths. Fundamentally electron heating is a collisional phenomenon whereby particle collisions provide the vital phase randomization and stochastization mechanism needed to generate both a local (or ohmic) heating component, and a non-local (or hybrid) heating component.
Collisionless drift-tearing modes in the magnetopause
NASA Technical Reports Server (NTRS)
Gladd, N. T.
1990-01-01
The linear stability properties of collisionless drift-tearing modes are analyzed in a modified Harris equilibrium model of the magnetopause. Particular attention is paid to the relevance of the parametric behavior of growth rates to the 'magnetic percolation' theory of flux transfer event formation (Galeev et al., 1986). Numerical methods are used to solve the drift-tearing eigenmode equations and the results are compared with those previously obtained by analytical methods. The analytical results are found to correctly model important parametric dependencies but to typically overestimate the rate of growth. The eigenmode equations are numerically difficult, and an integration scheme utilizing Ricatti transforms is developed to affect their solution.
Bowen; Sharif
1997-03-15
A Galerkin finite-element approach combined with an error estimator and automatic mesh refinement has been used to provide a flexible numerical solution of the Poisson-Boltzmann equation. A Newton sequence technique was used to solve the nonlinear equations arising from the finite-element discretization procedure. Errors arising from the finite-element solution due to mesh refinement were calculated using the Zienkiewicz-Zhu error estimator, and an automatic remeshing strategy was adopted to achieve a solution satisfying a preset quality. Examples of the performance of the error estimator in adaptive mesh refinement are presented. The adaptive finite-element scheme presented in this study has proved to be an effective technique in minimizing errors in finite-element solutions for a given problem, in particular those of complex geometries. As an example, numerical solutions are presented for the case of a charged spherical particle at various distances from a charged cylindrical pore in a charged planar surface. Such a scheme provides a quantification of the significance of electrostatic interactions for an important industrial technology-membrane separation processes.
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.
Boltzmann-Electron Model in Aleph.
Hughes, Thomas Patrick; Hooper, Russell
2014-11-01
We apply the Boltzmann-electron model in the electrostatic, particle-in-cell, finite- element code Aleph to a plasma sheath. By assuming a Boltzmann energy distribution for the electrons, the model eliminates the need to resolve the electron plasma fre- quency, and avoids the numerical "grid instability" that can cause unphysical heating of electrons. This allows much larger timesteps to be used than with kinetic electrons. Ions are treated with the standard PIC algorithm. The Boltzmann-electron model re- quires solution of a nonlinear Poisson equation, for which we use an iterative Newton solver (NOX) from the Trilinos Project. Results for the spatial variation of density and voltage in the plasma sheath agree well with an analytic model
TEMPEST Simulations of Collisionless Damping of Geodesic-Acoustic Mode in Edge Plasma Pedestal
Xu, X; Xiong, Z; Nevins, W; McKee, G
2007-05-31
The fully nonlinear 4D TEMPEST gyrokinetic continuum code produces frequency, collisionless damping of geodesic-acoustic mode (GAM) and zonal flow with fully nonlinear Boltzmann electrons for the inverse aspect ratio {epsilon}-scan and the tokamak safety factor q-scan in homogeneous plasmas. The TEMPEST simulation shows that GAM exists in edge plasma pedestal for steep density and temperature gradients, and an initial GAM relaxes to the standard neoclassical residual, rather than Rosenbluth-Hinton residual due to the presence of ion-ion collisions. The enhanced GAM damping explains experimental BES measurements on the edge q scaling of the GAM amplitude.
TEMPEST Simulations of Collisionless Damping of Geodesic-Acoustic Mode in Edge Plasma Pedestal
Xu, X Q; Xiong, Z; Nevins, W M; McKee, G R
2007-05-30
The fully nonlinear (full-f) 4D TEMPEST gyrokinetic continuum code produces frequency, collisionless damping of GAM and zonal flow with fully nonlinear Boltzmann electrons for the inverse aspect ratio {epsilon}-scan and the tokamak safety factor q-scan in homogeneous plasmas. The TEMPEST simulation shows that GAM exists in edge plasma pedestal for steep density and temperature gradients, and an initial GAM relaxes to the standard neoclassical residual, rather than Rosenbluth-Hinton residual due to the presence of ion-ion collisions. The enhanced GAM damping explains experimental BES measurements on the edge q scaling of the GAM amplitude.
Tempest Simulations of Collisionless Damping of the Geodesic-Acoustic Mode in Edge-Plasma Pedestals
Xu, X. Q.; Xiong, Z.; Nevins, W. M.; Gao, Z.; McKee, G. R.
2008-05-30
The fully nonlinear (full-f) four-dimensional TEMPEST gyrokinetic continuum code correctly produces the frequency and collisionless damping of geodesic-acoustic modes (GAMs) and zonal flow, with fully nonlinear Boltzmann electrons for the inverse aspect ratio {epsilon} scan and the tokamak safety factor q scan in homogeneous plasmas. TEMPEST simulations show that the GAMs exist in the edge pedestal for steep density and temperature gradients in the form of outgoing waves. The enhanced GAM damping may explain experimental beam emission spectroscopy measurements on the edge q scaling of the GAM amplitude.
Tempest Simulations of Collisionless Damping of the Geodesic-Acoustic Mode in Edge-Plasma Pedestals
NASA Astrophysics Data System (ADS)
Xu, X. Q.; Xiong, Z.; Gao, Z.; Nevins, W. M.; McKee, G. R.
2008-05-01
The fully nonlinear (full-f) four-dimensional TEMPEST gyrokinetic continuum code correctly produces the frequency and collisionless damping of geodesic-acoustic modes (GAMs) and zonal flow, with fully nonlinear Boltzmann electrons for the inverse aspect ratio γ scan and the tokamak safety factor q scan in homogeneous plasmas. TEMPEST simulations show that the GAMs exist in the edge pedestal for steep density and temperature gradients in the form of outgoing waves. The enhanced GAM damping may explain experimental beam emission spectroscopy measurements on the edge q scaling of the GAM amplitude.
Lattice Boltzmann model for wave propagation.
Zhang, Jianying; Yan, Guangwu; Shi, Xiubo
2009-08-01
A lattice Boltzmann model for two-dimensional wave equation is proposed by using the higher-order moment method. The higher-order moment method is based on the solution of a series of partial differential equations obtained by using multiscale technique and Chapman-Enskog expansion. In order to obtain the lattice Boltzmann model for the wave equation with higher-order accuracy of truncation errors, we removed the second-order dissipation term and the third-order dispersion term by employing the moments up to fourth order. The reversibility in time appears owing to the absence of the second-order dissipation term and the third-order dispersion term. As numerical examples, some classical examples, such as interference, diffraction, and wave passing through a convex lens, are simulated. The numerical results show that this model can be used to simulate wave propagation.
The role of entropy in collisionless evolution
NASA Astrophysics Data System (ADS)
Barnes, Eric
2011-04-01
Understanding the path to mechanical equilibria for collisionless systems is a topic with a rich history. We are investigating the part that entropy plays in determining the outcome of collisionless evolution. I will discuss some previous work that lays a foundation for our current studies. With that framework in place, I will explain a modification to the entropy maximization procedure and compare the results to previous ideas. It is possible that entropy maxima are not achievable, and we argue that entropy production rates can influence collisionless system evolution. This work has been supported by NASA ATP grant NNX07AG86G.
Monte Carlo variance reduction approaches for non-Boltzmann tallies
Booth, T.E.
1992-12-01
Quantities that depend on the collective effects of groups of particles cannot be obtained from the standard Boltzmann transport equation. Monte Carlo estimates of these quantities are called non-Boltzmann tallies and have become increasingly important recently. Standard Monte Carlo variance reduction techniques were designed for tallies based on individual particles rather than groups of particles. Experience with non-Boltzmann tallies and analog Monte Carlo has demonstrated the severe limitations of analog Monte Carlo for many non-Boltzmann tallies. In fact, many calculations absolutely require variance reduction methods to achieve practical computation times. Three different approaches to variance reduction for non-Boltzmann tallies are described and shown to be unbiased. The advantages and disadvantages of each of the approaches are discussed.
Sugimoto, Yu; Kitazumi, Yuki; Shirai, Osamu; Yamamoto, Masahiro; Kano, Kenji
2016-03-31
To understand electrostatic interactions in biomolecules, the bimolecular rate constants (k) between redox enzymes and charged substrates (in this study, redox mediators in the electrode reaction) were evaluated at various ionic strengths (I) for the mediated bioelectrocatalytic reaction. The k value between bilirubin oxidase (BOD) and positively charged mediators increased with I, while that between BOD and negatively charged mediators decreased with I. The opposite trend was observed for the reaction of glucose oxidase (GOD). In the case of noncharged mediators, the k value was independent of I for both BOD and GOD. These results reflect the electrostatic interactions between the enzymes and the mediators. Furthermore, we estimated k/k° (k° being the thermodynamic rate constant) by numerical simulation (finite element method) based on the Poisson-Boltzmann (PB) equation. By considering the charges of individual atoms involved in the amino acids around the substrate binding sites in the enzymes, the simulated k/k° values well reproduced the experimental data. In conclusion, k/k° can be predicted by PB-based simulation as long as the crystal structure of the enzyme and the substrate binding site are known. PMID:26956542
Dispersion discontinuities of strong collisionless shocks
NASA Technical Reports Server (NTRS)
Coroniti, F. V.
1970-01-01
Linear fluid equations are used to estimate wave train properties of strong collisionless shocks. Fast shocks exhibit several dispersion changes with increasing Mach number. For perpendicular propagation into a finite-beta plasma, an ion cyclotron radius trailing wave train exists only for (M sub F)2 is smaller than 2. Oblique fast shocks have a leading ion inertia wave train if M sub A is smaller than root of M(+)/M(-) cos theta/2 and a trailing electron inertia train if M sub A is greater than root of M(+)/M(-) cos theta/2. If the downstream sound speed exceeds the flow speed, linear wave theory predicts a trailing ion acoustic structure which probably resides within the magnetic shock. For a turbulent shock model in which an effective electron-ion collision frequency exceeds the lower hybrid frequency, ions decouple from the magnetic field; the shock wave train now trails with electron inertia and electron gyroradius lengths. Comparisons of this turbulent model and observations on the earth's bow shock are made.
Turbulent dynamo in a collisionless plasma.
Rincon, François; Califano, Francesco; Schekochihin, Alexander A; Valentini, Francesco
2016-04-12
Magnetic fields pervade the entire universe and affect the formation and evolution of astrophysical systems from cosmological to planetary scales. The generation and dynamical amplification of extragalactic magnetic fields through cosmic times (up to microgauss levels reported in nearby galaxy clusters, near equipartition with kinetic energy of plasma motions, and on scales of at least tens of kiloparsecs) are major puzzles largely unconstrained by observations. A dynamo effect converting kinetic flow energy into magnetic energy is often invoked in that context; however, extragalactic plasmas are weakly collisional (as opposed to magnetohydrodynamic fluids), and whether magnetic field growth and sustainment through an efficient turbulent dynamo instability are possible in such plasmas is not established. Fully kinetic numerical simulations of the Vlasov equation in a 6D-phase space necessary to answer this question have, until recently, remained beyond computational capabilities. Here, we show by means of such simulations that magnetic field amplification by dynamo instability does occur in a stochastically driven, nonrelativistic subsonic flow of initially unmagnetized collisionless plasma. We also find that the dynamo self-accelerates and becomes entangled with kinetic instabilities as magnetization increases. The results suggest that such a plasma dynamo may be realizable in laboratory experiments, support the idea that intracluster medium turbulence may have significantly contributed to the amplification of cluster magnetic fields up to near-equipartition levels on a timescale shorter than the Hubble time, and emphasize the crucial role of multiscale kinetic physics in high-energy astrophysical plasmas. PMID:27035981
Turbulent dynamo in a collisionless plasma
Rincon, François; Califano, Francesco; Schekochihin, Alexander A.; Valentini, Francesco
2016-01-01
Magnetic fields pervade the entire universe and affect the formation and evolution of astrophysical systems from cosmological to planetary scales. The generation and dynamical amplification of extragalactic magnetic fields through cosmic times (up to microgauss levels reported in nearby galaxy clusters, near equipartition with kinetic energy of plasma motions, and on scales of at least tens of kiloparsecs) are major puzzles largely unconstrained by observations. A dynamo effect converting kinetic flow energy into magnetic energy is often invoked in that context; however, extragalactic plasmas are weakly collisional (as opposed to magnetohydrodynamic fluids), and whether magnetic field growth and sustainment through an efficient turbulent dynamo instability are possible in such plasmas is not established. Fully kinetic numerical simulations of the Vlasov equation in a 6D-phase space necessary to answer this question have, until recently, remained beyond computational capabilities. Here, we show by means of such simulations that magnetic field amplification by dynamo instability does occur in a stochastically driven, nonrelativistic subsonic flow of initially unmagnetized collisionless plasma. We also find that the dynamo self-accelerates and becomes entangled with kinetic instabilities as magnetization increases. The results suggest that such a plasma dynamo may be realizable in laboratory experiments, support the idea that intracluster medium turbulence may have significantly contributed to the amplification of cluster magnetic fields up to near-equipartition levels on a timescale shorter than the Hubble time, and emphasize the crucial role of multiscale kinetic physics in high-energy astrophysical plasmas. PMID:27035981
Expansion techniques for collisionless stellar dynamical simulations
Meiron, Yohai; Li, Baile; Holley-Bockelmann, Kelly; Spurzem, Rainer
2014-09-10
We present graphics processing unit (GPU) implementations of two fast force calculation methods based on series expansions of the Poisson equation. One method is the self-consistent field (SCF) method, which is a Fourier-like expansion of the density field in some basis set; the other method is the multipole expansion (MEX) method, which is a Taylor-like expansion of the Green's function. MEX, which has been advocated in the past, has not gained as much popularity as SCF. Both are particle-field methods and optimized for collisionless galactic dynamics, but while SCF is a 'pure' expansion, MEX is an expansion in just the angular part; thus, MEX is capable of capturing radial structure easily, while SCF needs a large number of radial terms. We show that despite the expansion bias, these methods are more accurate than direct techniques for the same number of particles. The performance of our GPU code, which we call ETICS, is profiled and compared to a CPU implementation. On the tested GPU hardware, a full force calculation for one million particles took ∼0.1 s (depending on expansion cutoff), making simulations with as many as 10{sup 8} particles fast for a comparatively small number of nodes.
Turbulent dynamo in a collisionless plasma
NASA Astrophysics Data System (ADS)
Rincon, François; Califano, Francesco; Schekochihin, Alexander A.; Valentini, Francesco
2016-04-01
Magnetic fields pervade the entire universe and affect the formation and evolution of astrophysical systems from cosmological to planetary scales. The generation and dynamical amplification of extragalactic magnetic fields through cosmic times (up to microgauss levels reported in nearby galaxy clusters, near equipartition with kinetic energy of plasma motions, and on scales of at least tens of kiloparsecs) are major puzzles largely unconstrained by observations. A dynamo effect converting kinetic flow energy into magnetic energy is often invoked in that context; however, extragalactic plasmas are weakly collisional (as opposed to magnetohydrodynamic fluids), and whether magnetic field growth and sustainment through an efficient turbulent dynamo instability are possible in such plasmas is not established. Fully kinetic numerical simulations of the Vlasov equation in a 6D-phase space necessary to answer this question have, until recently, remained beyond computational capabilities. Here, we show by means of such simulations that magnetic field amplification by dynamo instability does occur in a stochastically driven, nonrelativistic subsonic flow of initially unmagnetized collisionless plasma. We also find that the dynamo self-accelerates and becomes entangled with kinetic instabilities as magnetization increases. The results suggest that such a plasma dynamo may be realizable in laboratory experiments, support the idea that intracluster medium turbulence may have significantly contributed to the amplification of cluster magnetic fields up to near-equipartition levels on a timescale shorter than the Hubble time, and emphasize the crucial role of multiscale kinetic physics in high-energy astrophysical plasmas.
Turbulent dynamo in a collisionless plasma.
Rincon, François; Califano, Francesco; Schekochihin, Alexander A; Valentini, Francesco
2016-04-12
Magnetic fields pervade the entire universe and affect the formation and evolution of astrophysical systems from cosmological to planetary scales. The generation and dynamical amplification of extragalactic magnetic fields through cosmic times (up to microgauss levels reported in nearby galaxy clusters, near equipartition with kinetic energy of plasma motions, and on scales of at least tens of kiloparsecs) are major puzzles largely unconstrained by observations. A dynamo effect converting kinetic flow energy into magnetic energy is often invoked in that context; however, extragalactic plasmas are weakly collisional (as opposed to magnetohydrodynamic fluids), and whether magnetic field growth and sustainment through an efficient turbulent dynamo instability are possible in such plasmas is not established. Fully kinetic numerical simulations of the Vlasov equation in a 6D-phase space necessary to answer this question have, until recently, remained beyond computational capabilities. Here, we show by means of such simulations that magnetic field amplification by dynamo instability does occur in a stochastically driven, nonrelativistic subsonic flow of initially unmagnetized collisionless plasma. We also find that the dynamo self-accelerates and becomes entangled with kinetic instabilities as magnetization increases. The results suggest that such a plasma dynamo may be realizable in laboratory experiments, support the idea that intracluster medium turbulence may have significantly contributed to the amplification of cluster magnetic fields up to near-equipartition levels on a timescale shorter than the Hubble time, and emphasize the crucial role of multiscale kinetic physics in high-energy astrophysical plasmas.
Physics of collisionless phase mixing
Tsiklauri, D.; Haruki, T.
2008-11-15
Previous studies of phase mixing of ion cyclotron (IC), Alfvenic, waves in the collisionless regime have established the generation of parallel electric field and hence acceleration of electrons in the regions of transverse density inhomogeneity. However, outstanding issues were left open. Here we use the 2.5 D, relativistic, fully electromagnetic particle-in-cell code and an analytic magnetohydrodynamic (MHD) formulation, to establish the following points: (i) Using the generalized Ohm's law we find that the parallel electric field is supported mostly by the electron pressure tensor, with a smaller contribution from the electron inertia term. (ii) The generated parallel electric field and the fraction of accelerated electrons are independent of the IC wave frequency remaining at a level of six orders of magnitude larger than the Dreicer value and approximately 20%, respectively. The generated parallel electric field and the fraction of accelerated electrons increase with the increase of IC wave amplitude. The generated parallel electric field seems to be independent of plasma beta, while the fraction of accelerated electrons strongly increases with the decrease of plasma beta (for plasma beta of 0.0001 the fraction of accelerated electrons can be as large as 47%). (iii) In the collisionless regime IC wave dissipation length (that is defined as the distance over which the wave damps) variation with the driving frequency shows a deviation from the analytical MHD result, which we attribute to a possible frequency dependence of the effective resistivity. (iv) Effective anomalous resistivity, inferred from our numerical simulations, is at least four orders of magnitude larger than the classical Spitzer value.
Analytical collisionless damping rate of geodesic acoustic mode
NASA Astrophysics Data System (ADS)
Ren, H.; Xu, X. Q.
2016-10-01
Collisionless damping of geodesic acoustic mode (GAM) is analytically investigated by considering the finite-orbit-width (FOW) resonance effect to the 3rd order in the gyro-kinetic equations. A concise and transparent expression for the damping rate is presented for the first time. Good agreement is found between the analytical damping rate and the previous TEMPEST simulation result (Xu 2008 et al Phys. Rev. Lett. 100 215001) for systematic q scans. Our result also shows that it is of sufficient accuracy and has to take into account the FOW effect to the 3rd order.
NASA Astrophysics Data System (ADS)
Kulchytskyy, Bohdan; Andriyash, Evgeny; Amin, Mohammed; Melko, Roger
The field of machine learning has been revolutionized by the recent improvements in the training of deep networks. Their architecture is based on a set of stacked layers of simpler modules. One of the most successful building blocks, known as a restricted Boltzmann machine, is an energetic model based on the classical Ising Hamiltonian. In our work, we investigate the benefits of quantum effects on the learning capacity of Boltzmann machines by extending its underlying Hamiltonian with a transverse field. For this purpose, we employ exact and stochastic training procedures on data sets with physical origins.
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.
Entropy in the sense of Boltzmann and Poincaré
NASA Astrophysics Data System (ADS)
Vedenyapin, V. V.; Adzhiev, S. Z.
2014-12-01
The H-theorem is proved for generalized equations of chemical kinetics, and important physical examples of such generalizations are considered: a discrete model of the quantum kinetic equations (the Uehling-Uhlenbeck equations) and a quantum Markov process (a quantum random walk). The time means are shown to coincide with the Boltzmann extremals for these equations and for the Liouville equation. Bibliography: 41 titles.
Fermion particle production in semiclassical Boltzmann-Vlasov transport theory
Dawson, John F.; Mihaila, Bogdan; Cooper, Fred
2009-07-01
We present numerical solutions of the semiclassical Boltzmann-Vlasov equation for fermion particle-antiparticle production by strong electric fields in boost-invariant coordinates in (1+1) and (3+1) dimensional QED. We compare the Boltzmann-Vlasov results with those of recent quantum field theory calculations and find good agreement. We conclude that extending the Boltzmann-Vlasov approach to the case of QCD should allow us to do a thorough investigation of how backreaction affects recent results on the dependence of the transverse momentum distribution of quarks and antiquarks on a second Casimir invariant of color SU(3)
Hong, X; Gao, H
2014-06-15
Purpose: The Linear Boltzmann Transport Equation (LBTE) solved through statistical Monte Carlo (MC) method provides the accurate dose calculation in radiotherapy. This work is to investigate the alternative way for accurately solving LBTE using deterministic numerical method due to its possible advantage in computational speed from MC. Methods: Instead of using traditional spherical harmonics to approximate angular scattering kernel, our deterministic numerical method directly computes angular scattering weights, based on a new angular discretization method that utilizes linear finite element method on the local triangulation of unit angular sphere. As a Result, our angular discretization method has the unique advantage in positivity, i.e., to maintain all scattering weights nonnegative all the time, which is physically correct. Moreover, our method is local in angular space, and therefore handles the anisotropic scattering well, such as the forward-peaking scattering. To be compatible with image-guided radiotherapy, the spatial variables are discretized on the structured grid with the standard diamond scheme. After discretization, the improved sourceiteration method is utilized for solving the linear system without saving the linear system to memory. The accuracy of our 3D solver is validated using analytic solutions and benchmarked with Geant4, a popular MC solver. Results: The differences between Geant4 solutions and our solutions were less than 1.5% for various testing cases that mimic the practical cases. More details are available in the supporting document. Conclusion: We have developed a 3D LBTE solver based on a new angular discretization method that guarantees the positivity of scattering weights for physical correctness, and it has been benchmarked with Geant4 for photon dose calculation.
Collisionless Reconnection and Electron Demagnetization
NASA Astrophysics Data System (ADS)
Scudder, J. D.
Observable, dimensionless properties of the electron diffusion region of collisionless magnetic reconnection are motivated and benchmarked in two and three dimensional Particle In Cell (PIC) simulations as appropriate for measurements with present state of the art spacecraft. The dimensionless quantities of this paper invariably trace their origin to breaking the magnetization of the thermal electrons. Several observable proxies are also motivated for the rate of frozen flux violation and a parameter \\varLambda _{\\varPhi } that when greater than unity is associated with close proximity to the analogue of the saddle point region of 2D reconnection usually called the electron diffusion region. Analogous regions to the electron diffusion region of 2D reconnection with \\varLambda _{\\varPhi } > 1 have been identified in 3D simulations. 10-20 disjoint diffusion regions are identified and the geometrical patterns of their locations illustrated. First examples of associations between local observables based on electron demagnetization and global diagnostics (like squashing) are also presented. A by product of these studies is the development of a single spacecraft determinations of gradient scales in the plasma.
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.
L2-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians
NASA Astrophysics Data System (ADS)
Ha, Seung-Yeal; Xiao, Qinghua; Xiong, Linjie; Zhao, Huijiang
2013-12-01
We present a L2-stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L2-distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and Xiao, Q.-H., "A revisiting to the L2-stability theory of the Boltzmann equation near global Maxwellians," (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., "L2 stability theory of the Boltzmann equation near a global Maxwellian," Arch. Ration. Mech. Anal. 197, 657-688 (2010)] on the L2-stability of the Boltzmann equation to the Boltzmann equation coupled with self-consistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. ["Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space," Commun. Pure Appl. Math. 24, 1497-1546 (2011)] and Guo ["The Vlasov-Maxwell-Boltzmann system near Maxwellians," Invent. Math. 153(3), 593-630 (2003)] satisfy a uniform L2-stability estimate. This is the first result on the L2-stability of the Boltzmann equation coupled with self-consistent field equations in three dimensions.
Modeling combined collisional/collisionless plasma interpenetration
Thomas, V.A.
1997-04-01
This paper describes one technique by which multifluid modeling capability can be achieved within the context of a Lagrangean single-fluid model. This technique is applied to the interpenetration of laser-produced, substantially collisionless plasmas. A single-fluid model by itself cannot simulate the interpenetration of a collisionless plasma correctly, but must be augmented with some other tool. One tool that can calculate collisionless plasma interpenetration correctly is ISIS, a particle code for plasma simulations which includes appropriate collision models. However, ISIS does not have the necessary physics to do the laser deposition, the atomic physics, the radiation transport, and does not possess a realistic electron temperature model. With appropriate integration of the single-fluid code and ISIS, a new capability is achieved which allows simulation of the colliding plasma problem, a problem that neither code can properly simulate individually.
Boltzmann-type control of opinion consensus through leaders
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
Magnetic reconnection in collisionless plasmas - Prescribed fields
NASA Technical Reports Server (NTRS)
Burkhart, G. R.; Drake, J. F.; Chen, J.
1990-01-01
The structure of the dissipation region during magnetic reconnection in collisionless plasma is investigated by examining a prescribed two-dimensional magnetic x line configuration with an imposed inductive electric field E(y). The calculations represent an extension of recent MHD simulations of steady state reconnection (Biskamp, 1986; Lee and Fu, 1986) to the collisionless kinetic regime. It is shown that the structure of the x line reconnection configuration depends on only two parameters: a normalized inductive field and a parameter R which represents the opening angle of the magnetic x lines.
Physics of collisionless shocks: theory and simulation
NASA Astrophysics Data System (ADS)
Stockem Novo, A.; Bret, A.; Fonseca, R. A.; Silva, L. O.
2016-01-01
Collisionless shocks occur in various fields of physics. In the context of space and astrophysics they have been investigated for many decades. However, a thorough understanding of shock formation and particle acceleration is still missing. Collisionless shocks can be distinguished into electromagnetic and electrostatic shocks. Electromagnetic shocks are of importance mainly in astrophysical environments and they are mediated by the Weibel or filamentation instability. In such shocks, charged particles gain energy by diffusive shock acceleration. Electrostatic shocks are characterized by a strong electrostatic field, which leads to electron trapping. Ions are accelerated by reflection from the electrostatic potential. Shock formation and particle acceleration will be discussed in theory and simulations.
Collisionless damping of dust-acoustic waves in a charge varying dusty plasma with nonextensive ions
Amour, Rabia; Tribeche, Mouloud
2014-12-15
The charge variation induced nonlinear dust-acoustic wave damping in a charge varying dusty plasma with nonextensive ions is considered. It is shown that the collisionless damping due to dust charge fluctuation causes the nonlinear dust acoustic wave propagation to be described by a damped Korteweg-de Vries (dK-dV) equation the coefficients of which depend sensitively on the nonextensive parameter q. The damping term, solely due to the dust charge variation, is affected by the ion nonextensivity. For the sake of completeness, the possible effects of nonextensivity and collisionless damping on weakly nonlinear wave packets described by the dK-dV equation are succinctly outlined by deriving a nonlinear Schrödinger-like equation with a complex nonlinear coefficient.
Simplex-in-cell technique for collisionless plasma simulations
NASA Astrophysics Data System (ADS)
Kates-Harbeck, Julian; Totorica, Samuel; Zrake, Jonathan; Abel, Tom
2016-01-01
We extend the simplex-in-cell (SIC) technique recently introduced in the context of collisionless dark matter fluids [1,2] to the case of collisionless plasmas. The six-dimensional phase space distribution function f (x , v) is represented by an ensemble of three-dimensional manifolds, which we refer to as sheets. The electric potential field is obtained by solving the Poisson equation on a uniform mesh, where the charge density is evaluated by a spatial projection of the phase space sheets. The SIC representation of phase space density facilitates robust, high accuracy numerical evolution of the Vlasov-Poisson system using significantly fewer tracer particles than comparable particle-in-cell (PIC) approaches by reducing the numerical shot-noise associated with the latter. We introduce the SIC formulation and describe its implementation in a new code, which we validate using standard test problems including plasma oscillations, Landau damping, and two stream instabilities in one dimension. Merits of the new scheme are shown to include higher accuracy and faster convergence rates in the number of particles. We finally motivate and outline the efficient application of SIC to higher dimensional problems.
Critical assessment of the Boltzmann approach to active systems.
Thüroff, Florian; Weber, Christoph A; Frey, Erwin
2013-11-01
Generic models of propelled particle systems posit that the emergence of polar order is driven by the competition between local alignment and noise. Although this notion has been confirmed employing the Boltzmann equation, the range of applicability of this equation remains elusive. We introduce a broad class of mesoscopic collision rules and analyze the prerequisites for the emergence of polar order in the framework of kinetic theory. Our findings suggest that a Boltzmann approach is appropriate for weakly aligning systems but is incompatible with experiments on cluster forming systems.
Transition from Collisionless to Collisional MRI
Prateek Sharma; Gregory W. Hammett; Eliot Quataert
2003-07-24
Recent calculations by Quataert et al. (2002) found that the growth rates of the magnetorotational instability (MRI) in a collisionless plasma can differ significantly from those calculated using MHD. This can be important in hot accretion flows around compact objects. In this paper, we study the transition from the collisionless kinetic regime to the collisional MHD regime, mapping out the dependence of the MRI growth rate on collisionality. A kinetic closure scheme for a magnetized plasma is used that includes the effect of collisions via a BGK operator. The transition to MHD occurs as the mean free path becomes short compared to the parallel wavelength 2*/k(sub)||. In the weak magnetic field regime where the Alfven and MRI frequencies w are small compared to the sound wave frequency k(sub)||c(sub)0, the dynamics are still effectively collisionless even if omega << v, so long as the collision frequency v << k(sub)||c(sub)0; for an accretion flow this requires n less than or approximately equal to *(square root of b). The low collisionality regime not only modifies the MRI growth rate, but also introduces collisionless Landau or Barnes damping of long wavelength modes, which may be important for the nonlinear saturation of the MRI.
Investigation of the kinetic model equations
NASA Astrophysics Data System (ADS)
Liu, Sha; Zhong, Chengwen
2014-03-01
Currently the Boltzmann equation and its model equations are widely used in numerical predictions for dilute gas flows. The nonlinear integro-differential Boltzmann equation is the fundamental equation in the kinetic theory of dilute monatomic gases. By replacing the nonlinear fivefold collision integral term by a nonlinear relaxation term, its model equations such as the famous Bhatnagar-Gross-Krook (BGK) equation are mathematically simple. Since the computational cost of solving model equations is much less than that of solving the full Boltzmann equation, the model equations are widely used in predicting rarefied flows, multiphase flows, chemical flows, and turbulent flows although their predictions are only qualitatively right for highly nonequilibrium flows in transitional regime. In this paper the differences between the Boltzmann equation and its model equations are investigated aiming at giving guidelines for the further development of kinetic models. By comparing the Boltzmann equation and its model equations using test cases with different nonequilibrium types, two factors (the information held by nonequilibrium moments and the different relaxation rates of high- and low-speed molecules) are found useful for adjusting the behaviors of modeled collision terms in kinetic regime. The usefulness of these two factors are confirmed by a generalized model collision term derived from a mathematical relation between the Boltzmann equation and BGK equation that is also derived in this paper. After the analysis of the difference between the Boltzmann equation and the BGK equation, an attempt at approximating the collision term is proposed.
Electrostatic forces in the Poisson-Boltzmann systems.
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. PMID:24028101
Low-frequency instabilities of collisionless plasma and the 16-moment approximation
Dzhalilov, N. S. Kuznetsov, V. D.
2013-12-15
Using the 16-moment equations that take into account heat fluxes in anisotropic collisionless plasma, the properties of magnetohydrodynamic (MHD) instabilities are investigated. For all instabilities occurring in the MHD approach (the normal incompressible firehose instability, the second compressible almost longitudinal firehose instability, and the almost transverse mirror instability of slow magnetosonic modes, as well as thermal instability caused by the heat flux directed along the magnetic field), their kinetic analogs are considered. The kinetic dispersion relation in the low-frequency range in the vicinity of the ion thermal velocity is analyzed. The flow of plasma ions along the magnetic field is taken into account. The thresholds and instability growth rates obtained in the MHD and kinetic approaches are found to be in good agreement. This indicates that the 16-moment MHD equations adequately describe the dynamics of collisionless plasma.
Nonlinear Particle Pinch in Collisionless Trapped Electron Mode Turbulence
NASA Astrophysics Data System (ADS)
Terry, P. W.; Gatto, R.
2005-10-01
Collisionless trapped electron mode turbulence is shown to have an anomalous particle pinch fundamentally unlike pinches identified previously. It arises from a nonlinear fluctuation eigenmode, placing it outside the purview of quasilinear theory. The nonlinear eigenmode develops because the nonlinearity excites a damped linear eigenmode, changing the density- potential correlation. The flux is solved from spectrum balance equations in a complete basis spanning the fluctuation space under a joint expansion in collision frequency and instability threshold parameter. The solution accounts for saturation by anisotropic energy transfer to zonal wavenumbers of the damped eigenmode. To lowest order the pinch is a convective-like flux driven by temperature gradient. It arises from the damped eigenmode energy and the real part of the correlation between damped and growing eigenmodes. The pinch is slightly smaller than the outwardly directed flux associated with the growing eigenmode, making the flux a small fraction of the quasilinear value. Work supported by US DOE.
Lattice Boltzmann model for numerical relativity.
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. PMID:26986435
Lattice Boltzmann model for numerical relativity.
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.
Lattice Boltzmann model for numerical relativity
NASA Astrophysics Data System (ADS)
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.
Momentum transfer of a Boltzmann-lattice fluid with boundaries
NASA Astrophysics Data System (ADS)
Bouzidi, M'hamed; Firdaouss, Mouaouia; Lallemand, Pierre
2001-11-01
We study the velocity boundary condition for curved boundaries in the lattice Boltzmann equation (LBE). We propose a LBE boundary condition for moving boundaries by combination of the "bounce-back" scheme and spatial interpolations of first or second order. The proposed boundary condition is a simple, robust, efficient, and accurate scheme. Second-order accuracy of the boundary condition is demonstrated for two cases: (1) time-dependent two-dimensional circular Couette flow and (2) two-dimensional steady flow past a periodic array of circular cylinders (flow through the porous media of cylinders). For the former case, the lattice Boltzmann solution is compared with the analytic solution of the Navier-Stokes equation. For the latter case, the lattice Boltzmann solution is compared with a finite-element solution of the Navier-Stokes equation. The lattice Boltzmann solutions for both flows agree very well with the solutions of the Navier-Stokes equations. We also analyze the torque due to the momentum transfer between the fluid and the boundary for two initial conditions: (a) impulsively started cylinder and the fluid at rest, and (b) uniformly rotating fluid and the cylinder at rest.
NASA Technical Reports Server (NTRS)
Barnes, A.
1983-01-01
An exact nonlinear solution is found to the relativistic kinetic and electrodynamic equations (in their hydromagnetic limit) that describes the large-amplitude fast-mode magnetoacoustic wave propagating normal to the magnetic field in a collisionless, previously uniform plasma. It is pointed out that a wave of this kind will be generated by transverse compression of any collisionless plasma. The solution is in essence independent of the detailed form of the particle momentum distribution functions. The solution is obtained, in part, through the method of characteristics; the wave exhibits the familiar properties of steepening and shock formation. A detailed analysis is given of the ultrarelativistic limit of this wave.
Lessons on collisionless reconnection from quantum fluids
NASA Astrophysics Data System (ADS)
Narita, Yasuhito; Baumjohann, Wolfgang
2014-12-01
Magnetic reconnection in space plasmas remains a challenge in physics in that the phenomenon is associated with the breakdown of frozen-in magnetic field in a collisionless medium. Such a topology change can also be found in superfluidity, known as the quantum vortex reconnection. We give a plasma physicists' view of superfluidity to obtain insights on essential processes in collisionless reconnection, including discussion of the kinetic and fluid pictures, wave dynamics, and time reversal asymmetry. The most important lesson from the quantum fluid is the scenario that reconnection is controlled by the physics of topological defects on the microscopic scale, and by the physics of turbulence on the macroscopic scale. Quantum vortex reconnection is accompanied by wave emission in the form of Kelvin waves and sound waves, which imprints the time reversal asymmetry.
Supermagnetosonic Jets behind a Collisionless Quasiparallel Shock
Hietala, H.; Vainio, R.; Laitinen, T. V.; Vaivads, A.; Andreeova, K.; Palmroth, M.; Pulkkinen, T. I.; Koskinen, H. E. J.; Lucek, E. A.; Reme, H.
2009-12-11
The downstream region of a collisionless quasiparallel shock is structured containing bulk flows with high kinetic energy density from a previously unidentified source. We present Cluster multispacecraft measurements of this type of supermagnetosonic jet as well as of a weak secondary shock front within the sheath, that allow us to propose the following generation mechanism for the jets: The local curvature variations inherent to quasiparallel shocks can create fast, deflected jets accompanied by density variations in the downstream region. If the speed of the jet is super(magneto)sonic in the reference frame of the obstacle, a second shock front forms in the sheath closer to the obstacle. Our results can be applied to collisionless quasiparallel shocks in many plasma environments.
Hassanein, A.; Konkashbaev, I.
1999-03-15
The structure of a collisionless scrape-off-layer (SOL) plasma in tokamak reactors is being studied to define the electron distribution function and the corresponding sheath potential between the divertor plate and the edge plasma. The collisionless model is shown to be valid during the thermal phase of a plasma disruption, as well as during the newly desired low-recycling normal phase of operation with low-density, high-temperature, edge plasma conditions. An analytical solution is developed by solving the Fokker-Planck equation for electron distribution and balance in the SOL. The solution is in good agreement with numerical studies using Monte-Carlo methods. The analytical solutions provide an insight to the role of different physical and geometrical processes in a collisionless SOL during disruptions and during the enhanced phase of normal operation over a wide range of parameters.
Boltzmann's H theorem for systems with frictional dissipation.
Bizarro, João P S
2011-03-01
By use of Boltzmann's equation to describe an ensemble of particles under the influence of a friction force, Boltzmann's H theorem is refined to explicitly include frictional dissipation, the accompanying fluctuations being modeled via an added diffusive, Fokker-Planck term. If the friction force per particle mass is proportional to velocity, as is the case with viscous drag with a friction coefficient γ, Boltzmann's H theorem for the time rate of change of the quantity H reads dH/dt ≤ γ. The classical formulation stating that H can never increase is thus replaced by the statement that H cannot increase at a rate higher than γ, a general result but of particular relevance when fluctuations are negligible and the system is far from equilibrium. When the particles are not far from thermal equilibrium, an alternative, more suitable expression emerges which can be written in the form of a Clausius inequality. PMID:21517545
Kolesnikov, R.A.; Krommes, J.A.
2005-09-22
The collisionless limit of the transition to ion-temperature-gradient-driven plasma turbulence is considered with a dynamical-systems approach. The importance of systematic analysis for understanding the differences in the bifurcations and dynamics of linearly damped and undamped systems is emphasized. A model with ten degrees of freedom is studied as a concrete example. A four-dimensional center manifold (CM) is analyzed, and fixed points of its dynamics are identified and used to predict a ''Dimits shift'' of the threshold for turbulence due to the excitation of zonal flows. The exact value of that shift in terms of physical parameters is established for the model; the effects of higher-order truncations on the dynamics are noted. Multiple-scale analysis of the CM equations is used to discuss possible effects of modulational instability on scenarios for the transition to turbulence in both collisional and collisionless cases.
Lattice Boltzmann method for adiabatic acoustics.
Li, Yanbing; Shan, Xiaowen
2011-06-13
The lattice Boltzmann method (LBM) has been proved to be a useful tool in many areas of computational fluid dynamics, including computational aero-acoustics (CAA). However, for historical reasons, its applications in CAA have been largely restricted to simulations of isothermal (Newtonian) sound waves. As the recent kinetic theory-based reformulation establishes a theoretical framework in which LBM can be extended to recover the full Navier-Stokes-Fourier (NS) equations and beyond, in this paper, we show that, at least at the low-frequency limit (sound frequency much less than molecular collision frequency), adiabatic sound waves can be accurately simulated by the LBM provided that the lattice and the distribution function ensure adequate recovery of the full NS equations.
NASA Technical Reports Server (NTRS)
Klimas, Alexander J.
1987-01-01
The solution of filtered Vlasov-Maxwell equations, rather than the Vlasov-Maxwell equations themselves, is shown to ameliorate the velocity space filamentation problem in collisionless plasma models. Exact field solutions and filtered velocity distribution functions are obtained without introducing errors. Proper selection of the filter width is demonstrated to inhibit development of velocity space filamentation and, it is conjectured, position space filamentation. The results of sample calculations of both filtered and nonfiltered field solutions illustrate a high degree of agreement between both solutions, with significant savings in computational time and memory requirements with the filtered solutions.
NASA Technical Reports Server (NTRS)
Collins, William
1989-01-01
The dispersion equation of Barnes (1966) is used to study the dissipation of asymptotic wave packets generated by localized periodic sources. The solutions of the equation are linear waves, damped by Landau and transit-time processes, in a collisionless warm plasma. For the case of an ideal MHD system, most of the waves emitted from a source are shown to cancel asympotically through destructive interference. The modes transporting significant flux to asymptotic distances are found to be Alfven waves and fast waves with theta (the angle between the magnetic field and the characteristics of the far-field waves) of about 0 and about pi/2.
Collisionless shocks in the heliosphere: A tutorial review
NASA Technical Reports Server (NTRS)
Stone, Robert G. (Editor); Tsurutani, Bruce T. (Editor)
1985-01-01
An update is presented on current knowledge of collisionless shocks in the heliosphere. The individual papers address: a quarter century of collisionless shock research, some macroscopic properties of shock waves in the heliosphere, microinstabilities and anomalous transport, and acceleration of energetic particles.
L{sup 2}-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians
Ha, Seung-Yeal Xiao, Qinghua; Xiong, Linjie Zhao, Huijiang
2013-12-15
We present a L{sup 2}-stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L{sup 2}-distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and Xiao, Q.-H., “A revisiting to the L{sup 2}-stability theory of the Boltzmann equation near global Maxwellians,” (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., “L{sup 2} stability theory of the Boltzmann equation near a global Maxwellian,” Arch. Ration. Mech. Anal. 197, 657–688 (2010)] on the L{sup 2}-stability of the Boltzmann equation to the Boltzmann equation coupled with self-consistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. [“Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,” Commun. Pure Appl. Math. 24, 1497–1546 (2011)] and Guo [“The Vlasov-Maxwell-Boltzmann system near Maxwellians,” Invent. Math. 153(3), 593–630 (2003)] satisfy a uniform L{sup 2}-stability estimate. This is the first result on the L{sup 2}-stability of the Boltzmann equation coupled with self-consistent field equations in three dimensions.
New Expression for Collisionless Magnetic Reconnection Rate
NASA Technical Reports Server (NTRS)
Klimas, Alexander J.
2014-01-01
For 2D, symmetric, anti-parallel, collisionless magnetic reconnection, a new expression for the reconnection rate in the electron diffusion region is introduced. It is shown that this expression can be derived in just a few simple steps from a physically intuitive starting point; the derivation is given in its entirety and the validity of each step is confirmed. The predictions of this expression are compared to the results of several long-duration, open-boundary PIC reconnection simulations to demonstrate excellent agreement.
Evidence for collisionless magnetic reconnection at Mars
NASA Astrophysics Data System (ADS)
Eastwood, J. P.; Brain, D. A.; Halekas, J. S.; Drake, J. F.; Phan, T. D.; Øieroset, M.; Mitchell, D. L.; Lin, R. P.; Acuña, M.
2008-01-01
Using data from Mars Global Surveyor (MGS) in combination with Particle-In-Cell (PIC) simulations of reconnection, we present the first direct evidence of collisionless magnetic reconnection at Mars. The evidence indicates that the spacecraft passed through the diffusion region where reconnection is initiated and observed the magnetic field signatures of differential electron and ion motion - the Hall magnetic field - that uniquely indicate the reconnection process. These are the first such in-situ reconnection observations at an astronomical body other than the Earth. Reconnection may be the source of Mars' recently discovered auroral activity and the changing boundaries of the closed regions of crustal magnetic field.
Thin-shell instability in collisionless plasma.
Dieckmann, M E; Ahmed, H; Doria, D; Sarri, G; Walder, R; Folini, D; Bret, A; Ynnerman, A; Borghesi, M
2015-09-01
Thin-shell instability is one process which can generate entangled structures in astrophysical plasma on collisional (fluid) scales. It is driven by a spatially varying imbalance between the ram pressure of the inflowing upstream plasma and the downstream's thermal pressure at a nonplanar shock. Here we show by means of a particle-in-cell simulation that an analog process can destabilize a thin shell formed by two interpenetrating, unmagnetized, and collisionless plasma clouds. The amplitude of the shell's spatial modulation grows and saturates after about ten inverse proton plasma frequencies, when the shell consists of connected piecewise linear patches.
Scattering of radiation in collisionless dusty plasmas
Tolias, P.; Ratynskaia, S.
2013-04-15
Scattering of electromagnetic waves in collisionless dusty plasmas is studied in the framework of a multi-component kinetic model. The investigation focuses on the spectral distribution of the scattered radiation. Pronounced dust signatures are identified in the coherent spectrum due to scattering from the shielding cloud around the dust grains, dust acoustic waves, and dust-ion acoustic waves. The magnitude and shape of the scattered signal near these spectral regions are determined with the aid of analytical expressions and its dependence on the dust parameters is investigated. The use of radiation scattering as a potential diagnostic tool for dust detection is discussed.
Cascaded proton acceleration by collisionless electrostatic shock
NASA Astrophysics Data System (ADS)
Xu, T. J.; Shen, B. F.; Zhang, X. M.; Yi, L. Q.; Wang, W. P.; Zhang, L. G.; Xu, J. C.; Zhao, X. Y.; Shi, Y.; Liu, C.; Pei, Z. K.
2015-07-01
A new scheme for proton acceleration by cascaded collisionless electrostatic shock (CES) is proposed. By irradiating a foil target with a moderate high-intensity laser beam, a stable CES field can be induced, which is employed as the accelerating field for the booster stage of proton acceleration. The mechanism is studied through simulations and theoretical analysis, showing that a 55 MeV seed proton beam can be further accelerated to 265 MeV while keeping a good energy spread. This scheme offers a feasible approach to produce proton beams with energy of hundreds of MeV by existing available high-intensity laser facilities.
Cascaded proton acceleration by collisionless electrostatic shock
Xu, T. J.; Shen, B. F. E-mail: zhxm@siom.ac.cn; Zhang, X. M. E-mail: zhxm@siom.ac.cn; Yi, L. Q.; Wang, W. P.; Zhang, L. G.; Xu, J. C.; Zhao, X. Y.; Shi, Y.; Liu, C.; Pei, Z. K.
2015-07-15
A new scheme for proton acceleration by cascaded collisionless electrostatic shock (CES) is proposed. By irradiating a foil target with a moderate high-intensity laser beam, a stable CES field can be induced, which is employed as the accelerating field for the booster stage of proton acceleration. The mechanism is studied through simulations and theoretical analysis, showing that a 55 MeV seed proton beam can be further accelerated to 265 MeV while keeping a good energy spread. This scheme offers a feasible approach to produce proton beams with energy of hundreds of MeV by existing available high-intensity laser facilities.
Collisionless Relaxation in Non-Neutral Plasmas
Levin, Yan; Pakter, Renato; Teles, Tarcisio N.
2008-02-01
A theoretical framework is presented which allows us to quantitatively predict the final stationary state achieved by a non-neutral plasma during a process of collisionless relaxation. As a specific application, the theory is used to study relaxation of charged-particle beams. It is shown that a fully matched beam relaxes to the Lynden-Bell distribution. However, when a mismatch is present and the beam oscillates, parametric resonances lead to a core-halo phase separation. The approach developed accounts for both the density and the velocity distributions in the final stationary state.
New expression for collisionless magnetic reconnection rate
Klimas, Alex
2015-04-15
For 2D, symmetric, anti-parallel, collisionless magnetic reconnection, new expressions for the reconnection rate in the electron diffusion region are introduced. It is shown that these expressions can be derived in just a few simple steps from a physically intuitive starting point; the derivations are given in their entirety, and the validity of each step is confirmed. The predictions of these expressions are compared to the results of several long-duration, open-boundary particle-in-cell reconnection simulations to demonstrate excellent agreement.
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
Poisson-Boltzmann-Nernst-Planck model
Zheng Qiong; Wei Guowei
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
Igor D. Kaganovich; Oleg V. Polomarov; Constantine E. Theodosiou
2004-01-30
In low-pressure discharges, where the electron mean free path is larger or comparable with the discharge length, the electron dynamics is essentially nonlocal. Moreover, the electron energy distribution function (EEDF) deviates considerably from a Maxwellian. Therefore, an accurate kinetic description of the low-pressure discharges requires knowledge of the nonlocal conductivity operator and calculation of the non-Maxwellian EEDF. The previous treatments made use of simplifying assumptions: a uniform density profile and a Maxwellian EEDF. In the present study a self-consistent system of equations for the kinetic description of nonlocal, nonuniform, nearly collisionless plasmas of low-pressure discharges is reported. It consists of the nonlocal conductivity operator and the averaged kinetic equation for calculation of the non-Maxwellian EEDF. This system was applied to the calculation of collisionless heating in capacitively and inductively coupled plasmas. In particular, the importance of accounting for the nonuniform plasma density profile for computing the current density profile and the EEDF is demonstrated. The enhancement of collisionless heating due to the bounce resonance between the electron motion in the potential well and the external radio-frequency electric field is investigated. It is shown that a nonlinear and self-consistent treatment is necessary for the correct description of collisionless heating.
Diamagnetic boundary layers - A kinetic theory. [for collisionless magnetized plasmas
NASA Technical Reports Server (NTRS)
Lemaire, J.; Burlaga, L. F.
1976-01-01
A kinetic theory is presented for boundary layers associated with MHD tangential 'discontinuities' in a collisionless magnetized plasma, such as those observed in the solar wind. The theory consists of finding self-consistent solutions of Vlasov's equation and Maxwell's equation for stationary one-dimensional boundary layers separating two Maxwellian plasma states. Layers in which the current is carried by electrons are found to have a thickness of the order of a few electron gyroradii, but the drift speed of the current-carrying electrons is found to exceed the Alfven speed, and accordingly such layers are not stable. Several types of layers in which the current is carried by protons are discussed; in particular, cases are considered in which the magnetic-field intensity, direction, or both, changed across the layer. In every case, the thickness was of the order of a few proton gyroradii, and the field changed smoothly, although the characteristics depended somewhat on the boundary conditions. The drift speed was always less than the Alfven speed, consistent with stability of such structures. These results are consistent with observations of boundary layers in the solar wind near 1 AU.
Temperature based Restricted Boltzmann Machines.
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-13
Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view.
Temperature based Restricted Boltzmann Machines
NASA Astrophysics Data System (ADS)
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-01
Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view.
Boltzmann's Approach to Statistical Mechanics
NASA Astrophysics Data System (ADS)
Goldstein, Sheldon
In the last quarter of the nineteenth century, Ludwig Boltzmann explained how irreversible macroscopic laws, in particular the second law of thermodynamics, originate in the time-reversible laws of microscopic physics. Boltzmann's analysis, the essence of which I shall review here, is basically correct. The most famous criticisms of Boltzmann's later work on the subject have little merit. Most twentieth century innovations - such as the identification of the state of a physical system with a probability distribution \\varrho on its phase space, of its thermodynamic entropy with the Gibbs entropy of \\varrho, and the invocation of the notions of ergodicity and mixing for the justification of the foundations of statistical mechanics - are thoroughly misguided.
NASA Astrophysics Data System (ADS)
Nomura, Yasunori
2015-10-01
Understanding the observed arrow of time is equivalent, under general assumptions, to explaining why Boltzmann brains do not overwhelm ordinary observers. It is usually thought that this provides a condition on the decay rate of every cosmologically accessible de Sitter vacuum, and that this condition is determined by the production rate of Boltzmann brains calculated using semiclassical theory built on each such vacuum. We argue, based on a recently developed picture of microscopic quantum gravitational degrees of freedom, that this thinking needs to be modified. In particular, depending on the structure of the fundamental theory, the decay rate of a de Sitter vacuum may not have to satisfy any condition except for the one imposed by the Poincaré recurrence. The framework discussed here also addresses the question of whether a Minkowski vacuum may produce Boltzmann brains.
Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids.
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work.
Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids.
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work. PMID:26986441
NASA Astrophysics Data System (ADS)
de Vega, H. J.; Sanchez, N. G.
2016-05-01
We find the distribution function f(E) for dark matter (DM) halos in galaxies and the corresponding equation of state from the (empirical) DM density profiles derived from observations. We solve for DM in galaxies the analogous of the Eddington equation originally used for the gas of stars in globular clusters. The observed density profiles are a good realistic starting point and the distribution functions derived from them are realistic. We do not make any assumption about the DM nature, the methods developed here apply to any DM kind, though all results are consistent with warm dark matter (WDM). With these methods we find: (i) Cored density profiles behaving quadratically for small distances ρ(r)= r → 0ρ(0) ‑ Kr2 produce distribution functions which are finite and positive at the halo center while cusped density profiles always produce divergent distribution functions at the center. (ii) Cored density profiles produce approximate thermal Boltzmann distribution functions for r ≲ 3rh where rh is the halo radius. (iii) Analytic expressions for the dispersion velocity and the pressure are derived yielding at each halo point an ideal DM gas equation of state with local temperature T(r) ≡ mv2(r)/3. T(r) turns out to be constant in the same region where the distribution function is thermal and exhibits the same temperature within the percent. The self-gravitating DM gas can thermalize despite being collisionless because it is an ergodic system. (iv) The DM halo can be consistently considered at local thermal equilibrium with: (a) a constant temperature T(r) = T0 for r ≲ 3rh, (b) a space dependent temperature T(r) for 3rh < r ≲ Rvirial, which slowly decreases with r. That is, the DM halo is realistically a collisionless self-gravitating thermal gas for r ≲ Rvirial. (v) T(r) outside the halo radius nicely follows the decrease of the circular velocity squared.
The collisionless magnetoviscous-thermal instability
Islam, Tanim
2014-05-20
It is likely that nearly all central galactic massive and supermassive black holes are nonradiative: their accretion luminosities are orders of magnitude below what can be explained by efficient black hole accretion within their ambient environments. These objects, of which Sagittarius A* is the best-known example, are also dilute (mildly collisional to highly collisionless) and optically thin. In order for accretion to occur, magnetohydrodynamic (MHD) instabilities must develop that not only transport angular momentum, but also gravitational energy generated through matter infall, outward. A class of new magnetohydrodynamical fluid instabilities—the magnetoviscous-thermal instability (MVTI)—was found to transport angular momentum and energy along magnetic field lines through large (fluid) viscosities and thermal conductivities. This paper describes the analog to the MVTI, the collisionless MVTI (CMVTI), that similarly transports energy and angular momentum outward, expected to be important in describing the flow properties of hot, dilute, and radiatively inefficient accretion flows around black holes. We construct a local equilibrium for MHD stability analysis in this differentially rotating disk. We then find and characterize specific instabilities expected to be important in describing their flow properties, and show their qualitative similarities to instabilities derived using the fluid formalism. We conclude with further work needed in modeling this class of accretion flow.
Zermelo, Boltzmann, and the recurrence paradox
NASA Astrophysics Data System (ADS)
Steckline, Vincent S.
1983-10-01
The papers exchanged by Ludwig Boltzmann and Ernst Zermelo concerning the recurrence paradox are summarized. The historical context of the paradox, Zermelo's proof of the paradox, his opinions of its consequences, Boltzmann's reply, and the ensuing discussion are described.
NASA Astrophysics Data System (ADS)
Gao, Dong-Ning; Qi, Xin; Hong, Xue-Ren; Yang, Xue; Duan, Wen-Shan; Yang, Lei; Yang
2014-06-01
Numerical and theoretical investigations are carried out for the stability of the dust acoustic waves (DAWs) under the transverse perturbation in a two-ion temperature magnetized and collisionless dusty plasma. The Zakharov-Kuznetsov (ZK) equation, modified ZK equation, and Extended ZK (EZK) equation of the DAWs are given by using the reductive perturbation technique. The cut-off frequency is obtained by applying higher-order transverse perturbations to the soliton solution of the EZK equation. The propagation velocity of solitary waves, the real cut-off frequency, as well as the growth rate of the higher-order perturbation to the solitary wave are obtained.
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.
Bang, Jin Young; Chung, Chin Wook
2009-09-15
In plasma, the Boltzmann relation is often used to connect the electron density to the plasma potential because it is not easy to calculate electric potentials on the basis of the Poisson equation due to the quasineutrality. From the Boltzmann relation, the electric potential can be simply obtained from the electron density or vice versa. However, the Boltzmann relation assumes that electrons are in thermal equilibrium and have a Maxwellian distribution, so it cannot be applied to non-Maxwellian distributions. In this paper, the Boltzmann relation for bi-Maxwellian distributions was newly derived from fluid equations and the comparison with the experimental results was given by measuring electron energy probability functions in an inductively coupled plasma. It was found that the spatial distribution of the electron density in bulk plasma is governed by the effective electron temperature, while that of the cold and hot electrons are governed by each electron temperature.
Boltzmann's constant: A laboratory experiment
NASA Astrophysics Data System (ADS)
Kruglak, Haym
1989-03-01
The mean-square displacement of a latex microsphere is determined from its projection on a TV monitor. The distribution of displacement is shown to be Gaussian. Boltzmann's constant, calculated from the pooled data of several observers, is in excellent agreement with the accepted value. The experiment is designed for one laboratory period in the advanced undergraduate laboratory.
Lattice Boltzmann model for incompressible flows through porous media.
Guo, Zhaoli; Zhao, T S
2002-09-01
In this paper a lattice Boltzmann model is proposed for isothermal incompressible flow in porous media. The key point is to include the porosity into the equilibrium distribution, and add a force term to the evolution equation to account for the linear and nonlinear drag forces of the medium (the Darcy's term and the Forcheimer's term). Through the Chapman-Enskog procedure, the generalized Navier-Stokes equations for incompressible flow in porous media are derived from the present lattice Boltzmann model. The generalized two-dimensional Poiseuille flow, Couette flow, and lid-driven cavity flow are simulated using the present model. It is found the numerical results agree well with the analytical and/or the finite-difference solutions.
Acoustic multipole sources for the lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Viggen, Erlend Magnus
2013-02-01
By including an oscillating particle source term, acoustic multipole sources can be implemented in the lattice Boltzmann method. The effect of this source term on the macroscopic conservation equations is found using a Chapman-Enskog expansion. In a lattice with q particle velocities, the source term can be decomposed into q orthogonal multipoles. More complex sources may be formed by superposing these basic multipoles. Analytical solutions found from the macroscopic equations and an analytical lattice Boltzmann wavenumber are compared with inviscid multipole simulations, finding very good agreement except close to singularities in the analytical solutions. Unlike the BGK operator, the regularized collision operator is proven capable of accurately simulating two-dimensional acoustic generation and propagation at zero viscosity.
Boltzmann transport calculation of collinear spin transport on short timescales
NASA Astrophysics Data System (ADS)
Nenno, Dennis M.; Kaltenborn, Steffen; Schneider, Hans Christian
2016-09-01
A spin-dependent Boltzmann transport equation is used to describe charge and spin dynamics resulting from the excitation of hot electrons in a ferromagnet/normal metal heterostructure. As the microscopic Boltzmann equation works with k -dependent distribution functions, it can describe far-from-equilibrium excitations, which are outside the scope of drift-diffusion theories. We study different scenarios for spin-dependent carrier injection into a nonmagnetic metal using an effectively two-dimensional phase space. While the charge signal is robust for various excitation schemes, the shape of the resulting spin current/density depends critically on the interplay between transport and scattering, and on the energetic distribution of the injected carriers. Our results imply that the energy dependence of the injected hot electrons has a decisive effect on the spin dynamics.
Evidence for Collisionless Magnetic Reconnection at Mars
NASA Astrophysics Data System (ADS)
Brain, D.; Eastwood, J.; Halekas, J.; Drake, J.; Phan, T.; Oieroset, M.; Mitchell, D.; Lin, R.; Acuna, M.
2007-12-01
Magnetic reconnection is a fundamental plasma process that enables the rapid conversion of magnetic to particle energy and is important in astrophysics as well as solar, space and planetary physics. Using data from the Mars Global Surveyor (MGS) spacecraft in combination with simulations of reconnection, we present the first direct evidence of collisionless magnetic reconnection at Mars. The evidence indicates that the spacecraft passed through the diffusion region where reconnection is initiated and observed the magnetic field signatures of differential electron and ion motion that uniquely indicate the reconnection process. These are the first such in- situ reconnection observations at an astronomical body other than the Earth. Reconnection may be the source of Mars" recently discovered auroral activity and the changing boundaries of the closed regions of crustal magnetic field.
Collisionless Dynamics and the Cosmic Web
NASA Astrophysics Data System (ADS)
Hahn, Oliver
2016-10-01
I review the nature of three-dimensional collapse in the Zeldovich approximation, how it relates to the underlying nature of the three-dimensional Lagrangian manifold and naturally gives rise to a hierarchical structure formation scenario that progresses through collapse from voids to pancakes, filaments and then halos. I then discuss how variations of the Zeldovich approximation (based on the gravitational or the velocity potential) have been used to define classifications of the cosmic large-scale structure into dynamically distinct parts. Finally, I turn to recent efforts to devise new approaches relying on tessellations of the Lagrangian manifold to follow the fine-grained dynamics of the dark matter fluid into the highly non-linear regime and both extract the maximum amount of information from existing simulations as well as devise new simulation techniques for cold collisionless dynamics.
Weak collisionless shocks in laser-plasmas
NASA Astrophysics Data System (ADS)
Cairns, R. A.; Bingham, R.; Trines, R. G. M.; Norreys, P.
2015-04-01
We obtain a theory describing laminar shock-like structures in a collisionless plasma and examine the parameter limits, in terms of the ion sound Mach number and the electron/ion temperature ratio, within which these structures exist. The essential feature is the inclusion of finite ion temperature with the result that some ions are reflected from a potential ramp. This destroys the symmetry between upstream and downstream regions that would otherwise give the well-known ion solitary wave solution. We have shown earlier (Cairns et al 2014 Phys. Plasmas 21 022112) that such structures may be relevant to problems such as the existence of strong, localized electric fields observed in laser compressed pellets and laser acceleration of ions. Here we present results on the way in which these structures may produce species separation in fusion targets and suggest that it may be possible to use shock ion acceleration for fast ignition.
Collisionless electron heating in periodic arrays of inductively coupled plasmas
Czarnetzki, U.; Tarnev, Kh.
2014-12-15
A novel mechanism of collisionless heating in large planar arrays of small inductive coils operated at radio frequencies is presented. In contrast to the well-known case of non-local heating related to the transversal conductivity, when the electrons move perpendicular to the planar coil, we investigate the problem of electrons moving in a plane parallel to the coils. Two types of periodic structures are studied. Resonance velocities where heating is efficient are calculated analytically by solving the Vlasov equation. Certain scaling parameters are identified. The concept is further investigated by a single particle simulation based on the ergodic principle and combined with a Monte Carlo code allowing for collisions with Argon atoms. Resonances, energy exchange, and distribution functions are obtained. The analytical results are confirmed by the numerical simulation. Pressure and electric field dependences are studied. Stochastic heating is found to be most efficient when the electron mean free path exceeds the size of a single coil cell. Then the mean energy increases approximately exponentially with the electric field amplitude.
Landau, Case, van Kampen and Collisionless Fluid Closures
NASA Astrophysics Data System (ADS)
Joseph, Ilon
2015-11-01
Landau damping represents a fundamental paradox within plasma physics. The equations of motion of classical particles and fields are symmetric under time-reversal; yet, the open system formed by integration over velocity space is not invariant and damping results from phase-mixing. Here, it is shown that the Case-van Kampen theorem can be extended to magnetized plasmas: the linear eigenfunctions provide a complete representation of the particle distribution function and exponentially damped and growing eigenmodes must appear in pairs. The numerical Case-van Kampen transformation can performed efficiently in Fourier velocity space and allows fast timescales in the evolution of the system to be treated using exponential integration. On the other hand, fluid moments require integration over velocity space, and, thus, representation of Landau damping requires explicit introduction of the arrow of time through a collisionless damping operator. This operator captures linear phenomena at the cost of damping nonlinear phenomena such as the plasma echo. Numerical comparisons of these two rather different representations will be presented. LLNL-ABS-674917 prepared by LLNL under Contract DE-AC52-07NA27344.
Collisionless magnetic reconnection under anisotropic MHD approximation
NASA Astrophysics Data System (ADS)
Hirabayashi, Kota; Hoshino, Masahiro
We study the formation of slow-mode shocks in collisionless magnetic reconnection by using one- and two-dimensional collisionless magneto-hydro-dynamic (MHD) simulations based on the double adiabatic approximation, which is an important step to bridge the gap between the Petschek-type MHD reconnection model accompanied by a pair of slow shocks and the observational evidence of the rare occasion of in-situ slow shock observation. According to our results, a pair of slow shocks does form in the reconnection layer. The resultant shock waves, however, are quite weak compared with those in an isotropic MHD from the point of view of the plasma compression and the amount of the magnetic energy released across the shock. Once the slow shock forms, the downstream plasma are heated in highly anisotropic manner and a firehose-sense (P_{||}>P_{⊥}) pressure anisotropy arises. The maximum anisotropy is limited by the marginal firehose criterion, 1-(P_{||}-P_{⊥})/B(2) =0. In spite of the weakness of the shocks, the resultant reconnection rate is kept at the same level compared with that in the corresponding ordinary MHD simulations. It is also revealed that the sequential order of propagation of the slow shock and the rotational discontinuity, which appears when the guide field component exists, changes depending on the magnitude of the guide field. Especially, when no guide field exists, the rotational discontinuity degenerates with the contact discontinuity remaining at the position of the initial current sheet, while with the slow shock in the isotropic MHD. Our result implies that the slow shock does not necessarily play an important role in the energy conversion in the reconnection system and is consistent with the satellite observation in the Earth's magnetosphere.
The microphysics of collisionless shock waves
NASA Astrophysics Data System (ADS)
Marcowith, A.; Bret, A.; Bykov, A.; Dieckman, M. E.; O'C Drury, L.; Lembège, B.; Lemoine, M.; Morlino, G.; Murphy, G.; Pelletier, G.; Plotnikov, I.; Reville, B.; Riquelme, M.; Sironi, L.; Stockem Novo, A.
2016-04-01
Collisionless shocks, that is shocks mediated by electromagnetic processes, are customary in space physics and in astrophysics. They are to be found in a great variety of objects and environments: magnetospheric and heliospheric shocks, supernova remnants, pulsar winds and their nebulæ, active galactic nuclei, gamma-ray bursts and clusters of galaxies shock waves. Collisionless shock microphysics enters at different stages of shock formation, shock dynamics and particle energization and/or acceleration. It turns out that the shock phenomenon is a multi-scale non-linear problem in time and space. It is complexified by the impact due to high-energy cosmic rays in astrophysical environments. This review adresses the physics of shock formation, shock dynamics and particle acceleration based on a close examination of available multi-wavelength or in situ observations, analytical and numerical developments. A particular emphasis is made on the different instabilities triggered during the shock formation and in association with particle acceleration processes with regards to the properties of the background upstream medium. It appears that among the most important parameters the background magnetic field through the magnetization and its obliquity is the dominant one. The shock velocity that can reach relativistic speeds has also a strong impact over the development of the micro-instabilities and the fate of particle acceleration. Recent developments of laboratory shock experiments has started to bring some new insights in the physics of space plasma and astrophysical shock waves. A special section is dedicated to new laser plasma experiments probing shock physics.
The microphysics of collisionless shock waves.
Marcowith, A; Bret, A; Bykov, A; Dieckman, M E; Drury, L O'C; Lembège, B; Lemoine, M; Morlino, G; Murphy, G; Pelletier, G; Plotnikov, I; Reville, B; Riquelme, M; Sironi, L; Novo, A Stockem
2016-04-01
Collisionless shocks, that is shocks mediated by electromagnetic processes, are customary in space physics and in astrophysics. They are to be found in a great variety of objects and environments: magnetospheric and heliospheric shocks, supernova remnants, pulsar winds and their nebulæ, active galactic nuclei, gamma-ray bursts and clusters of galaxies shock waves. Collisionless shock microphysics enters at different stages of shock formation, shock dynamics and particle energization and/or acceleration. It turns out that the shock phenomenon is a multi-scale non-linear problem in time and space. It is complexified by the impact due to high-energy cosmic rays in astrophysical environments. This review adresses the physics of shock formation, shock dynamics and particle acceleration based on a close examination of available multi-wavelength or in situ observations, analytical and numerical developments. A particular emphasis is made on the different instabilities triggered during the shock formation and in association with particle acceleration processes with regards to the properties of the background upstream medium. It appears that among the most important parameters the background magnetic field through the magnetization and its obliquity is the dominant one. The shock velocity that can reach relativistic speeds has also a strong impact over the development of the micro-instabilities and the fate of particle acceleration. Recent developments of laboratory shock experiments has started to bring some new insights in the physics of space plasma and astrophysical shock waves. A special section is dedicated to new laser plasma experiments probing shock physics. PMID:27007555
Saltwater Intrusion Simulation in Heterogeneous Aquifer Using Lattice Boltzmann Method
NASA Astrophysics Data System (ADS)
Servan-Camas, B.; Tsai, F. T.
2006-12-01
This study develops a saltwater intrusion simulation model using a lattice Boltzmann method (LBM) in a two- dimensional coastal confined aquifer. The saltwater intrusion phenomenon is described by density-varied groundwater flow and mass transport equations, where a freshwater-saltwater mixing zone is considered. Although primarily developed using the mesoscopic approach to solve macroscopic fluid dynamic problems (e.g. Navier-Stoke equation), LBM is able to be adopted to solve physical-based diffusion-type governing equations as for the groundwater flow and mass transport equations. The challenge of using LBM in saltwater intrusion modeling is to recover hydraulic conductivity heterogeneity. In this study, the Darcy equation and the advection-dispersion equation (ADE) are recovered in the lattice Boltzmann modeling. Specifically, the hydraulic conductivity heterogeneity is represented by the speed of sound in LBM. Under the consideration on the steady-state groundwater flow due to low storativity, in each time step the flow problem is modified to be a Poisson equation and solved by LBM. Nevertheless, the groundwater flow is still a time-marching problem with spatial-temporal variation in salinity concentration as well as density. The Henry problem is used to compare the LBM results against the Henry analytic solution and SUTRA result. Also, we show that LBM is capable of handling the Dirichlet, Neumann, and Cauchy concentration boundary conditions at the sea side. Finally, we compare the saltwater intrusion results using LBM in the Henry problem when heterogeneous hydraulic conductivity is considered.
Charging-delay induced dust acoustic collisionless shock wave: Roles of negative ions
Ghosh, Samiran; Bharuthram, R.; Khan, Manoranjan; Gupta, M. R.
2006-11-15
The effects of charging-delay and negative ions on nonlinear dust acoustic waves are investigated. It has been found that the charging-delay induced anomalous dissipation causes generation of dust acoustic collisionless shock waves in an electronegative dusty plasma. The small but finite amplitude wave is governed by a Korteweg-de Vries Burger equation in which the Burger term arises due to the charging-delay. Numerical investigations reveal that the charging-delay induced dissipation and shock strength decreases (increases) with the increase of negative ion concentration (temperature)
Liu, Chang; Dodin, Ilya Y.
2015-08-15
The nonlinear frequency shift is derived in a transparent asymptotic form for intense Langmuir waves in general collisionless plasma. The formula describes both fluid and kinetic effects simultaneously. The fluid nonlinearity is expressed, for the first time, through the plasma dielectric function, and the kinetic nonlinearity accounts for both smooth distributions and trapped-particle beams. Various known limiting scalings are reproduced as special cases. The calculation avoids differential equations and can be extended straightforwardly to other nonlinear plasma waves.
Kinetic model for the collisionless sheath of a collisional plasma
NASA Astrophysics Data System (ADS)
Tang, Xian-Zhu; Guo, Zehua
2016-08-01
Collisional plasmas typically have mean-free-path still much greater than the Debye length, so the sheath is mostly collisionless. Once the plasma density, temperature, and flow are specified at the sheath entrance, the profile variation of electron and ion density, temperature, flow speed, and conductive heat fluxes inside the sheath is set by collisionless dynamics, and can be predicted by an analytical kinetic model distribution. These predictions are contrasted here with direct kinetic simulations, showing good agreement.
Global Scale-Invariant Dissipation in Collisionless Plasma Turbulence
Kiyani, K. H.; Chapman, S. C.; Khotyaintsev, Yu. V.; Dunlop, M. W.; Sahraoui, F.
2009-08-14
A higher-order multiscale analysis of the dissipation range of collisionless plasma turbulence is presented using in situ high-frequency magnetic field measurements from the Cluster spacecraft in a stationary interval of fast ambient solar wind. The observations, spanning five decades in temporal scales, show a crossover from multifractal intermittent turbulence in the inertial range to non-Gaussian monoscaling in the dissipation range. This presents a strong observational constraint on theories of dissipation mechanisms in turbulent collisionless plasmas.
Multi-valued Boltzmann machine
Lin, C.T.; Lee, C.S.G.
1995-04-01
The idea of Hopfield network is based on the Ising spin glass model in which each spin has only two possible states: up and down. This paper generalizes these ideas to multivalue case based on the XY spin glass model in which each spin can be in any direction in a plane. Simply using the gradient descent method and the analog Hopfield network, two different analog connectionist structures and their corresponding evolving rules are first designed to transform the XY spin glass model to distributed computational models. Since these two structures can easily get stuck in local minima, a multivalued Boltzmann machine is proposed which adopts the discrete planar spin glass model for the local minimum problem. The multivalued Boltzmann machine can be applied to the mobile robot navigation problem by defining proper artificial magnetic field on the traverse terrain. This new approach has shown to have several advantages over existing graph search and potential field techniques. 28 refs.
Multiscale dynamics based on kinetic simulation of collisionless magnetic reconnection
NASA Astrophysics Data System (ADS)
Fujimoto, Keizo; Takamoto, Makoto
2016-07-01
Magnetic reconnection is a natural energy converter which allows explosive energy release of the magnetic field energy into plasma kinetic energy. The reconnection processes inherently involve multi-scale process. The breaking of the field lines takes place predominantly in a small region called the diffusion region formed near the x-line, while the fast plasma jets resulting from reconnection extend to a distance far beyond the ion kinetic scales from the x-line. There has been a significant gap in understanding of macro-scale and micro-scale processes. The macro-scale model of reconnection has been developed using the magnetohydrodynamics (MHD) equations, while the micro-scale processes around the x-line have been based on kinetic equations including the ion and electron inertia. The problem is that these two kinds of model have significant discrepancies. It has been believed without any guarantee that the microscopic model near the x-line would connect to the macroscopic model far downstream of the x-line. In order to bridge the gap between the macro and micro-scale processes, we have performed large-scale particle-in-cell simulations with the adaptive mesh refinement. The simulation results suggest that the microscopic processes around the x-line do not connect to the previous MHD model even in the region far downstream of the x-line. The slow mode shocks and the associated plasma acceleration do not appear at the exhaust boundary of kinetic reconnection. Instead, the ions are accelerated due to the Speiser motion in the current layer extending to a distance beyond the kinetic scales. The different acceleration mechanisms between the ions and electrons lead to the Hall current system in broad area of the exhaust. Therefore, the previous MHD model could be inappropriate for collisionless magnetic reconnection. Ref. K. Fujimoto & M. Takamoto, Phys. Plasmas, 23, 012903 (2016).
Temperature based Restricted Boltzmann Machines.
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-01
Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view. PMID:26758235
Temperature based Restricted Boltzmann Machines
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-01
Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view. PMID:26758235
Gravitational Instability in Collisionless Cosmological Pancakes
NASA Astrophysics Data System (ADS)
Valinia, Azita; Shapiro, Paul R.; Martel, Hugo; Vishniac, Ethan T.
1997-04-01
The gravitational instability of cosmological pancakes composed of collisionless dark matter in an Einstein-de Sitter universe is investigated numerically to demonstrate that pancakes are unstable with respect to fragmentation and the formation of filaments. A ``pancake'' is defined here as the nonlinear outcome of the growth of a one-dimensional, sinusoidal, plane-wave, adiabatic density perturbation. We have used high-resolution, two-dimensional, N-body simulations by the particle mesh (PM) method to study the response of pancakes to perturbation by either symmetric (density) or antisymmetric (bending or rippling) modes, with corresponding wavevectors ks and ka transverse to the wavevector kp of the unperturbed pancake plane wave. We consider dark matter that is initially ``cold'' (i.e., with no random thermal velocity in the initial conditions). We also investigate the effect of a finite, random, isotropic, initial velocity dispersion (i.e., initial thermal velocity) on the fate of pancake collapse and instability. Our results include the following: (1) For ``cold'' initial conditions, pancakes are gravitationally unstable with respect to all perturbations of wavenumber k >~ 1 (where k = λp/λ, and λp and λ are the wavelengths of the unperturbed pancake and of the perturbation, respectively). This is contrary to the expectations of an approximate, thin-sheet energy argument applied to the results of one-dimensional pancake simulations. The latter predicts that unstable wavenumbers are restricted to the range kmin < k < kmax, where perturbations with k < kmin ~ 1 are stabilized by Hubble expansion, while those with k > kmax > 1 are stabilized by the one-dimensional velocity dispersion of the collisionless particles along the direction of pancake collapse, within the region of shell crossing. (2) Shortly after the pancake first reaches a nonlinear state of collapse, the dimensionless growth rate of the perturbation of pancake surface density by unstable
Simulating Electric Double Layer Capacitance by Using Lattice Boltzmann Method
NASA Astrophysics Data System (ADS)
Sun, Ning; Gersappe, Dilip
2015-03-01
By using the Lattice Boltzmann Method (LBM) we studied diffuse-charge dynamics in electrochemical systems. We use the LBM to solve Poisson-Nernst-Planck equations (PNP) and Modified Poisson-Nernst-Planck equations (MPNP). The isotropic permittivity of electrolyte is modeled using the Booth model. The results show that both steric effect (MPNP) and isotropic permittivity (Booth model) can have large influence on diffuse-charge dynamics, especially when electrolyte concentration or applied potential is high. This model can be applied to simulate electric double layer capacitance of super capacitors with complex geometry and also incorporate other effects such as heat convection in a modular manner.
Accounting for adsorption and desorption in lattice Boltzmann simulations.
Levesque, Maximilien; Duvail, Magali; Pagonabarraga, Ignacio; Frenkel, Daan; Rotenberg, Benjamin
2013-07-01
We report a Lattice-Boltzmann scheme that accounts for adsorption and desorption in the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. Lattice Boltzmann simulations made it possible to solve numerically the coupled Navier-Stokes equations of fluid dynamics and Nernst-Planck equations of electrokinetics in complex, heterogeneous media. With the moment propagation scheme, it became possible to extract the effective diffusion and dispersion coefficients of tracers, or solutes, of any charge, e.g., in porous media. Nevertheless, the dynamical properties of tracers depend on the tracer-surface affinity, which is not purely electrostatic and also includes a species-specific contribution. In order to capture this important feature, we introduce specific adsorption and desorption processes in a lattice Boltzmann scheme through a modified moment propagation algorithm, in which tracers may adsorb and desorb from surfaces through kinetic reaction rates. The method is validated on exact results for pure diffusion and diffusion-advection in Poiseuille flows in a simple geometry. We finally illustrate the importance of taking such processes into account in the time-dependent diffusion coefficient in a more complex porous medium. PMID:23944584
Accounting for adsorption and desorption in lattice Boltzmann simulations.
Levesque, Maximilien; Duvail, Magali; Pagonabarraga, Ignacio; Frenkel, Daan; Rotenberg, Benjamin
2013-07-01
We report a Lattice-Boltzmann scheme that accounts for adsorption and desorption in the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. Lattice Boltzmann simulations made it possible to solve numerically the coupled Navier-Stokes equations of fluid dynamics and Nernst-Planck equations of electrokinetics in complex, heterogeneous media. With the moment propagation scheme, it became possible to extract the effective diffusion and dispersion coefficients of tracers, or solutes, of any charge, e.g., in porous media. Nevertheless, the dynamical properties of tracers depend on the tracer-surface affinity, which is not purely electrostatic and also includes a species-specific contribution. In order to capture this important feature, we introduce specific adsorption and desorption processes in a lattice Boltzmann scheme through a modified moment propagation algorithm, in which tracers may adsorb and desorb from surfaces through kinetic reaction rates. The method is validated on exact results for pure diffusion and diffusion-advection in Poiseuille flows in a simple geometry. We finally illustrate the importance of taking such processes into account in the time-dependent diffusion coefficient in a more complex porous medium.
Boltzmann-Fokker-Planck calculations using standard discrete-ordinates codes
Morel, J.E.
1987-01-01
The Boltzmann-Fokker-Planck (BFP) equation can be used to describe both neutral and charged-particle transport. Over the past several years, the author and several collaborators have developed methods for representing Fokker-Planck operators with standard multigroup-Legendre cross-section data. When these data are input to a standard S/sub n/ code such as ONETRAN, the code actually solves the Boltzmann-Fokker-Planck equation rather than the Boltzmann equation. This is achieved wihout any modification to the S/sub n/ codes. Because BFP calculations can be more demanding from a numerical viewpoint than standard neutronics calculations, we have found it useful to implement new quadrature methods ad convergence acceleration methods in the standard discrete-ordinates code, ONETRAN. We discuss our BFP cross-section representation techniques, our improved quadrature and acceleration techniques, and present results from BFP coupled electron-photon transport calculations performed with ONETRAN. 19 refs., 7 figs.
Master Equation for a Quantum Particle in a Gas
Hornberger, Klaus
2006-08-11
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
Master equation for a quantum particle in a gas.
Hornberger, Klaus
2006-08-11
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
The Lattice Boltzmann Method applied to neutron transport
Erasmus, B.; Van Heerden, F. A.
2013-07-01
In this paper the applicability of the Lattice Boltzmann Method to neutron transport is investigated. One of the main features of the Lattice Boltzmann method is the simultaneous discretization of the phase space of the problem, whereby particles are restricted to move on a lattice. An iterative solution of the operator form of the neutron transport equation is presented here, with the first collision source as the starting point of the iteration scheme. A full description of the discretization scheme is given, along with the quadrature set used for the angular discretization. An angular refinement scheme is introduced to increase the angular coverage of the problem phase space and to mitigate lattice ray effects. The method is applied to a model problem to investigate its applicability to neutron transport and the results are compared to a reference solution calculated, using MCNP. (authors)
Tether-Induced Airglow: Collisionless Effects
NASA Technical Reports Server (NTRS)
Mishin, E. V.; Khazanov, G. V.
2006-01-01
Martinez-Sanchez and Sanmartin [1997] showed that a bare conducting tether can be used as a source of an energetic electron beam. Interacting with the E region atmosphere, the beam should produce airglow thus making possible to deduce the neutral density on a continuous basis. Fujii et al. [2005] suggested that this idea be tested in a specially-designed sounding rocket experiment. We show that collisionless beam-plasma interactions (BPI) complement direct impact, leading to appreciable green-line (557.7 nm) emissions in the F region. In the E region, BPI develops near the entry in the valley, resulting in a narrow layer of strongly-elevated and airglow. Besides, neutralizing electric currents carried by ionospheric electrons in the valley can become unstable or even insufficient to compensate the beam current. Developing plasma waves inhibit neutralizing currents. In the extreme case, the beam might be locked in the valley (the 'virtual cathode'). In addition to optical observations, these effects can also be observed by radiophysical means.
Driving Weibel-mediated collisionless shocks with NIF
NASA Astrophysics Data System (ADS)
Fiuza, Frederico; Spitkovsky, Anatoly; Ryutov, Dmitri; Ross, Steven; Huntington, Channing; Mori, Warren; Silva, Luis; Park, Hye-Sook; Remington, Bruce
2013-10-01
Collisionless shocks are ubiquitous in astrophysical plasmas and are known to be responsible for particle acceleration; however, the microphysics underlying shock formation and particle acceleration is not yet fully understood. High-power lasers are bringing the study of collisionless shocks into the realm of laboratory experiments. In particular, the National Ignition Facility allows for the generation of collisionless plasma flows that are hundreds of ion skin-depths long and provides ideal conditions for the study of Weibel-mediated shocks. We have performed detailed 2D and 3D particle-in-cell simulations with OSIRIS to explore the laboratory conditions associated with counter-streaming high-velocity plasma flows for realistic profiles. We have modeled the proton radiography of the interaction for self-consistent fields and determined the experimental signatures of the generation of Weibel B-fields and collisionless shocks. We will discuss the importance of modeling realistic ion to electron mass ratios and of taking into account Biermann battery B-fields. Our work identifies the conditions for the formation of collisionless shocks in laboratory, both in unmagnetized and magnetized scenarios, showing the possibility of observing for the first time Weibel-mediated shocks in near future experiments.
Shearing Box Simulations of the MRI in a Collisionless Plasma
Sharma, Prateek; Hammett, Gregory, W.; Quataert, Eliot; Stone, James, M.
2005-08-31
We describe local shearing box simulations of turbulence driven by the magnetorotational instability (MRI) in a collisionless plasma. Collisionless effects may be important in radiatively inefficient accretion flows, such as near the black hole in the Galactic Center. The MHD version of ZEUS is modified to evolve an anisotropic pressure tensor. A fluid closure approximation is used to calculate heat conduction along magnetic field lines. The anisotropic pressure tensor provides a qualitatively new mechanism for transporting angular momentum in accretion flows (in addition to the Maxwell and Reynolds stresses). We estimate limits on the pressure anisotropy due to pitch angle scattering by kinetic instabilities. Such instabilities provide an effective ''collision'' rate in a collisionless plasma and lead to more MHD-like dynamics. We find that the MRI leads to efficient growth of the magnetic field in a collisionless plasma, with saturation amplitudes comparable to those in MHD. In the saturated state, the anisotropic stress is comparable to the Maxwell stress, implying that the rate of angular momentum transport may be moderately enhanced in a collisionless plasma.
ENTROPY PRODUCTION IN COLLISIONLESS SYSTEMS. II. ARBITRARY PHASE-SPACE OCCUPATION NUMBERS
Barnes, Eric I.; Williams, Liliya L. R. E-mail: llrw@astro.umn.edu
2012-04-01
We present an analysis of two thermodynamic techniques for determining equilibria of self-gravitating systems. One is the Lynden-Bell (LB) entropy maximization analysis that introduced violent relaxation. Since we do not use the Stirling approximation, which is invalid at small occupation numbers, our systems have finite mass, unlike LB's isothermal spheres. (Instead of Stirling, we utilize a very accurate smooth approximation for ln x{exclamation_point}.) The second analysis extends entropy production extremization to self-gravitating systems, also without the use of the Stirling approximation. In addition to the LB statistical family characterized by the exclusion principle in phase space, and designed to treat collisionless systems, we also apply the two approaches to the Maxwell-Boltzmann (MB) families, which have no exclusion principle and hence represent collisional systems. We implicitly assume that all of the phase space is equally accessible. We derive entropy production expressions for both families and give the extremum conditions for entropy production. Surprisingly, our analysis indicates that extremizing entropy production rate results in systems that have maximum entropy, in both LB and MB statistics. In other words, both thermodynamic approaches lead to the same equilibrium structures.
Entropy Production in Collisionless Systems. II. Arbitrary Phase-space Occupation Numbers
NASA Astrophysics Data System (ADS)
Barnes, Eric I.; Williams, Liliya L. R.
2012-04-01
We present an analysis of two thermodynamic techniques for determining equilibria of self-gravitating systems. One is the Lynden-Bell (LB) entropy maximization analysis that introduced violent relaxation. Since we do not use the Stirling approximation, which is invalid at small occupation numbers, our systems have finite mass, unlike LB's isothermal spheres. (Instead of Stirling, we utilize a very accurate smooth approximation for ln x!.) The second analysis extends entropy production extremization to self-gravitating systems, also without the use of the Stirling approximation. In addition to the LB statistical family characterized by the exclusion principle in phase space, and designed to treat collisionless systems, we also apply the two approaches to the Maxwell-Boltzmann (MB) families, which have no exclusion principle and hence represent collisional systems. We implicitly assume that all of the phase space is equally accessible. We derive entropy production expressions for both families and give the extremum conditions for entropy production. Surprisingly, our analysis indicates that extremizing entropy production rate results in systems that have maximum entropy, in both LB and MB statistics. In other words, both thermodynamic approaches lead to the same equilibrium structures.
Dependence of Langmuir probe data on distance from the axis of a collisionless plasma
Knappmiller, Scott; Robertson, Scott
2007-03-15
A cylindrical Langmuir probe in a low-density, collisionless plasma (density {approx}10{sup 8} cm{sup -3}, electron temperature 0.2 eV) has been scanned radially through the presheath region to determine the effect of distance from the axis on the current-voltage characteristic. In the ion part of the probe characteristic, the collected ion current decreases with distance from the axis as a consequence of ion acceleration by the presheath. The part of the ion current from charge-exchange collisions remains relatively constant. In the electron part of the probe characteristic, the collected current decreases with distance from the axis, consistent with the existence of a small potential barrier from the presheath between the axis and the probe. The electron temperature from the slope of the probe characteristic is nearly constant across the presheath region. The plasma potential from the Langmuir probe characteristic is also nearly constant, indicating that the probe analysis finds the plasma potential on the axis, even when the probe is not on the axis. The plasma potential from an emissive probe shows an approximately parabolic profile. The plasma potential from the emissive probe and the Boltzmann relation give nearly the same density profile in the presheath that is obtained from the Langmuir probe data.
NASA Astrophysics Data System (ADS)
Falceta-Gonçalves, D.; Kowal, G.
2015-07-01
In this work we report on a numerical study of the cosmic magnetic field amplification due to collisionless plasma instabilities. The collisionless magnetohydrodynamic equations derived account for the pressure anisotropy that leads, in specific conditions, to the firehose and mirror instabilities. We study the time evolution of seed fields in turbulence under the influence of such instabilities. An approximate analytical time evolution of the magnetic field is provided. The numerical simulations and the analytical predictions are compared. We found that (i) amplification of the magnetic field was efficient in firehose-unstable turbulent regimes, but not in the mirror-unstable models (ii) the growth rate of the magnetic energy density is much faster than the turbulent dynamo and (iii) the efficient amplification occurs at small scales. The analytical prediction for the correlation between the growth timescales and pressure anisotropy is confirmed by the numerical simulations. These results reinforce the idea that pressure anisotropies—driven naturally in a turbulent collisionless medium, e.g., the intergalactic medium, could efficiently amplify the magnetic field in the early universe (post-recombination era), previous to the collapse of the first large-scale gravitational structures. This mechanism, though fast for the small-scale fields (∼kpc scales), is unable to provide relatively strong magnetic fields at large scales. Other mechanisms that were not accounted for here (e.g., collisional turbulence once instabilities are quenched, velocity shear, or gravitationally induced inflows of gas into galaxies and clusters) could operate afterward to build up large-scale coherent field structures in the long time evolution.
Particle simulation of collisionless reconnection using TRISTAN
NASA Astrophysics Data System (ADS)
Kotzé, P. B.; Nishikawa, K.-I.; Büchner, J.
Magnetic reconnection is an important mechanism in the dynamics of the magnetosphere in facilitating the change in magnetospheric topology in response to the orientation of the interplanetary magnetic field (IMF). In the magnetosphere the classical collision rate is small, while the inertia of the electrons allows the frozen-in flux constraint to be broken. At small values of resistivity, this dissipation region then controls the rate of reconnection by forming an elongated Sweet-Parker layer, with an inflow velocity νi into the x-line that scales like: νi = δ ∆ νA νA (1) where δ and ∆ are the width (controlled by resistivity) and length (macroscopic) of the dissipation region respectively and νA is the Alfvén velocity. The scale length around the x-line where the electrons become demagnetised is of the order of the electron skin depth c/ωpe. This region is however much smaller than the ion inertial length c/ωpi, below which the Hall terms in the kinetic Ohm's law become important. Within this distance from the x-line the ions decouple from the electrons and are accelerated away at Alfv´enic velocities (Burkhart et al., 1990) The dynamics of the system at the scale length of the electron dissipation layer is therefore linked to Hall physics, making it a critical ingredient in determining collisionless reconnection rates. Particle simulation techniques have been used to investigate magnetic reconnection in 2-D for a Harris sheet equilibrium. A set of parameters are chosen as well as the dimensions of the computational domain, the boundary conditions and the initial amplitude and form of a seed magnetic island to start the reconnection process. Some preliminary results will be given in this paper.
Student Understanding of the Boltzmann Factor
ERIC Educational Resources Information Center
Smith, Trevor I.; Mountcastle, Donald B.; Thompson, John R.
2015-01-01
We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data…
Recent advances in lattice Boltzmann methods
Chen, S.; Doolen, G.D.; He, X.; Nie, X.; Zhang, R.
1998-12-31
In this paper, the authors briefly present the basic principles of lattice Boltzmann method and summarize recent advances of the method, including the application of the lattice Boltzmann method for fluid flows in MEMS and simulation of the multiphase mixing and turbulence.
A Semianalytical Ion Current Model for Radio Frequency Driven Collisionless Sheaths
NASA Technical Reports Server (NTRS)
Bose, Deepak; Govindan, T. R.; Meyyappan, M.; Arnold, Jim (Technical Monitor)
2001-01-01
We propose a semianalytical ion dynamics model for a collisionless radio frequency biased sheath. The model uses bulk plasma conditions and electrode boundary condition to predict ion impact energy distribution and electrical properties of the sheath. The proposed model accounts for ion inertia and ion current modulation at bias frequencies that are of the same order of magnitude as the ion plasma frequency. A relaxation equation for ion current oscillations is derived which is coupled with a damped potential equation in order to model ion inertia effects. We find that inclusion of ion current modulation in the sheath model shows marked improvements in the predictions of sheath electrical properties and ion energy distribution function.
ICPP: Collisionless shock and supernova remnant simulation experiments on VULCAN.
NASA Astrophysics Data System (ADS)
Woolsey, Nigel C.
2000-10-01
The VULCAN laser at the Central Laser Facility is used for laboratory-based simulations of collisionless shocks. One of the most difficult aspects of collisionless shock behaviour, the role of the magnetic field, is to be tested directly against experiment. Preliminary experiments to generate strong magnetic fields using a laser-driven mm-scale Helmholtz coil, and the formation of collisionless colliding plasmas using two counter-streaming exploding foil plasmas will be discussed. We consider the scaling of the hydrodynamics and magnetic field of the these experiments to those in supernova remnants (SNR) impacting the interstellar medium (ISM). This is achieved by ensuring the experiment and the SNR-ISM exhibit similar values of key dimensionless parameters. Work supported in part by EPSRC, CLF Direct Access, CEC-ERB FMR XCT 980168, Euratom and the UK DTI.
Low Frequency Waves at and Upstream of Collisionless Shocks
NASA Astrophysics Data System (ADS)
Wilson, L. B.
2016-02-01
This chapter focuses on the range of low frequency electromagnetic modes observed at and upstream of collisionless shocks in the heliosphere. It discusses a specific class of whistler mode wave observed immediately upstream of collisionless shock ramps, called a whistler precursor. Though these modes have been (and are often) observed upstream of quasi-parallel shocks, the authors limit their discussion to those observed upstream of quasi-perpendicular shocks. The chapter discusses the various ion velocity distributions observed at and upstream of collisionless shocks. It also introduces some terminology and relevant instabilities for ion foreshock waves. The chapter discusses the most common ultra-low frequency (ULF) wave types, their properties, and their free energy sources. It discusses modes that are mostly Alfvénic (i.e., mostly transverse but can be compressive) in nature.
In situ detection of collisionless reconnection in the Earth's magnetotail.
Oieroset, M; Phan, T D; Fujimoto, M; Lin, R P; Lepping, R P
2001-07-26
Magnetic reconnection is the process by which magnetic field lines of opposite polarity reconfigure to a lower-energy state, with the release of magnetic energy to the surroundings. Reconnection at the Earth's dayside magnetopause and in the magnetotail allows the solar wind into the magnetosphere. It begins in a small 'diffusion region', where a kink in the newly reconnected lines produces jets of plasma away from the region. Although plasma jets from reconnection have previously been reported, the physical processes that underlie jet formation have remained poorly understood because of the scarcity of in situ observations of the minuscule diffusion region. Theoretically, both resistive and collisionless processes can initiate reconnection, but which process dominates in the magnetosphere is still debated. Here we report the serendipitous encounter of the Wind spacecraft with an active reconnection diffusion region, in which are detected key processes predicted by models of collisionless reconnection. The data therefore demonstrate that collisionless reconnection occurs in the magnetotail.
High-order hydrodynamics via lattice Boltzmann methods.
Colosqui, Carlos E
2010-02-01
In this work, closure of the Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) moment hierarchy is accomplished via projection of the distribution function f onto a space H(N) spanned by N-order Hermite polynomials. While successive order approximations retain an increasing number of leading-order moments of f , the presented procedure produces a hierarchy of (single) N-order partial-differential equations providing exact analytical description of the hydrodynamics rendered by ( N-order) lattice Boltzmann-BGK (LBBGK) simulation. Numerical analysis is performed with LBBGK models and direct simulation Monte Carlo for the case of a sinusoidal shear wave (Kolmogorov flow) in a wide range of Weissenberg number Wi=taunuk(2) (i.e., Knudsen number Kn=lambdak=square root Wi); k is the wave number, [corrected] tau is the relaxation time of the system, and lambda approximately tauc(s) is the mean-free path, where c(s) is the speed of sound. The present results elucidate the applicability of LBBGK simulation under general nonequilibrium conditions.
PARTICLE ACCELERATION DURING MAGNETOROTATIONAL INSTABILITY IN A COLLISIONLESS ACCRETION DISK
Hoshino, Masahiro
2013-08-20
Particle acceleration during the magnetorotational instability (MRI) in a collisionless accretion disk was investigated by using a particle-in-cell simulation. We discuss the important role that magnetic reconnection plays not only on the saturation of MRI but also on the relativistic particle generation. The plasma pressure anisotropy of p > p{sub ||} induced by the action of MRI dynamo leads to rapid growth in magnetic reconnection, resulting in the fast generation of nonthermal particles with a hard power-law spectrum. This efficient particle acceleration mechanism involved in a collisionless accretion disk may be a possible model to explain the origin of high-energy particles observed around massive black holes.
Lattice Boltzmann LES for MHD Turbulence
NASA Astrophysics Data System (ADS)
Flint, Chris; Vahala, George; Vahala, Linda; Soe, Min
2015-11-01
Dellar's lattice Boltzmann (LB) model of 2D incompressible MHD introduced both a scalar velocity and vector magnetic distribution functions, which automatically enforces div B = 0 through the trace of an antisymmetric perturbed tensor. In the Smagorinsky LES model, the filtered Reynolds stresses are modeled by mean field gradient terms, with ad hoc closure eddy transport terms. Ansumali et. al. have developed an LES for Navier-Stokes turbulence by filtering the underlying mesoscopic LB. The filtered LB equations are then subjected to the Chapman-Enskog expansion. A Smagorinsky-like LES is recovered with no ad hoc assumptions other than the subgrid terms contribute only at the transport time scales. Here we extend these ideas to 2D MHD turbulence. The DNS data base is being generated from a multiple relaxation time (MRT) model with a quasi-entropic analytic scheme introduced recently by Karlin et. al. (2014) based on splitting the moment representation into various subgroups. Work supported by NSF, DoD.
Three-dimensional lattice Boltzmann model for compressible flows.
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.
Lattice Boltzmann model for the simulation of multicomponent mixtures.
Arcidiacono, S; Karlin, I V; Mantzaras, J; Frouzakis, C E
2007-10-01
A lattice Boltzmann (LB) model for the simulation of realistic multicomponent mixtures is constructed. In the hydrodynamic limit, the LB model recovers the equations of continuum mechanics within the mixture-averaged diffusion approximation. The present implementation can be used to simulate realistic mixtures with arbitrary Schmidt numbers and molecular masses of the species. The model is applied to the mixing of two opposed jets of different concentrations and the results are in excellent agreement with a continuum model. An application to the simulation of mixtures in microflows is also presented. Results compare well with existing kinetic theory predictions of the slip coefficient for mixtures in a Couette flow.
Poisson-Boltzmann theory for two parallel uniformly charged plates
NASA Astrophysics Data System (ADS)
Xing, Xiangjun
2011-04-01
We solve the nonlinear Poisson-Boltzmann equation for two parallel and like-charged plates both inside a symmetric electrolyte, and inside a 2:1 asymmetric electrolyte, in terms of Weierstrass elliptic functions. From these solutions we derive the functional relation between the surface charge density, the plate separation, and the pressure between plates. For the one plate problem, we obtain exact expressions for the electrostatic potential and for the renormalized surface charge density, both in symmetric and in asymmetric electrolytes. For the two plate problems, we obtain new exact asymptotic results in various regimes.
A lattice Boltzmann method for dilute polymer solutions.
Singh, Shiwani; Subramanian, Ganesh; Ansumali, Santosh
2011-06-13
We present a lattice Boltzmann approach for the simulation of non-Newtonian fluids. The method is illustrated for the specific case of dilute polymer solutions. With the appropriate local equilibrium distribution, phase-space dynamics on a lattice, driven by a Bhatnagar-Gross-Krook (BGK) relaxation term, leads to a solution of the Fokker-Planck equation governing the probability density of polymer configurations. Results for the bulk rheological characteristics for steady and start-up shear flow are presented, and compare favourably with those obtained using Brownian dynamics simulations. The new method is less expensive than stochastic simulation techniques, particularly in the range of small to moderate Weissenberg numbers (Wi).
Boltzmann: The Genius of Disorder
NASA Astrophysics Data System (ADS)
Mussardo, G.; Merlone, A.
2010-07-01
The tragedy and greatness of the contribution of Ludwig Boltzmann cannot be understood without taking into account for the relevant scientific developments that took place in the nineteenth century, one of the most eventful periods in the history of science. The kinetic theory opened a new theoretical perspective in understanding natural phenomena. The introduction of new categories of order and disorder changed radically the point of view of those physicists that accepted Boltzmann’s thesis and led, at the same time, to strong opposition to the Viennese Scientist. In this article, we present the academic situation, scientific theories, and disputes involving the Boltzmann’s theories. A short introduction on the birth of the atomistic theories opens the article, while a view on the evolution of the concept of temperature and the definition of its unit quantity closes it.
Reassessing the single relaxation time Lattice Boltzmann method for the simulation of Darcy’s flows
NASA Astrophysics Data System (ADS)
Prestininzi, Pietro; Montessori, Andrea; La Rocca, Michele; Succi, Sauro
2016-09-01
It is shown that the single relaxation time (SRT) version of the Lattice Boltzmann (LB) equation permits to compute the permeability of Darcy’s flows in porous media within a few percent accuracy. This stands in contrast with previous claims of inaccuracy, which we relate to the lack of recognition of the physical dependence of the permeability on the Knudsen number.
Reprint of : The Boltzmann--Langevin approach: A simple quantum-mechanical derivation
NASA Astrophysics Data System (ADS)
Nagaev, K. E.
2016-08-01
We present a simple quantum-mechanical derivation of correlation function of Langevin sources in the semiclassical Boltzmann-Langevin equation. The specific case of electron-phonon scattering is considered. It is shown that the assumption of weak scattering leads to the Poisson nature of the scattering fluxes.
Kwok, Dixon T.K.
2008-05-10
A robust and stable numerical algorithm is developed for the hybrid method of particle-in-cell ions and Boltzmann distribution of electrons. A different approach to estimate the electron density reference and its proper potential reference is developed to overcome the problems of instability and divergence of previous approaches. The electron density reference is precisely calculated, the tolerance criterion is well-defined, and convergence is guaranteed by applying bi-section golden rule. To increase the rate of convergence, an external loop is incorporated with the bi-section golden rule to vary the brackets. The validity of the method is proved by comparing the simulated result with well-known analytical formula. The simulated sheath potential at a floating wall fit well to the analytic result. The collisionless ion kinetic energy acquired from the voltage difference between the pre-sheath and ion sheath does not violate the Bohm sheath criterion. For work that focuses on the plasma process at the ion sheath and not on the generation of plasma, this method saves simulation time by avoiding time consuming particle or kinetic model of electrons. The new approach reproduces the ion density profile at the ion sheaths region of a plasma with bi-Maxwellian electrons coupling with radio-frequency (RF) signal by introducing two Boltzmann relations to describe the cold and hot thermal electrons for the first time.
NASA Astrophysics Data System (ADS)
Wang, Liang; Hakim, Ammar H.; Bhattacharjee, A.; Germaschewski, K.
2015-01-01
We introduce an extensible multi-fluid moment model in the context of collisionless magnetic reconnection. This model evolves full Maxwell equations and simultaneously moments of the Vlasov-Maxwell equation for each species in the plasma. Effects like electron inertia and pressure gradient are self-consistently embedded in the resulting multi-fluid moment equations, without the need to explicitly solving a generalized Ohm's law. Two limits of the multi-fluid moment model are discussed, namely, the five-moment limit that evolves a scalar pressures for each species and the ten-moment limit that evolves the full anisotropic, non-gyrotropic pressure tensor for each species. We first demonstrate analytically and numerically that the five-moment model reduces to the widely used Hall magnetohydrodynamics (Hall MHD) model under the assumptions of vanishing electron inertia, infinite speed of light, and quasi-neutrality. Then, we compare ten-moment and fully kinetic particle-in-cell (PIC) simulations of a large scale Harris sheet reconnection problem, where the ten-moment equations are closed with a local linear collisionless approximation for the heat flux. The ten-moment simulation gives reasonable agreement with the PIC results regarding the structures and magnitudes of the electron flows, the polarities and magnitudes of elements of the electron pressure tensor, and the decomposition of the generalized Ohm's law. Possible ways to improve the simple local closure towards a nonlocal fully three-dimensional closure are also discussed.
Wang, Liang Germaschewski, K.; Hakim, Ammar H.; Bhattacharjee, A.
2015-01-15
We introduce an extensible multi-fluid moment model in the context of collisionless magnetic reconnection. This model evolves full Maxwell equations and simultaneously moments of the Vlasov-Maxwell equation for each species in the plasma. Effects like electron inertia and pressure gradient are self-consistently embedded in the resulting multi-fluid moment equations, without the need to explicitly solving a generalized Ohm's law. Two limits of the multi-fluid moment model are discussed, namely, the five-moment limit that evolves a scalar pressures for each species and the ten-moment limit that evolves the full anisotropic, non-gyrotropic pressure tensor for each species. We first demonstrate analytically and numerically that the five-moment model reduces to the widely used Hall magnetohydrodynamics (Hall MHD) model under the assumptions of vanishing electron inertia, infinite speed of light, and quasi-neutrality. Then, we compare ten-moment and fully kinetic particle-in-cell (PIC) simulations of a large scale Harris sheet reconnection problem, where the ten-moment equations are closed with a local linear collisionless approximation for the heat flux. The ten-moment simulation gives reasonable agreement with the PIC results regarding the structures and magnitudes of the electron flows, the polarities and magnitudes of elements of the electron pressure tensor, and the decomposition of the generalized Ohm's law. Possible ways to improve the simple local closure towards a nonlocal fully three-dimensional closure are also discussed.
Collisionless plasma expansion into vacuum: two new twists on an old problem
NASA Astrophysics Data System (ADS)
Arefiev, Alexey
2008-11-01
Plasma expansion into vacuum is a generic problem with a broad range of applications. Of particular interest are those regimes where the expanding plasma consists of energetic electrons and cold ions. The expansion is then caused by electron pressure and serves as an energy transfer mechanism from electrons to ions. Collisional plasma expansion is similar to the gas-dynamic expansion, with the fluid description applicable, whereas collisionless plasma expansion requires a kinetic treatment, especially for the energetic electrons. The collisionless expansion is often described under the assumption that the electron distribution is Maxwellian [1]. However, this assumption is not universally relevant, since the expansion may lead to a significant distortion of the electron distribution function. Also, non-Maxwellian electrons may force the quasineutrality condition to break down. This talk presents two problems [2,3] which illustrate the above kinetic effects. The first one is the problem of a magnetic nozzle that transforms an incoming subsonic plasma flow into a supersonic jet. The second is the problem of an expanding nanoplasma (cluster) with a two-component electron distribution. In the nozzle problem, a magnetic mirror, together with the expanding plasma boundary, generates a trapped electron population downstream. This population is decoupled from the plasma source and, consequently, it undergoes adiabatic cooling. The resulting distortion of the electron distribution function is a new element not captured by the usually used Boltzmann relation. In the cluster problem, the key feature is the initial two-component electron distribution with a cold majority and a hot minority both occupying the same volume prior to the expansion. The cluster problem exhibits a breakdown of quasineutrality manifested by a double-layer inside the flow. Both problems are illustrated with closed-form analytical solutions [2,3]. This work was supported by the US DOE NNSA under
Ergodicity, ensembles, irreversibility in Boltzmann and beyond
NASA Astrophysics Data System (ADS)
Gallavotti, Giovanni
1995-03-01
The contents of a not too well-known paper by Boltzmann are critically examined. The etymology of the word ergodic and its implications are discussed. A connection with the modern theory of Ruelle is attempted.
Lattice Boltzmann simulation of a fluid flow around a triangular unit of three isothermal cylinders
NASA Astrophysics Data System (ADS)
Alinejad, J.
2016-01-01
The lattice Boltzmann method is employed to simulate heat transfer in the flow past three arrangements of elliptical and circular cylinders under an isothermal boundary condition. The lattice Boltzmann equations and the Bhatnagar-Gross-Krook model are used to simulate two-dimensional forced convection at 30 ≤ Re ≤ 100 and Pr = 0.71. Pressure distributions, isotherms, and streamlines are obtained. Vortex shedding maps are observed in detail for several cases. The present results are in good agreement with available experimental and numerical data.
Student understanding of the Boltzmann factor
NASA Astrophysics Data System (ADS)
Smith, Trevor I.; Mountcastle, Donald B.; Thompson, John R.
2015-12-01
[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations of student discussions about the Boltzmann factor and its derivation during the tutorial development process. This additional information informed modifications that improved students' abilities to complete the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. These data also show an increase in students' appreciation of the origin and significance of the Boltzmann factor during the student discussions. Our findings provide evidence that working in groups to better understand the physical origins of the canonical probability distribution helps students gain a better understanding of when the Boltzmann factor is applicable and how to use it appropriately in answering relevant questions.
Maxwell-Vlasov equations as a continuous Hamiltonian system
Morrison, P.J.
1980-11-01
The well-known Maxwell-Vlasov equations that describe a collisionless plasma are cast into Hamiltonian form. The dynamical variables are the physical although noncanonical variables E, B, and f. We present a Poisson bracket which acts on these variables and the energy functional to produce the equations of motion.
NASA Technical Reports Server (NTRS)
Matsuda, Y.
1974-01-01
A low-noise plasma simulation model is developed and applied to a series of linear and nonlinear problems associated with electrostatic wave propagation in a one-dimensional, collisionless, Maxwellian plasma, in the absence of magnetic field. It is demonstrated that use of the hybrid simulation model allows economical studies to be carried out in both the linear and nonlinear regimes with better quantitative results, for comparable computing time, than can be obtained by conventional particle simulation models, or direct solution of the Vlasov equation. The characteristics of the hybrid simulation model itself are first investigated, and it is shown to be capable of verifying the theoretical linear dispersion relation at wave energy levels as low as .000001 of the plasma thermal energy. Having established the validity of the hybrid simulation model, it is then used to study the nonlinear dynamics of monochromatic wave, sideband instability due to trapped particles, and satellite growth.
Three-dimensional lattice Boltzmann model for electrodynamics.
Mendoza, M; Muñoz, J D
2010-11-01
In this paper we introduce a three-dimensional Lattice-Boltzmann model that recovers in the continuous limit the Maxwell equations in materials. In order to build conservation equations with antisymmetric tensors, like the Faraday law, the model assigns four auxiliary vectors to each velocity vector. These auxiliary vectors, when combined with the distribution functions, give the electromagnetic fields. The evolution is driven by the usual Bhatnager-Gross-Krook (BGK) collision rule, but with a different form for the equilibrium distribution functions. This lattice Bhatnager-Gross-Krook (LBGK) model allows us to consider for both dielectrics and conductors with realistic parameters, and therefore it is adequate to simulate the most diverse electromagnetic problems, like the propagation of electromagnetic waves (both in dielectric media and in waveguides), the skin effect, the radiation pattern of a small dipole antenna and the natural frequencies of a resonant cavity, all with 2% accuracy. Actually, it shows to be one order of magnitude faster than the original Finite-difference time-domain (FDTD) formulation by Yee to reach the same accuracy. It is, therefore, a valuable alternative to simulate electromagnetic fields and opens lattice Boltzmann for a broad spectrum of new applications in electrodynamics.
NASA Technical Reports Server (NTRS)
Lie-Svendsen, O.; Leer, E.
1995-01-01
We have studied the evolution of the velocity distribution function of a test population of electrons in the solar corona and inner solar wind region, using a recently developed kinetic model. The model solves the time dependent, linear transport equation, with a Fokker-Planck collision operator to describe Coulomb collisions between the 'test population' and a thermal background of charged particles, using a finite differencing scheme. The model provides information on how non-Maxwellian features develop in the distribution function in the transition region from collision dominated to collisionless flow. By taking moments of the distribution the evolution of higher order moments, such as the heat flow, can be studied.
Linearized lattice Boltzmann method for micro- and nanoscale flow and heat transfer.
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.
A lattice Boltzmann model for multiphase flows with large density ratio
NASA Astrophysics Data System (ADS)
Zheng, H. W.; Shu, C.; Chew, Y. T.
2006-10-01
A lattice Boltzmann model for simulating multiphase flows with large density ratios is described in this paper. The method is easily implemented. It does not require solving the Poisson equation and does not involve the complex treatments of derivative terms. The interface capturing equation is recovered without any additional terms as compared to other methods [M.R. Swift, W.R. Osborn, J.M. Yeomans, Lattice Boltzmann simulation of liquid-gas and binary fluid systems, Phys. Rev. E 54 (1996) 5041-5052; T. Inamuro, T. Ogata, S. Tajima, N. Konishi, A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys. 198 (2004) 628-644; T. Lee, C.-L. Lin, A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio, J. Comput. Phys. 206 (2005) 16-47]. Besides, it requires less discrete velocities. As a result, its efficiency could be greatly improved, especially in 3D applications. It is validated by several cases: a bubble in a stationary flow and the capillary wave. The numerical surface tension obtained from the Laplace law and the interface profile agrees very well with the respective analytical solution. The method is further verified by its application to capillary wave and the bubble rising under buoyancy with comparison to other methods. All the numerical experiments show that the present approach can be used to model multiphase flows with large density ratios.
Linearized lattice Boltzmann method for micro- and nanoscale flow and heat transfer.
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. PMID:26274307
Linearized lattice Boltzmann method for micro- and nanoscale flow and heat transfer
NASA Astrophysics Data System (ADS)
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.
Comment on ‘A low-uncertainty measurement of the Boltzmann constant’
NASA Astrophysics Data System (ADS)
Macnaughton, Donald B.
2016-02-01
The International Committee for Weights and Measures has projected a major revision of the International System of Units in which all the base units will be defined by fixing the values of certain fundamental constants of nature. To assist, de Podesta et al recently experimentally obtained a precise new estimate of the Boltzmann constant. This estimate is proposed as a basis for the redefinition of the unit of temperature, the kelvin. The present paper reports a reanalysis of de Podesta et al’s data that reveals systematic non-random patterns in the residuals of the key fitted model equation. These patterns violate the assumptions underlying the analysis and thus they raise questions about the validity of de Podesta et al’s estimate of the Boltzmann constant. An approach is discussed to address these issues, which should lead to an accurate estimate of the Boltzmann constant with a lower uncertainty.
Phase segregation via Vlasov-Boltzmann particle dynamics
Bastea, S
1999-01-19
In order to better understand and model the phase segregation of binary fluids we opted for a mesoscopic description that proves to be simplifying both conceptually and computationally. The system that we studied is a mixture of two kinds of particles. All particles interact with each other through strong short-range interactions modeled by hard spheres with the same mass and diameter. There is also a smooth long-range repulsion between particles of different kinds. At low overall densities and weak enough repulsion the natural dynamical description for this system is given in terms of two coupled, energy and momentum conserving Vlasov- Boltzmann equations, making it what we call a dynamical mean-field model. The computational scheme that we used is a combination of direct sim- ulation Monte Carlo (DSMC) and particle-in-the-cell (PIC) evolution, that inherits the efficiency and robustness of these two algorithms. The DSMC is a stochastic algorithm due to Bird that consistently incorporates the as- sumptions behind the Boltzmann equation into the particle dynamics. The method is essentially the following: the physical space is divided into a net- work of cells containing typically tens of particles and the free flow of the particles over a small time interval {Delta}t is followed by representative collisions among pairs of particles sharing the same cell. The typical linear dimension of a cell is a fraction of the mean free path between collisions. The PIC method for integrating the equations of motion was first used to deal with the l/r potential in plasma physics. It takes advantage of the simple form of the Vlasov potential, which is a product in Fourier space, by calculating the densities on a grid through some weighting, then the potentials and forces on the same grid, and finally interpolating the forces at the position of each particle. These two methods can be naturally brought together by replacing the free flow of the DSMC procedure by motion in the
NASA Astrophysics Data System (ADS)
Gulati, Mamta; Saini, Tarun Deep
2016-07-01
The short-wave asymptotics (WKB) of spiral density waves in self-gravitating stellar discs is well suited for the study of the dynamics of tightly-wound wavepackets. But the textbook WKB theory is not well adapted to the study of the linear eigenmodes in a collisionless self-gravitating disc because of the transcendental nature of the dispersion relation. We present a modified WKB theory of spiral density waves, for collisionless discs in the epicyclic limit, in which the perturbed gravitational potential is related to the perturbed surface density by the Poisson integral in Kalnaj's logarithmic spiral form. An integral equation is obtained for the surface density perturbation, which is seen to also reduce to the standard WKB dispersion relation. Although our formulation is general and applies to all discs, we present our analysis only for nearly Keplerian, low-mass, self-gravitating discs revolving around massive central objects, and derive an integral equation governing the slow precessional modes of such discs. For a prograde disc, the integral kernel turns out be real and symmetric, implying that all slow modes are stable. We apply the slow mode integral equation to two unperturbed disc profiles, the Jalali-Tremaine annular discs, and the Kuzmin disc. We determine eigenvalues and eigenfunctions for both m = 1 and m = 2 slow modes for these profiles and discuss their properties. Our results compare well with those of Jalali-Tremaine.
Lattice Boltzmann simulations of a viscoelastic shear-thinning fluid.
Papenkort, S; Voigtmann, Th
2015-07-28
We present a hybrid lattice Boltzmann algorithm for the simulation of flow glass-forming fluids, characterized by slow structural relaxation, at the level of the Navier-Stokes equation. The fluid is described in terms of a nonlinear integral constitutive equation, relating the stress tensor locally to the history of flow. As an application, we present results for an integral nonlinear Maxwell model that combines the effects of (linear) viscoelasticity and (nonlinear) shear thinning. We discuss the transient dynamics of velocities, shear stresses, and normal stress differences in planar pressure-driven channel flow, after switching on (startup) and off (cessation) of the driving pressure. This transient dynamics depends nontrivially on the channel width due to an interplay between hydrodynamic momentum diffusion and slow structural relaxation. PMID:26233150
Lattice Boltzmann Method for Spacecraft Propellant Slosh Simulation
NASA Technical Reports Server (NTRS)
Orr, Jeb S.; Powers, Joseph F.; Yang, Hong Q.
2015-01-01
A scalable computational approach to the simulation of propellant tank sloshing dynamics in microgravity is presented. In this work, we use the lattice Boltzmann equation (LBE) to approximate the behavior of two-phase, single-component isothermal flows at very low Bond numbers. Through the use of a non-ideal gas equation of state and a modified multiple relaxation time (MRT) collision operator, the proposed method can simulate thermodynamically consistent phase transitions at temperatures and density ratios consistent with typical spacecraft cryogenic propellants, for example, liquid oxygen. Determination of the tank forces and moments relies upon the global momentum conservation of the fluid domain, and a parametric wall wetting model allows tuning of the free surface contact angle. Development of the interface is implicit and no interface tracking approach is required. Numerical examples illustrate the method's application to predicting bulk fluid motion including lateral propellant slosh in low-g conditions.
Lattice Boltzmann Method for Spacecraft Propellant Slosh Simulation
NASA Technical Reports Server (NTRS)
Orr, Jeb S.; Powers, Joseph F.; Yang, Hong Q
2015-01-01
A scalable computational approach to the simulation of propellant tank sloshing dynamics in microgravity is presented. In this work, we use the lattice Boltzmann equation (LBE) to approximate the behavior of two-phase, single-component isothermal flows at very low Bond numbers. Through the use of a non-ideal gas equation of state and a modified multiple relaxation time (MRT) collision operator, the proposed method can simulate thermodynamically consistent phase transitions at temperatures and density ratios consistent with typical spacecraft cryogenic propellants, for example, liquid oxygen. Determination of the tank forces and moments is based upon a novel approach that relies on the global momentum conservation of the closed fluid domain, and a parametric wall wetting model allows tuning of the free surface contact angle. Development of the interface is implicit and no interface tracking approach is required. A numerical example illustrates the method's application to prediction of bulk fluid behavior during a spacecraft ullage settling maneuver.
Progress in developing Poisson-Boltzmann equation solvers
Li, Chuan; Li, Lin; Petukh, Marharyta; Alexov, Emil
2013-01-01
This review outlines the recent progress made in developing more accurate and efficient solutions to model electrostatics in systems comprised of bio-macromolecules and nano-objects, the last one referring to objects that do not have biological function themselves but nowadays are frequently used in biophysical and medical approaches in conjunction with bio-macromolecules. The problem of modeling macromolecular electrostatics is reviewed from two different angles: as a mathematical task provided the specific definition of the system to be modeled and as a physical problem aiming to better capture the phenomena occurring in the real experiments. In addition, specific attention is paid to methods to extend the capabilities of the existing solvers to model large systems toward applications of calculations of the electrostatic potential and energies in molecular motors, mitochondria complex, photosynthetic machinery and systems involving large nano-objects. PMID:24199185
Global existence of solutions for a model Boltzmann equation
NASA Astrophysics Data System (ADS)
Cercignani, C.
1987-12-01
A model recently introduced by Ianiro and Lebowitz is shown to have a global solution for initial data having a finite H-functional and belonging to L {/υ 1}( L {/x ∞}). Methods previously introduced by Tartar to deal with discrete velocity models are used.
Global existence of solutions for a model Boltzmann equation
Cercignani, C.
1987-12-01
A model recently introduced by Ianiro and Lebowitz is shown to have a global solution for initial data having a finite H-functional and belonging to L/sub v//sup 1/ (L/sub x//sup infinity/). Methods previously introduced by Tartar to deal with discrete velocity models are used.
The ion polytropic coefficient in a collisionless sheath containing hot ions
NASA Astrophysics Data System (ADS)
Lin, Binbin; Xiang, Nong; Ou, Jing
2016-08-01
The fluid approach has been widely used to study plasma sheath dynamics. For a sheath containing hot ions whose temperature is greater than the electron's, how to truncate the fluid hierarchy chain equations while retaining to the fullest extent of the kinetic effects is always a difficult problem. In this paper, a one-dimensional, collisionless sheath containing hot ions is studied via particle-in-cell simulations. By analyzing the ion energy equation and taking the kinetic effects into account, we have shown that the ion polytropic coefficient in the vicinity of the sheath edge is approximately constant so that the state equation with the modified polytropic coefficient can be used to close the hierarchy chain of the ion fluid equations. The value of the polytropic coefficient strongly depends on the hot ion temperature and its concentration in the plasma. The semi-analytical model is given to interpret the simulation results. As an application, the kinetic effects on the ion saturation current density in the probe theory are discussed.
Collisionless shock experiments with lasers and observation of Weibel instabilitiesa)
NASA Astrophysics Data System (ADS)
Park, H.-S.; Huntington, C. M.; Fiuza, F.; Drake, R. P.; Froula, D. H.; Gregori, G.; Koenig, M.; Kugland, N. L.; Kuranz, C. C.; Lamb, D. Q.; Levy, M. C.; Li, C. K.; Meinecke, J.; Morita, T.; Petrasso, R. D.; Pollock, B. B.; Remington, B. A.; Rinderknecht, H. G.; Rosenberg, M.; Ross, J. S.; Ryutov, D. D.; Sakawa, Y.; Spitkovsky, A.; Takabe, H.; Turnbull, D. P.; Tzeferacos, P.; Weber, S. V.; Zylstra, A. B.
2015-05-01
Astrophysical collisionless shocks are common in the universe, occurring in supernova remnants, gamma ray bursts, and protostellar jets. They appear in colliding plasma flows when the mean free path for ion-ion collisions is much larger than the system size. It is believed that such shocks could be mediated via the electromagnetic Weibel instability in astrophysical environments without pre-existing magnetic fields. Here, we present laboratory experiments using high-power lasers and investigate the dynamics of high-Mach-number collisionless shock formation in two interpenetrating plasma streams. Our recent proton-probe experiments on Omega show the characteristic filamentary structures of the Weibel instability that are electromagnetic in nature with an inferred magnetization level as high as ˜1% [C. M. Huntington et al., "Observation of magnetic field generation via the weibel instability in interpenetrating plasma flows," Nat. Phys. 11, 173-176 (2015)]. These results imply that electromagnetic instabilities are significant in the interaction of astrophysical conditions.
Collisionless shock experiments with lasers and observation of Weibel instabilities
Park, H. -S.; Huntington, C. M.; Fiuza, F.; Drake, R. P.; Froula, D. H.; Gregori, G.; Koenig, M.; Kugland, N. L.; Kuranz, C. C.; Lamb, D. Q.; Levy, M. C.; Li, C. K.; Meinecke, J.; Morita, T.; Petrasso, R. D.; Pollock, B. B.; Remington, B. A.; Rinderknecht, H. G.; Rosenberg, M.; Ross, J. S.; Ryutov, D. D.; Sakawa, Y.; Spitkovsky, A.; Takabe, H.; Turnbull, D. P.; Tzeferacos, P.; Weber, S. V.; Zylstra, A. B.
2015-05-13
Astrophysical collisionless shocks are common in the universe, occurring in supernova remnants, gamma ray bursts, and protostellar jets. They appear in colliding plasma flows when the mean free path for ion-ion collisions is much larger than the system size. It is believed that such shocks could be mediated via the electromagnetic Weibel instability in astrophysical environments without preexisting magnetic fields. Here, we present laboratory experiments using high-power lasers and investigate the dynamics of high-Mach-number collisionless shock formation in two interpenetrating plasma streams. Our recent proton-probe experiments on Omega show the characteristic filamentary structures of the Weibel instability that are electromagnetic in nature with an inferred magnetization level as high as ~1% These results imply that electromagnetic instabilities are significant in the interaction of astrophysical conditions.
Exploring the nature of collisionless shocks under laboratory conditions
Stockem, A.; Fiuza, F.; Bret, A.; Fonseca, R. A.; Silva, L. O.
2014-01-01
Collisionless shocks are pervasive in astrophysics and they are critical to understand cosmic ray acceleration. Laboratory experiments with intense lasers are now opening the way to explore and characterise the underlying microphysics, which determine the acceleration process of collisionless shocks. We determine the shock character – electrostatic or electromagnetic – based on the stability of electrostatic shocks to transverse electromagnetic fluctuations as a function of the electron temperature and flow velocity of the plasma components, and we compare the analytical model with particle-in-cell simulations. By making the connection with the laser parameters driving the plasma flows, we demonstrate that shocks with different and distinct underlying microphysics can be explored in the laboratory with state-of-the-art laser systems. PMID:24488212
The role of microturbulence on collisionless reconnection. [in magnetospheric plasmas
NASA Technical Reports Server (NTRS)
Papadopoulos, K.
1980-01-01
The linear, non-linear and anomalous transport properties associated with various microinstabilities driven by cross field currents in reconnecting geometries are reviewed. An assessment of their role in collisionless tearing based on analytic theory, computer simulations and experimental evidence, supports the dominant role of lower hybrid waves. The relevance of microturbulence on macroscopic stationary and time dependent models of merging is presented. It is concluded that a fluid-numerical simulation approach that includes (at each space and time step) the effects of anomalous transport in a self consistent manner, similar to the one used for laboratory collisionless shocks, represents the best method for studying and modeling the details of the reconnection process.
Evolution of velocity dispersion along cold collisionless flows
NASA Astrophysics Data System (ADS)
Banik, Nilanjan; Sikivie, Pierre
2016-05-01
The infall of cold dark matter onto a galaxy produces cold collisionless flows and caustics in its halo. If a signal is found in the cavity detector of dark matter axions, the flows will be readily apparent as peaks in the energy spectrum of photons from axion conversion, allowing the densities, velocity vectors and velocity dispersions of the flows to be determined. We discuss the evolution of velocity dispersion along cold collisionless flows in one and two dimensions. A technique is presented for obtaining the leading behavior of the velocity dispersion near caustics. The results are used to derive an upper limit on the energy dispersion of the big flow from the sharpness of its nearby caustic and a prediction for the dispersions in its velocity components.
Collisionless shock formation, spontaneous electromagnetic fluctuations, and streaming instabilities
Bret, A.; Stockem, A.; Fiuza, F.; Silva, L. O.; Narayan, R.
2013-04-15
Collisionless shocks are ubiquitous in astrophysics and in the lab. Recent numerical simulations and experiments have shown how they can arise from the encounter of two collisionless plasma shells. When the shells interpenetrate, the overlapping region turns unstable, triggering the shock formation. As a first step towards a microscopic understanding of the process, we analyze here in detail the initial instability phase. On the one hand, 2D relativistic Particle-In-Cell simulations are performed where two symmetric initially cold pair plasmas collide. On the other hand, the instabilities at work are analyzed, as well as the field at saturation and the seed field which gets amplified. For mildly relativistic motions and onward, Weibel modes govern the linear phase. We derive an expression for the duration of the linear phase in good agreement with the simulations. This saturation time constitutes indeed a lower-bound for the shock formation time.
Collisionless shock experiments with lasers and observation of Weibel instabilities
Park, H. -S.; Huntington, C. M.; Fiuza, F.; Drake, R. P.; Froula, D. H.; Gregori, G.; Koenig, M.; Kugland, N. L.; Kuranz, C. C.; Lamb, D. Q.; et al
2015-05-13
Astrophysical collisionless shocks are common in the universe, occurring in supernova remnants, gamma ray bursts, and protostellar jets. They appear in colliding plasma flows when the mean free path for ion-ion collisions is much larger than the system size. It is believed that such shocks could be mediated via the electromagnetic Weibel instability in astrophysical environments without preexisting magnetic fields. Here, we present laboratory experiments using high-power lasers and investigate the dynamics of high-Mach-number collisionless shock formation in two interpenetrating plasma streams. Our recent proton-probe experiments on Omega show the characteristic filamentary structures of the Weibel instability that are electromagneticmore » in nature with an inferred magnetization level as high as ~1% These results imply that electromagnetic instabilities are significant in the interaction of astrophysical conditions.« less
Evolution of velocity dispersion along cold collisionless flows
Banik, Nilanjan; Sikivie, Pierre
2016-05-01
We found that the infall of cold dark matter onto a galaxy produces cold collisionless flows and caustics in its halo. If a signal is found in the cavity detector of dark matter axions, the flows will be readily apparent as peaks in the energy spectrum of photons from axion conversion, allowing the densities, velocity vectors and velocity dispersions of the flows to be determined. We also discuss the evolution of velocity dispersion along cold collisionless flows in one and two dimensions. A technique is presented for obtaining the leading behaviour of the velocity dispersion near caustics. The results aremore » used to derive an upper limit on the energy dispersion of the Big Flow from the sharpness of its nearby caustic, and a prediction for the dispersions in its velocity components.« less
Multiple eigenmodes of geodesic acoustic mode in collisionless plasmas
Gao Zhe; Itoh, K.; Sanuki, H.; Dong, J. Q.
2006-10-15
We report a series of eigenmodes of the geodesic acoustic mode (GAM), which includes the standard GAM, a branch of low-frequency mode, and a series of ion sound wave-like modes. The case of T{sub i}>>T{sub e} is investigated, and eigenfrequencies of these modes are obtained analytically from a linear gyrokinetic model in collisionless plasmas with a rigid constant electrostatic potential around a magnetic surface.
Collisionless Reconnection in an Electron-Positron Plasma
Bessho, N.; Bhattacharjee, A.
2005-12-09
Electromagnetic particle-in-cell simulations of fast collisionless reconnection in a two-dimensional electron-positron plasma (without an equilibrium guide field) are presented. A generalized Ohm's law in which the Hall current cancels out exactly is given. It is suggested that the key to fast reconnection in this plasma is the localization caused by the off-diagonal components of the pressure tensors, which produce an effect analogous to a spatially localized resistivity.
How to Patch Active Plasma and Collisionless Sheath: Practical Guide
Kaganovich, Igor D.
2002-08-22
Most plasmas have a very thin sheath compared with the plasma dimension. This necessitates separate calculations of the plasma and sheath. The Bohm criterion provides the boundary condition for calculation of plasma profiles. To calculate sheath properties, a value of electric field at the plasma-sheath interface has to be specified in addition to the Bohm criterion. The value of the boundary electric field and robust procedure to approximately patch plasma and collisionless sheath with a very good accuracy are reported.
Vlasov simulations of collisionless magnetic reconnection without background density
NASA Astrophysics Data System (ADS)
Schmitz, H.; Grauer, R.
2008-02-01
A standard starting point for the simulation of collisionless reconnection is the Harris equilibrium which is made up of a current sheet that separates two regions of opposing magnetic field. Magnetohydrodynamic simulations of collisionless reconnection usually include a homogeneous background density for reasons of numerical stability. While, in some cases, this is a realistic assumption, the background density may introduce new effects both due to the more involved structure of the distribution function or due to the fact that the Alfvèn speed remains finite far away from the current sheet. We present a fully kinetic Vlasov simulation of the perturbed Harris equilibrium using a Vlasov code. Parameters are chosen to match the Geospace Environment Modeling (GEM) Magnetic Reconnection Challenge but excluding the background density. This allows to compare with earlier simulations [Schmitz H, Grauer R. Kinetic Vlasov simulations of collisionless magnetic reconnection. Phys Plasmas 2006;13:092309] which include the background density. It is found that the absence of a background density causes the reconnection rate to be higher. On the other hand, the time until the onset of reconnection is hardly affected. Again the off diagonal elements of the pressure tensor are found to be important on the X-line but with modified importance for the individual terms.
Accuracy of non-Newtonian Lattice Boltzmann simulations
NASA Astrophysics Data System (ADS)
Conrad, Daniel; Schneider, Andreas; Böhle, Martin
2015-11-01
This work deals with the accuracy of non-Newtonian Lattice Boltzmann simulations. Previous work for Newtonian fluids indicate that, depending on the numerical value of the dimensionless collision frequency Ω, additional artificial viscosity is introduced, which negatively influences the accuracy. Since the non-Newtonian fluid behavior is incorporated through appropriate modeling of the dimensionless collision frequency, a Ω dependent error EΩ is introduced and its influence on the overall error is investigated. Here, simulations with the SRT and the MRT model are carried out for power-law fluids in order to numerically investigate the accuracy of non-Newtonian Lattice Boltzmann simulations. A goal of this accuracy analysis is to derive a recommendation for an optimal choice of the time step size and the simulation Mach number, respectively. For the non-Newtonian case, an error estimate for EΩ in the form of a functional is derived on the basis of a series expansion of the Lattice Boltzmann equation. This functional can be solved analytically for the case of the Hagen-Poiseuille channel flow of non-Newtonian fluids. With the help of the error functional, the prediction of the global error minimum of the velocity field is excellent in regions where the EΩ error is the dominant source of error. With an optimal simulation Mach number, the simulation is about one order of magnitude more accurate. Additionally, for both collision models a detailed study of the convergence behavior of the method in the non-Newtonian case is conducted. The results show that the simulation Mach number has a major impact on the convergence rate and second order accuracy is not preserved for every choice of the simulation Mach number.
NASA Astrophysics Data System (ADS)
Wang, Y.; Shu, C.; Huang, H. B.; Teo, C. J.
2015-01-01
A multiphase lattice Boltzmann flux solver (MLBFS) is proposed in this paper for incompressible multiphase flows with low- and large-density-ratios. In the solver, the flow variables at cell centers are given from the solution of macroscopic governing differential equations (Navier-Stokes equations recovered by multiphase lattice Boltzmann (LB) model) by the finite volume method. At each cell interface, the viscous and inviscid fluxes are evaluated simultaneously by local reconstruction of solution for the standard lattice Boltzmann equation (LBE). The forcing terms in the governing equations are directly treated by the finite volume discretization. The phase interfaces are captured by solving the phase-field Cahn-Hilliard equation with a fifth order upwind scheme. Unlike the conventional multiphase LB models, which restrict their applications on uniform grids with fixed time step, the MLBFS has the capability and advantage to simulate multiphase flows on non-uniform grids. The proposed solver is validated by several benchmark problems, such as two-phase co-current flow, Taylor-Couette flow in an annulus, Rayleigh-Taylor instability, and droplet splashing on a thin film at density ratio of 1000 with Reynolds numbers ranging from 20 to 1000. Numerical results show the reliability of the proposed solver for multiphase flows with high density ratio and high Reynolds number.
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.
Revised lattice Boltzmann model for traffic flow with equilibrium traffic pressure
NASA Astrophysics Data System (ADS)
Shi, Wei; Lu, Wei-Zhen; Xue, Yu; He, Hong-Di
2016-02-01
A revised lattice Boltzmann model concerning the equilibrium traffic pressure is proposed in this study to tackle the phase transition phenomena of traffic flow system. The traditional lattice Boltzmann model has limitation to investigate the complex traffic phase transitions due to its difficulty for modeling the equilibrium velocity distribution. Concerning this drawback, the equilibrium traffic pressure is taken into account to derive the equilibrium velocity distribution in the revised lattice Boltzmann model. In the proposed model, a three-dimensional velocity-space is assumed to determine the equilibrium velocity distribution functions and an alternative, new derivative approach is introduced to deduct the macroscopic equations with the first-order accuracy level from the lattice Boltzmann model. Based on the linear stability theory, the stability conditions of the corresponding macroscopic equations can be obtained. The outputs indicate that the stability curve is divided into three regions, i.e., the stable region, the neutral stability region, and the unstable region. In the stable region, small disturbance appears in the initial uniform flow and will vanish after long term evolution, while in the unstable region, the disturbance will be enlarged and finally leads to the traffic system entering the congested state. In the neutral stability region, small disturbance does not vanish with time and maintains its amplitude in the traffic system. Conclusively, the stability of traffic system is found to be enhanced as the equilibrium traffic pressure increases. Finally, the numerical outputs of the proposed model are found to be consistent with the recognized, theoretical results.
Electron precipitation in solar flares - Collisionless effects
NASA Technical Reports Server (NTRS)
Vlahos, L.; Rowland, H. L.
1984-01-01
A large fraction of the electrons which are accelerated during the impulsive phase of solar flares stream towards the chromosphere and are unstable to the growth of plasma waves. The linear and nonlinear evolution of plasma waves as a function of time is analyzed with a set of rate equations that follows, in time, the nonlinearly coupled system of plasma waves-ion fluctuations. As an outcome of the fast transfer of wave energy from the beam to the ambient plasma, nonthermal electron tails are formed which can stabilize the anomalous Doppler resonance instability responsible for the pitch angle scattering of the beam electrons. The non-collisional losses of the precipitating electrons are estimated, and the observational implication of these results are discussed.
Lattice Boltzmann Method for 3-D Flows with Curved Boundary
NASA Technical Reports Server (NTRS)
Mei, Renwei; Shyy, Wei; Yu, Dazhi; Luo, Li-Shi
2002-01-01
In this work, we investigate two issues that are important to computational efficiency and reliability in fluid dynamics applications of the lattice, Boltzmann equation (LBE): (1) Computational stability and accuracy of different lattice Boltzmann models and (2) the treatment of the boundary conditions on curved solid boundaries and their 3-D implementations. Three athermal 3-D LBE models (D3QI5, D3Ql9, and D3Q27) are studied and compared in terms of efficiency, accuracy, and robustness. The boundary treatment recently developed by Filippova and Hanel and Met et al. in 2-D is extended to and implemented for 3-D. The convergence, stability, and computational efficiency of the 3-D LBE models with the boundary treatment for curved boundaries were tested in simulations of four 3-D flows: (1) Fully developed flows in a square duct, (2) flow in a 3-D lid-driven cavity, (3) fully developed flows in a circular pipe, and (4) a uniform flow over a sphere. We found that while the fifteen-velocity 3-D (D3Ql5) model is more prone to numerical instability and the D3Q27 is more computationally intensive, the 63Q19 model provides a balance between computational reliability and efficiency. Through numerical simulations, we demonstrated that the boundary treatment for 3-D arbitrary curved geometry has second-order accuracy and possesses satisfactory stability characteristics.
Polyelectrolyte Microcapsules: Ion Distributions from a Poisson-Boltzmann Model
NASA Astrophysics Data System (ADS)
Tang, Qiyun; Denton, Alan R.; Rozairo, Damith; Croll, Andrew B.
2014-03-01
Recent experiments have shown that polystyrene-polyacrylic-acid-polystyrene (PS-PAA-PS) triblock copolymers in a solvent mixture of water and toluene can self-assemble into spherical microcapsules. Suspended in water, the microcapsules have a toluene core surrounded by an elastomer triblock shell. The longer, hydrophilic PAA blocks remain near the outer surface of the shell, becoming charged through dissociation of OH functional groups in water, while the shorter, hydrophobic PS blocks form a networked (glass or gel) structure. Within a mean-field Poisson-Boltzmann theory, we model these polyelectrolyte microcapsules as spherical charged shells, assuming different dielectric constants inside and outside the capsule. By numerically solving the nonlinear Poisson-Boltzmann equation, we calculate the radial distribution of anions and cations and the osmotic pressure within the shell as a function of salt concentration. Our predictions, which can be tested by comparison with experiments, may guide the design of microcapsules for practical applications, such as drug delivery. This work was supported by the National Science Foundation under Grant No. DMR-1106331.
Fast lattice Boltzmann solver for relativistic hydrodynamics.
Mendoza, M; Boghosian, B M; Herrmann, H J; Succi, S
2010-07-01
A lattice Boltzmann formulation for relativistic fluids is presented and numerically validated through quantitative comparison with recent hydrodynamic simulations of relativistic fluids. In order to illustrate its capability to handle complex geometries, the scheme is also applied to the case of a three-dimensional relativistic shock wave, generated by a supernova explosion, impacting on a massive interstellar cloud. This formulation opens up the possibility of exporting the proven advantages of lattice Boltzmann methods, namely, computational efficiency and easy handling of complex geometries, to the context of (mildly) relativistic fluid dynamics at large, from quark-gluon plasmas up to supernovae with relativistic outflows.
Entropic lattice Boltzmann model for compressible flows.
Frapolli, N; Chikatamarla, S S; Karlin, I V
2015-12-01
We present a lattice Boltzmann model (LBM) that covers the entire range of fluid flows, from low Mach weakly compressible to transonic and supersonic flows. One of the most restrictive limitations of the lattice Boltzmann method, the low Mach number limit, is overcome here by three fundamental changes to the LBM scheme: use of an appropriately chosen multispeed lattice, accurate evaluation of the equilibrium, and the entropic relaxation for the collision. The range of applications is demonstrated through the simulation of a bow shock in front of an airfoil and the simulation of decaying compressible turbulence with shocklets.
Fast lattice Boltzmann solver for relativistic hydrodynamics.
Mendoza, M; Boghosian, B M; Herrmann, H J; Succi, S
2010-07-01
A lattice Boltzmann formulation for relativistic fluids is presented and numerically validated through quantitative comparison with recent hydrodynamic simulations of relativistic fluids. In order to illustrate its capability to handle complex geometries, the scheme is also applied to the case of a three-dimensional relativistic shock wave, generated by a supernova explosion, impacting on a massive interstellar cloud. This formulation opens up the possibility of exporting the proven advantages of lattice Boltzmann methods, namely, computational efficiency and easy handling of complex geometries, to the context of (mildly) relativistic fluid dynamics at large, from quark-gluon plasmas up to supernovae with relativistic outflows. PMID:20867451
Stefan-Boltzmann Law for Massive Photons
NASA Astrophysics Data System (ADS)
Moreira, E. S.; Ribeiro, T. G.
2016-08-01
This paper generalizes the Stefan-Boltzmann law to include massive photons. A crucial ingredient to obtain the correct formula for the radiance is to realize that a massive photon does not travel at the speed of (massless) light. It follows that, contrary to what could be expected, the radiance is not proportional to the energy density times the speed of light.
Full Eulerian lattice Boltzmann model for conjugate heat transfer
NASA Astrophysics Data System (ADS)
Hu, Yang; Li, Decai; Shu, Shi; Niu, Xiaodong
2015-12-01
In this paper a full Eulerian lattice Boltzmann model is proposed for conjugate heat transfer. A unified governing equation with a source term for the temperature field is derived. By introducing the source term, we prove that the continuity of temperature and its normal flux at the interface is satisfied automatically. The curved interface is assumed to be zigzag lines. All physical quantities are recorded and updated on a Cartesian grid. As a result, any complicated treatment near the interface is avoided, which makes the proposed model suitable to simulate the conjugate heat transfer with complex interfaces efficiently. The present conjugate interface treatment is validated by several steady and unsteady numerical tests, including pure heat conduction, forced convection, and natural convection problems. Both flat and curved interfaces are also involved. The obtained results show good agreement with the analytical and/or finite volume results.
Finite-difference lattice-Boltzmann methods for binary fluids.
Xu, Aiguo
2005-06-01
We investigate two-fluid Bhatnagar-Gross-Krook (BGK) kinetic methods for binary fluids. The developed theory works for asymmetric as well as symmetric systems. For symmetric systems it recovers Sirovich's theory and is summarized in models A and B. For asymmetric systems it contributes models C, D, and E which are especially useful when the total masses and/or local temperatures of the two components are greatly different. The kinetic models are discretized based on an octagonal discrete velocity model. The discrete-velocity kinetic models and the continuous ones are required to describe the same hydrodynamic equations. The combination of a discrete-velocity kinetic model and an appropriate finite-difference scheme composes a finite-difference lattice Boltzmann method. The validity of the formulated methods is verified by investigating (i) uniform relaxation processes, (ii) isothermal Couette flow, and (iii) diffusion behavior. PMID:16089910
Full Eulerian lattice Boltzmann model for conjugate heat transfer.
Hu, Yang; Li, Decai; Shu, Shi; Niu, Xiaodong
2015-12-01
In this paper a full Eulerian lattice Boltzmann model is proposed for conjugate heat transfer. A unified governing equation with a source term for the temperature field is derived. By introducing the source term, we prove that the continuity of temperature and its normal flux at the interface is satisfied automatically. The curved interface is assumed to be zigzag lines. All physical quantities are recorded and updated on a Cartesian grid. As a result, any complicated treatment near the interface is avoided, which makes the proposed model suitable to simulate the conjugate heat transfer with complex interfaces efficiently. The present conjugate interface treatment is validated by several steady and unsteady numerical tests, including pure heat conduction, forced convection, and natural convection problems. Both flat and curved interfaces are also involved. The obtained results show good agreement with the analytical and/or finite volume results. PMID:26764851
Lattice Boltzmann method for mixtures at variable Schmidt number.
Monteferrante, Michele; Melchionna, Simone; Marconi, Umberto Marini Bettolo
2014-07-01
When simulating multicomponent mixtures via the Lattice Boltzmann Method, it is desirable to control the mutual diffusivity between species while maintaining the viscosity of the solution fixed. This goal is herein achieved by a modification of the multicomponent Bhatnagar-Gross-Krook evolution equations by introducing two different timescales for mass and momentum diffusion. Diffusivity is thus controlled by an effective drag force acting between species. Numerical simulations confirm the accuracy of the method for neutral binary and charged ternary mixtures in bulk conditions. The simulation of a charged mixture in a charged slit channel show that the conductivity and electro-osmotic mobility exhibit a departure from the Helmholtz-Smoluchowski prediction at high diffusivity.
Lattice Boltzmann method for mixtures at variable Schmidt number.
Monteferrante, Michele; Melchionna, Simone; Marconi, Umberto Marini Bettolo
2014-07-01
When simulating multicomponent mixtures via the Lattice Boltzmann Method, it is desirable to control the mutual diffusivity between species while maintaining the viscosity of the solution fixed. This goal is herein achieved by a modification of the multicomponent Bhatnagar-Gross-Krook evolution equations by introducing two different timescales for mass and momentum diffusion. Diffusivity is thus controlled by an effective drag force acting between species. Numerical simulations confirm the accuracy of the method for neutral binary and charged ternary mixtures in bulk conditions. The simulation of a charged mixture in a charged slit channel show that the conductivity and electro-osmotic mobility exhibit a departure from the Helmholtz-Smoluchowski prediction at high diffusivity. PMID:25005272
A Boltzmann treatment for the vorton excess problem
Peter, Patrick; Ringeval, Christophe E-mail: christophe.ringeval@uclouvain.be
2013-05-01
We derive and solve a Boltzmann equation governing the cosmological evolution of the number density of current carrying cosmic string loops, whose centrifugally supported equilibrium configurations are also referred to as vortons. The phase space is three-dimensional and consists of the time variable, the loop size, and a conserved quantum number. Our approach includes gravitational wave emission, a possibly finite lifetime for the vortons and works with any initial loop distribution and for any loop production function. We then show how our results generalize previous approaches on the vorton excess problem by tracking down the time evolution of the various sub-populations of current-carrying loops in a string network.
Beyond Poisson–Boltzmann: fluctuations and fluid structure in a self-consistent theory
NASA Astrophysics Data System (ADS)
Buyukdagli, S.; Blossey, R.
2016-09-01
Poisson–Boltzmann (PB) theory is the classic approach to soft matter electrostatics and has been applied to numerous physical chemistry and biophysics problems. Its essential limitations are in its neglect of correlation effects and fluid structure. Recently, several theoretical insights have allowed the formulation of approaches that go beyond PB theory in a systematic way. In this topical review, we provide an update on the developments achieved in the self-consistent formulations of correlation-corrected Poisson–Boltzmann theory. We introduce a corresponding system of coupled non-linear equations for both continuum electrostatics with a uniform dielectric constant, and a structured solvent—a dipolar Coulomb fluid—including non-local effects. While the approach is only approximate and also limited to corrections in the so-called weak fluctuation regime, it allows us to include physically relevant effects, as we show for a range of applications of these equations.
Lattice Boltzmann method for bosons and fermions and the fourth-order Hermite polynomial expansion.
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.
Beyond Poisson-Boltzmann: fluctuations and fluid structure in a self-consistent theory.
Buyukdagli, S; Blossey, R
2016-09-01
Poisson-Boltzmann (PB) theory is the classic approach to soft matter electrostatics and has been applied to numerous physical chemistry and biophysics problems. Its essential limitations are in its neglect of correlation effects and fluid structure. Recently, several theoretical insights have allowed the formulation of approaches that go beyond PB theory in a systematic way. In this topical review, we provide an update on the developments achieved in the self-consistent formulations of correlation-corrected Poisson-Boltzmann theory. We introduce a corresponding system of coupled non-linear equations for both continuum electrostatics with a uniform dielectric constant, and a structured solvent-a dipolar Coulomb fluid-including non-local effects. While the approach is only approximate and also limited to corrections in the so-called weak fluctuation regime, it allows us to include physically relevant effects, as we show for a range of applications of these equations. PMID:27357125
Lattice Boltzmann method for bosons and fermions and the fourth-order Hermite polynomial expansion.
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. PMID:24827360
Kinetic theory of correlated fluids: from dynamic density functional to Lattice Boltzmann methods.
Marconi, Umberto Marini Bettolo; Melchionna, Simone
2009-07-01
Using methods of kinetic theory and liquid state theory we propose a description of the nonequilibrium behavior of molecular fluids, which takes into account their microscopic structure and thermodynamic properties. The present work represents an alternative to the recent dynamic density functional theory, which can only deal with colloidal fluids and is not apt to describe the hydrodynamic behavior of a molecular fluid. The method is based on a suitable modification of the Boltzmann transport equation for the phase space distribution and provides a detailed description of the local structure of the fluid and its transport coefficients. Finally, we propose a practical scheme to solve numerically and efficiently the resulting kinetic equation by employing a discretization procedure analogous to the one used in the Lattice Boltzmann method.
A new fast reconnection model in a collisionless regime
Tsiklauri, David
2008-11-15
Based on the first principles [i.e., (i) by balancing the magnetic field advection with the term containing electron pressure tensor nongyrotropic components in the generalized Ohm's law; (ii) using the conservation of mass; and (iii) assuming that the weak magnetic field region width, where electron meandering motion supports electron pressure tensor off-diagonal (nongyrotropic) components, is of the order of electron Larmor radius] a simple model of magnetic reconnection in a collisionless regime is formulated. The model is general, resembling its collisional Sweet-Parker analog in that it is not specific to any initial configuration, e.g., Harris-type tearing unstable current sheet, X-point collapse or otherwise. In addition to its importance from the fundamental point of view, the collisionless reconnection model offers a much faster reconnection rate [M{sub c{sup '}}{sub less}=(c/{omega}{sub pe}){sup 2}/(r{sub L,e}L)] than Sweet-Parker's classical one (M{sub sp}=S{sup -1/2}). The width of the diffusion region (current sheet) in the collisionless regime is found to be {delta}{sub c{sup '}}{sub less}=(c/{omega}{sub pe}){sup 2}/r{sub L,e}, which is independent of the global reconnection scale L and is only prescribed by microphysics (electron inertial length, c/{omega}{sub pe}, and electron Larmor radius, r{sub L,e}). Amongst other issues, the fastness of the reconnection rate alleviates, e.g., the problem of interpretation of solar flares by means of reconnection, as for the typical solar coronal parameters the obtained collisionless reconnection time can be a few minutes, as opposed to Sweet-Parker's equivalent value of less than a day. The new theoretical reconnection rate is compared to the Magnetic Reconnection Experiment device experimental data by Yamada et al. [Phys. Plasmas 13, 052119 (2006)] and Ji et al. [Geophys. Res. Lett. 35, 13106 (2008)], and a good agreement is obtained.
A mean field Ohm's law for collisionless plasmas
Biglari, H. ); Diamond, P.H. )
1993-11-01
A mean field Ohm's law valid for collisionless plasmas is derived kinetically. It is shown that contrary to conventional thinking, the resulting hyperresistivity is significantly smaller than its fluid counterpart due to the fact that the turbulent decorrelation rate is linked to the rapid electron ballistic motion rather than the slower nonlinear mixing time. Moreover, the off-diagonal contributions to the parallel electron momentum flux are shown to result in Ohm's law renormalizations that dwarf the current diffusivity and break radial parity symmetry.
Neoclassical Transport Caused by Collisionless Scattering across an Asymmetric Separatrix
Dubin, Daniel H. E.; Driscoll, C. F.; Tsidulko, Yu. A.
2010-10-29
Plasma loss due to apparatus asymmetries is a ubiquitous phenomenon in magnetic plasma confinement. When the plasma equilibrium has locally trapped particle populations partitioned by a separatrix from one another and from passing particles, the asymmetry transport is enhanced. The trapped and passing particle populations react differently to the asymmetries, leading to the standard 1/{nu} and {radical}({nu}) transport regimes of superbanana orbit theory as particles collisionally scatter from one orbit type to another. However, when the separatrix is itself asymmetric, particles can collisionlessly transit from trapped to passing and back, leading to enhanced transport.
Reconnection properties in collisionless plasma with open boundary conditions
Sun, H. E.; Ma, Z. W.; Huang, J.
2014-07-15
Collisionless magnetic reconnection in a Harris current sheet with different initial thicknesses is investigated using a 21/2 -D Darwin particle-in-cell simulation with the magnetosonic open boundary condition. It is found that the thicknesses of the ion dissipation region and the reconnection current sheet, when the reconnection rate E{sub r} reaches its first peak, are independent of the initial thickness of the current sheet; while the peak reconnection rate depends on it. The peak reconnection rate increases with decrease of the current sheet thickness as E{sub r}∼a{sup −1/2}, where a is the initial current sheet half-thickness.
Generation of collisionless shock in laser-produced plasmas
NASA Astrophysics Data System (ADS)
Fiuza, Frederico
2015-08-01
Collisionless shocks are ubiquitous in astrophysical environments and are tightly connected with magnetic-field amplification and particle acceleration. The fast progress in high-power laser technology is bringing the study of high Mach number shocks into the realm of laboratory plasmas, where in situ measurements can be made helping us understand the fundamental kinetic processes behind shocks. I will discuss the recent progress in laser-driven shock experiments at state-of-the-art facilities like NIF and Omega and how these results, together with ab initio massively parallel simulations, can impact our understanding of magnetic field amplification and particle acceleration in astrophysical plasmas.
Entropy production rate as a constraint for collisionless fluid closures
Fleurence, E.; Sarazin, Y.; Garbet, X.; Dif-Pradalier, G.; Ghendrih, Ph.; Grandgirard, V.; Ottaviani, M.
2006-11-30
A novel method is proposed to construct collisionless fluid closures accounting for some kinetic properties. The first dropped fluid moment is assumed to be a linear function of the lower order ones. Optimizing the agreement between the fluid and kinetic entropy production rates is used to constrain the coefficients of the linear development. This procedure is applied to a reduced version of the interchange instability. The closure, involving the absolute value of the wave vector, is non-local in real space. In this case, the linear instability thresholds are the same, and the linear growth rates exhibit similar characteristics. Such a method is applicable to other models and classes of instabilities.
NASA Astrophysics Data System (ADS)
Sardar, Sankirtan; Bandyopadhyay, Anup; Das, K. P.
2016-07-01
A three-dimensional KP (Kadomtsev Petviashvili) equation is derived here describing the propagation of weakly nonlinear and weakly dispersive dust ion acoustic wave in a collisionless unmagnetized plasma consisting of warm adiabatic ions, static negatively charged dust grains, nonthermal electrons, and isothermal positrons. When the coefficient of the nonlinear term of the KP-equation vanishes an appropriate modified KP (MKP) equation describing the propagation of dust ion acoustic wave is derived. Again when the coefficient of the nonlinear term of this MKP equation vanishes, a further modified KP equation is derived. Finally, the stability of the solitary wave solutions of the KP and the different modified KP equations are investigated by the small-k perturbation expansion method of Rowlands and Infeld [J. Plasma Phys. 3, 567 (1969); 8, 105 (1972); 10, 293 (1973); 33, 171 (1985); 41, 139 (1989); Sov. Phys. - JETP 38, 494 (1974)] at the lowest order of k, where k is the wave number of a long-wavelength plane-wave perturbation. The solitary wave solutions of the different evolution equations are found to be stable at this order.
Lattice Boltzmann formulation for conjugate heat transfer in heterogeneous media.
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
Lattice Boltzmann formulation for conjugate heat transfer in heterogeneous media
NASA Astrophysics Data System (ADS)
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
Polarizable Atomic Multipole Solutes in a Poisson-Boltzmann Continuum
Schnieders, Michael J.; Baker, Nathan A.; Ren, Pengyu; Ponder, Jay W.
2008-01-01
Modeling the change in the electrostatics of organic molecules upon moving from vacuum into solvent, due to polarization, has long been an interesting problem. In vacuum, experimental values for the dipole moments and polarizabilities of small, rigid molecules are known to high accuracy; however, it has generally been difficult to determine these quantities for a polar molecule in water. A theoretical approach introduced by Onsager used vacuum properties of small molecules, including polarizability, dipole moment and size, to predict experimentally known permittivities of neat liquids via the Poisson equation. Since this important advance in understanding the condensed phase, a large number of computational methods have been developed to study solutes embedded in a continuum via numerical solutions to the Poisson-Boltzmann equation (PBE). Only recently have the classical force fields used for studying biomolecules begun to include explicit polarization in their functional forms. Here we describe the theory underlying a newly developed Polarizable Multipole Poisson-Boltzmann (PMPB) continuum electrostatics model, which builds on the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force field. As an application of the PMPB methodology, results are presented for several small folded proteins studied by molecular dynamics in explicit water as well as embedded in the PMPB continuum. The dipole moment of each protein increased on average by a factor of 1.27 in explicit water and 1.26 in continuum solvent. The essentially identical electrostatic response in both models suggests that PMPB electrostatics offers an efficient alternative to sampling explicit solvent molecules for a variety of interesting applications, including binding energies, conformational analysis, and pKa prediction. Introduction of 150 mM salt lowered the electrostatic solvation energy between 2–13 kcal/mole, depending on the formal charge of the protein, but had only a
Theory of spatially non-symmetric kinetic equilibria for collisionless plasmas
Cremaschini, Claudio; Tessarotto, Massimo
2013-01-15
The problem posed by the possible existence/non-existence of spatially non-symmetric kinetic equilibria has remained unsolved in plasma theory. For collisionless magnetized plasmas, this involves the construction of stationary solutions of the Vlasov-Maxwell equations. In this paper, the issue is addressed for non-relativistic plasmas both in astrophysical and laboratory contexts. The treatment is based on a Lagrangian variational description of single-particle dynamics. Starting point is a non-perturbative formulation of gyrokinetic theory, which allows one to construct 'a posteriori' with prescribed order of accuracy an asymptotic representation for the magnetic moment. In terms of the relevant particle adiabatic invariants generalized bi-Maxwellian equilibria are proved to exist. These are shown to recover, under suitable assumptions, a Chapman-Enskog form which permits an analytical treatment of the corresponding fluid moments. In particular, the constrained posed by the Poisson and the Ampere equations are analyzed, both for quasi-neutral and non-neutral plasmas. The conditions of existence of the corresponding non-symmetric kinetic equilibria are investigated. As a notable feature, both astrophysical and laboratory plasmas are shown to exhibit, under suitable conditions, a kinetic dynamo, whereby the equilibrium magnetic field can be self-generated by the equilibrium plasma currents.
Gyrokinetic δ f simulation of collisionless and semi-collisional tearing mode instabilities
NASA Astrophysics Data System (ADS)
Wan, Weigang; Chen, Yang; Parker, Scott
2004-11-01
The evolution of collisionless and semi-collisional tearing mode instabilities is studied using a three-dimensional particle-in-cell simulation model that utilizes the δ f-method with the split-weight scheme to enhance the time step, and a novel algorithm(Y. Chen and S.E. Parker, J. Comput. Phys. 198), 463 (2003) to accurately solve the Ampere's equation for experimentally relevant β values, βfracm_im_e≫ 1. We use the model of drift-kinetic electrons and gyrokinetic ions. Linear simulation results are benchmarked with eigenmode analysis for the case of fixed ions. In small box simulations the ions response can be neglected but for large box simulations the ions response is important because the width of perturbed current is larger than ρ_i.The nonlinear dynamics of magnetic islands will be studied and the results will be compared with previous theoretical studiesfootnote J.F. Drake and Y. C. Lee, Phys. Rev. Lett. 39, 453 (1977) on the saturation level and the electron bounce frequency. A collision operator is included in the electron drift kinetic equation to study the simulation in the semi-collisional regime. The algebraical growth stage has been observed and compared quantitatively with theory. Our progress on three-dimensional simulations of tearing mode instabilities will be reported.
A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments
NASA Astrophysics Data System (ADS)
Fisicaro, G.; Genovese, L.; Andreussi, O.; Marzari, N.; Goedecker, S.
2016-01-01
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.
Formulation of equations of motion for complex spacecraft
NASA Astrophysics Data System (ADS)
Kane, T. R.; Levinson, D. A.
1980-04-01
Seven methods of the formulation of motion equations for complex spacecraft are discussed: (1) the use of momentum principles, (2) D'Alembert principle, (3) Lagrange equations, (4) Hamilton's canonical equations, (5) the Boltzmann-Hamel equations, (6) the Gibbs equations, and (7) a method introduced by Kane and Wang in 1965. It is shown that the last method considered leads most directly to the simplest equations.
Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions.
Lallemand, Pierre; Luo, Li-Shi
2003-09-01
The focus of the present work is to provide an analysis for the acoustic and thermal properties of the energy-conserving lattice Boltzmann models, and a solution to the numerical defects and instability associated with these models in two and three dimensions. We discover that a spurious algebraic coupling between the shear and energy modes of the linearized evolution operator is a defect universal to the energy-conserving Boltzmann models in two and three dimensions. This spurious mode coupling is highly anisotropic and may occur at small values of wave number k along certain directions, and it is a direct consequence of the following key features of the lattice Boltzmann equation: (1) its simple spatial-temporal dynamics, (2) the linearity of the relaxation modeling for collision operator, and (3) the energy-conservation constraint. To eliminate the spurious mode coupling, we propose a hybrid thermal lattice Boltzmann equation (HTLBE) in which the mass and momentum conservation equations are solved by using the multiple-relaxation-time model due to d'Humières, whereas the diffusion-advection equation for the temperature is solved separately by using finite-difference technique (or other means). Through the Chapman-Enskog analysis we show that the hydrodynamic equations derived from the proposed HTLBE model include the equivalent effect of gamma=C(P)/C(V) in both the speed and attenuation of sound. Appropriate coupling between the energy and velocity field is introduced to attain correct acoustics in the model. The numerical stability of the HTLBE scheme is analyzed by solving the dispersion equation of the linearized collision operator. We find that the numerical stability of the lattice Boltzmann scheme improves drastically once the spurious mode coupling is removed. It is shown that the HTLBE scheme is far superior to the existing thermal LBE schemes in terms of numerical stability, flexibility, and possible generalization for complex fluids. We also present
Collisionless shock experiments with lasers and observation of Weibel instabilities
Park, H.-S. Huntington, C. M.; Fiuza, F.; Levy, M. C.; Pollock, B. B.; Remington, B. A.; Ross, J. S.; Ryutov, D. D.; Turnbull, D. P.; Weber, S. V.; Drake, R. P.; Kuranz, C. C.; Froula, D. H.; Rosenberg, M.; Gregori, G.; Meinecke, J.; Koenig, M.; Kugland, N. L.; Lamb, D. Q.; Tzeferacos, P.; and others
2015-05-15
Astrophysical collisionless shocks are common in the universe, occurring in supernova remnants, gamma ray bursts, and protostellar jets. They appear in colliding plasma flows when the mean free path for ion-ion collisions is much larger than the system size. It is believed that such shocks could be mediated via the electromagnetic Weibel instability in astrophysical environments without pre-existing magnetic fields. Here, we present laboratory experiments using high-power lasers and investigate the dynamics of high-Mach-number collisionless shock formation in two interpenetrating plasma streams. Our recent proton-probe experiments on Omega show the characteristic filamentary structures of the Weibel instability that are electromagnetic in nature with an inferred magnetization level as high as ∼1% [C. M. Huntington et al., “Observation of magnetic field generation via the weibel instability in interpenetrating plasma flows,” Nat. Phys. 11, 173–176 (2015)]. These results imply that electromagnetic instabilities are significant in the interaction of astrophysical conditions.
Laboratory astrophysical collisionless shock experiments on Omega and NIF
NASA Astrophysics Data System (ADS)
Park, Hye-Sook; Ross, J. S.; Huntington, C. M.; Fiuza, F.; Ryutov, D.; Casey, D.; Drake, R. P.; Fiksel, G.; Froula, D.; Gregori, G.; Kugland, N. L.; Kuranz, C.; Levy, M. C.; Li, C. K.; Meinecke, J.; Morita, T.; Petrasso, R.; Plechaty, C.; Remington, B.; Sakawa, Y.; Spitkovsky, A.; Takabe, H.; Zylstra, A. B.
2016-03-01
We are performing scaled astrophysics experiments on Omega and on NIF. Laser driven counter-streaming interpenetrating supersonic plasma flows can be studied to understand astrophysical electromagnetic plasma phenomena in a controlled laboratory setting. In our Omega experiments, the counter-streaming flow plasma state is measured using Thomson scattering diagnostics, demonstrating the plasma flows are indeed super-sonic and in the collisionless regime. We observe a surprising additional electron and ion heating from ion drag force in the double flow experiments that are attributed to the ion drag force and electrostatic instabilities. [1] A proton probe is used to image the electric and magnetic fields. We observe unexpected large, stable and reproducible electromagnetic field structures that arise in the counter-streaming flows [2]. The Biermann battery magnetic field generated near the target plane, advected along the flows, and recompressed near the midplane explains the cause of such self-organizing field structures [3]. A D3He implosion proton probe image showed very clear filamentary structures; three-dimensional Particle-In-Cell simulations and simulated proton radiography images indicate that these filamentary structures are generated by Weibel instabilities and that the magnetization level (ratio of magnetic energy over kinetic energy in the system) is ∼0.01 [4]. These findings have very high astrophysical relevance and significant implications. We expect to observe true collisionless shock formation when we use >100 kJ laser energy on NIF.
NASA Astrophysics Data System (ADS)
Hosseinpour, M.; Mohammadi, M. A.; Biabani, S.; Biabani
2013-10-01
Collisionless magnetic reconnection via tearing instability in non-relativistic electron-positron (pair) plasma with an anisotropic pressure is investigated. The equilibrium magnetic field is considered to be sheared force-free, and a set of linearized collisionless Magnetohydrodynamics equations describes the evolution of reconnection dynamics. A linear analytical analysis, based on scaling, demonstrates that in such a pair plasma, breaking the frozen in flow constraint for field lines can be mainly provided by the non-gyrotropic pressure of electrons and positrons (rather than the particle bulk inertia) when the current sheet width is smaller than the particle Larmor radius (Δx < r L ). This condition is satisfied when β > d 2 (d = c/ω p is the particle skin-depth with the electron/positron frequency ω p and β = 8πP (0)/B 0 2 << 1). Meanwhile, on top of the Lorentz force and in the absence of the reconnection facilitating mechanism of the Hall effect, non-scalar pressure force can accelerate bulk plasma into the diffusion region at the scale lengths of the order of dx. Therefore, the respective regime of tearing instability proceeds much faster compared with the case of an isotropic pressure with a new dimensionless growth rate of (γτ A ) ~ d.
Macroscopic model and truncation error of discrete Boltzmann method
NASA Astrophysics Data System (ADS)
Hwang, Yao-Hsin
2016-10-01
A derivation procedure to secure the macroscopically equivalent equation and its truncation error for discrete Boltzmann method is proffered in this paper. Essential presumptions of two time scales and a small parameter in the Chapman-Enskog expansion are disposed of in the present formulation. Equilibrium particle distribution function instead of its original non-equilibrium form is chosen as key variable in the derivation route. Taylor series expansion encompassing fundamental algebraic manipulations is adequate to realize the macroscopically differential counterpart. A self-contained and comprehensive practice for the linear one-dimensional convection-diffusion equation is illustrated in details. Numerical validations on the incurred truncation error in one- and two-dimensional cases with various distribution functions are conducted to verify present formulation. As shown in the computational results, excellent agreement between numerical result and theoretical prediction are found in the test problems. Straightforward extensions to more complicated systems including convection-diffusion-reaction, multi-relaxation times in collision operator as well as multi-dimensional Navier-Stokes equations are also exposed in the Appendix to point out its expediency in solving complicated flow problems.
NASA Astrophysics Data System (ADS)
Treumann, R. A.; Baumjohann, W.
2015-10-01
The present review concerns the relevance of collisionless reconnection in the astrophysical context. Emphasis is put on recent developments in theory obtained from collisionless numerical simulations in two and three dimensions. It is stressed that magnetic reconnection is a universal process of particular importance under collisionless conditions, when both collisional and anomalous dissipation are irrelevant. While collisional (resistive) reconnection is a slow, diffusive process, collisionless reconnection is spontaneous. On any astrophysical time scale, it is explosive. It sets on when electric current widths become comparable to the leptonic inertial length in the so-called lepton (electron/positron) "diffusion region", where leptons de-magnetise. Here, the magnetic field contacts its oppositely directed partner and annihilates. Spontaneous reconnection breaks the original magnetic symmetry, violently releases the stored free energy of the electric current, and causes plasma heating and particle acceleration. Ultimately, the released energy is provided by mechanical motion of either the two colliding magnetised plasmas that generate the current sheet or the internal turbulence cascading down to lepton-scale current filaments. Spontaneous reconnection in such extended current sheets that separate two colliding plasmas results in the generation of many reconnection sites (tearing modes) distributed over the current surface, each consisting of lepton exhausts and jets which are separated by plasmoids. Volume-filling factors of reconnection sites are estimated to be as large as {<}10^{-5} per current sheet. Lepton currents inside exhausts may be strong enough to excite Buneman and, for large thermal pressure anisotropy, also Weibel instabilities. They bifurcate and break off into many small-scale current filaments and magnetic flux ropes exhibiting turbulent magnetic power spectra of very flat power-law shape W_b∝ k^{-α } in wavenumber k with power becoming as
Coupling relativistic viscous hydrodynamics to Boltzmann descriptions
Pratt, Scott; Torrieri, Giorgio
2010-10-15
Models of relativistic heavy-ion collisions typically involve both a hydrodynamic module to describe the high-density liquidlike phase and a Boltzmann module to simulate the low-density breakup phase, which is gaslike. Coupling the prescriptions is more complicated for viscous prescriptions if one wants to maintain continuity of the entire stress-energy tensor and currents. Derivations for the viscosity for a gas are reviewed, which then lead to expressions for changes in the phase-space occupation based on simple relaxation-time pictures of viscosity. These expressions are shown to consistently reproduce the nonequilibrium components of the stress-energy tensor. An algorithm for generating a Monte Carlo sampling of particles with which to initiate the Boltzmann calculations is also presented.
Lattice-Boltzmann simulations of droplet evaporation.
Ledesma-Aguilar, Rodrigo; Vella, Dominic; Yeomans, Julia M
2014-11-01
We study the utility and validity of lattice-Boltzmann (LB) simulations to explore droplet evaporation driven by a concentration gradient. Using a binary-fluid lattice-Boltzmann algorithm based on Cahn-Hilliard dynamics, we study the evaporation of planar films and 3D sessile droplets from smooth solid surfaces. Our results show that LB simulations accurately reproduce the classical regime of quasi-static dynamics. Beyond this limit, we show that the algorithm can be used to explore regimes where the evaporative and diffusive timescales are not widely separated, and to include the effect of boundaries of prescribed driving concentration. We illustrate the method by considering the evaporation of a droplet from a solid surface that is chemically patterned with hydrophilic and hydrophobic stripes. PMID:25186667
On boundary conditions in lattice Boltzmann methods
Chen, S.; Martinez, D. |; Mei, R.
1996-09-01
A lattice Boltzmann boundary condition for simulation of fluid flow using simple extrapolation is proposed. Numerical simulations, including two-dimensional Poiseuille flow, unsteady Couette flow, lid-driven square cavity flow, and flow over a column of cylinders for a range of Reynolds numbers, are carried out, showing that this scheme is of second order accuracy in space discretization. Applications of the method to other boundary conditions, including pressure condition and flux condition are discussed. {copyright} {ital 1996 American Institute of Physics.}
Lattice Boltzmann method and channel flow
NASA Astrophysics Data System (ADS)
Stensholt, Sigvat; Mongstad Hope, Sigmund
2016-07-01
Lattice Boltzmann methods are presented at an introductory level with a focus on fairly simple simulations that can be used to test and illustrate the model’s capabilities. Two scenarios are presented. The first is a simple laminar flow in a straight channel driven by a pressure gradient (Poiseuille flow). The second is a more complex, including a wedge where Moffatt vortices may be induced if the wedge is deep enough. Simulations of the Poiseuille flow scenario accurately capture the theoretical velocity profile. The experiment shows the location of the fluid-wall boundary and the effects viscosity has on the velocity and convergence time. The numerical capabilities of the lattice Boltzmann model are tested further by simulating the more complex Moffatt vortex scenario. The method reproduces with high accuracy the theoretical predction that Moffat vortices will not form in a wedge if the vertex angle exceeds 146°. Practical issues limitations of the lattice Boltzmann method are discussed. In particular the accuracy of the bounce-back boundary condition is first order dependent on the grid resolution.
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. PMID:23859813
Boltzmann electron PIC simulation of the E-sail effect
NASA Astrophysics Data System (ADS)
Janhunen, P.
2015-12-01
The solar wind electric sail (E-sail) is a planned in-space propulsion device that uses the natural solar wind momentum flux for spacecraft propulsion with the help of long, charged, centrifugally stretched tethers. The problem of accurately predicting the E-sail thrust is still somewhat open, however, due to a possible electron population trapped by the tether. Here we develop a new type of particle-in-cell (PIC) simulation for predicting E-sail thrust. In the new simulation, electrons are modelled as a fluid, hence resembling hybrid simulation, but in contrast to normal hybrid simulation, the Poisson equation is used as in normal PIC to calculate the self-consistent electrostatic field. For electron-repulsive parts of the potential, the Boltzmann relation is used. For electron-attractive parts of the potential we employ a power law which contains a parameter that can be used to control the number of trapped electrons. We perform a set of runs varying the parameter and select the one with the smallest number of trapped electrons which still behaves in a physically meaningful way in the sense of producing not more than one solar wind ion deflection shock upstream of the tether. By this prescription we obtain thrust per tether length values that are in line with earlier estimates, although somewhat smaller. We conclude that the Boltzmann PIC simulation is a new tool for simulating the E-sail thrust. This tool enables us to calculate solutions rapidly and allows to easily study different scenarios for trapped electrons.
Liu, Zhe; Jiang, Liwei; Zheng, Yisong
2016-01-01
The diagonal and Hall conductivities of graphene arising from the spin-orbit coupling impurity scattering are theoretically studied. Based on the continuous model, i.e. the massless Dirac equation, we derive analytical expressions of the conductivity tensor from both the Kubo and Boltzmann transport theories. By performing numerical calculations, we find that the Kubo quantum transport result of the diagonal conductivity within the self-consistent Born approximation exhibits an insulating gap around the Dirac point. And in this gap a well-defined quantized spin Hall plateau occurs. This indicates the realization of the quantum spin Hall state of graphene driven by the spin-orbit coupling impurities. In contrast, the semi-classical Boltzmann theory fails to predict such a topological insulating phase. The Boltzmann diagonal conductivity is nonzero even in the insulating gap, in which the Boltzmann spin Hall conductivity does not exhibit any quantized plateau. PMID:27029398
Liu, Zhe; Jiang, Liwei; Zheng, Yisong
2016-03-31
The diagonal and Hall conductivities of graphene arising from the spin-orbit coupling impurity scattering are theoretically studied. Based on the continuous model, i.e. the massless Dirac equation, we derive analytical expressions of the conductivity tensor from both the Kubo and Boltzmann transport theories. By performing numerical calculations, we find that the Kubo quantum transport result of the diagonal conductivity within the self-consistent Born approximation exhibits an insulating gap around the Dirac point. And in this gap a well-defined quantized spin Hall plateau occurs. This indicates the realization of the quantum spin Hall state of graphene driven by the spin-orbit coupling impurities. In contrast, the semi-classical Boltzmann theory fails to predict such a topological insulating phase. The Boltzmann diagonal conductivity is nonzero even in the insulating gap, in which the Boltzmann spin Hall conductivity does not exhibit any quantized plateau.
Collisionless microtearing modes in hot tokamaks: Effect of trapped electrons
Swamy, Aditya K.; Ganesh, R.; Brunner, S.; Vaclavik, J.; Villard, L.
2015-07-15
Collisionless microtearing modes have recently been found linearly unstable in sharp temperature gradient regions of large aspect ratio tokamaks. The magnetic drift resonance of passing electrons has been found to be sufficient to destabilise these modes above a threshold plasma β. A global gyrokinetic study, including both passing electrons as well as trapped electrons, shows that the non-adiabatic contribution of the trapped electrons provides a resonant destabilization, especially at large toroidal mode numbers, for a given aspect ratio. The global 2D mode structures show important changes to the destabilising electrostatic potential. The β threshold for the onset of the instability is found to be generally downshifted by the inclusion of trapped electrons. A scan in the aspect ratio of the tokamak configuration, from medium to large but finite values, clearly indicates a significant destabilizing contribution from trapped electrons at small aspect ratio, with a diminishing role at larger aspect ratios.
Perpendicular diffusion of energetic particles in collisionless plasmas
Shalchi, A.
2015-01-15
A fundamental problem in plasma and astrophysics is the interaction between energetic particles and magnetized plasmas. In the current paper, we focus on particle diffusion across the guide magnetic field. It is shown that the perpendicular diffusion coefficient depends only on the parallel diffusion coefficient and the Kubo number. Therefore, one can find four asymptotic limits depending on the values of these two parameters. These regimes are the quasilinear limit, the Kadomtsev and Pogutse limit, the scaling of Rechester and Rosenbluth, and the scaling found by Zybin and Istomin. In the current article, we focus on the Rechester and Rosenbluth scenario because this was not discovered before in the context of collisionless plasmas. Examples and applications are discussed as well. We show that an energy independent ratio of perpendicular and parallel diffusion coefficients can be found and that this ratio can be very small but also close to unity. This is exactly what one observes in the solar wind.
Kinetic dissipation and anisotropic heating in a turbulent collisionless plasma
Parashar, T. N.; Shay, M. A.; Cassak, P. A.; Matthaeus, W. H.
2009-03-15
The kinetic evolution of the Orszag-Tang vortex is studied using collisionless hybrid simulations. In magnetohydrodynamics (MHD) this configuration leads rapidly to broadband turbulence. At large length scales, the evolution of the hybrid simulations is very similar to MHD, with magnetic power spectra displaying scaling similar to a Kolmogorov scaling of -5/3. At small scales, differences from MHD arise, as energy dissipates into heat almost exclusively through the magnetic field. The magnetic energy spectrum of the hybrid simulation shows a break where linear theory predicts that the Hall term in Ohm's law becomes significant, leading to dispersive kinetic Alfven waves. A key result is that protons are heated preferentially in the plane perpendicular to the mean magnetic field, creating a proton temperature anisotropy of the type observed in the corona and solar wind.
Nonlinear Weibel Instability and Turbulence in Strong Collisionless Shocks
Medvedev, Mikhail M.
2008-08-31
This research project was devoted to studies of collisionless shocks, their properties, microphysics and plasma physics of underlying phenomena, such as Weibel instability and generation of small-scale fields at shocks, particle acceleration and transport in the generated random fields, radiation mechanisms from these fields in application to astrophysical phenomena and laboratory experiments (e.g., laser-plasma and beam-plasma interactions, the fast ignition and inertial confinement, etc.). Thus, this study is highly relevant to astrophysical sciences, the inertial confinement program and, in particular, the Fast Ignition concept, etc. It makes valuable contributions to the shock physics, nonlinear plasma theory, as well as to the basic plasma science, in general.
Scaling of Magnetic Reconnection in Relativistic Collisionless Pair Plasmas
NASA Technical Reports Server (NTRS)
Liu, Yi-Hsin; Guo, Fan; Daughton, William; Li, Hui; Hesse, Michael
2015-01-01
Using fully kinetic simulations, we study the scaling of the inflow speed of collisionless magnetic reconnection in electron-positron plasmas from the non-relativistic to ultra-relativistic limit. In the anti-parallel configuration, the inflow speed increases with the upstream magnetization parameter sigma and approaches the speed of light when sigma is greater than O(100), leading to an enhanced reconnection rate. In all regimes, the divergence of the pressure tensor is the dominant term responsible for breaking the frozen-in condition at the x-line. The observed scaling agrees well with a simple model that accounts for the Lorentz contraction of the plasma passing through the diffusion region. The results demonstrate that the aspect ratio of the diffusion region, modified by the compression factor of proper density, remains approximately 0.1 in both the non-relativistic and relativistic limits.
Hybrid numerical model of shock waves in collisionless plasma
NASA Astrophysics Data System (ADS)
Vshivkova, L.; Dudnikova, G.; Vshivkov, K.
2016-10-01
We present a 2D hybrid numerical plasma model of generation and structure of collisionless shock waves in plasma and ion acceleration on their front considering physical processes in supernova remnant shock precursor. In modeling a shock wave is generated by sending a supersonic flow against a reflecting wall. The consequent interaction between incoming and reflected plasma flows lead to formation of waves, the structure of which depends on a flow velocity. The hybrid approach reduces the computational expenses relative to a fully kinetic one, and on the other hand, permits to model ions with a greater accuracy than the magnetohydrodynamics (MHD) allows. Also, another important advantage of the hybrid approach is the possibility to study the important instabilities on an ion time scale, neglecting the modes associated with electrons. In the current work a new computational scheme where stability condition allows carry out computations on more wide set of computational and physical parameters is presented.
ENTROPY PRODUCTION IN COLLISIONLESS SYSTEMS. III. RESULTS FROM SIMULATIONS
Barnes, Eric I.; Egerer, Colin P. E-mail: egerer.coli@uwlax.edu
2015-05-20
The equilibria formed by the self-gravitating, collisionless collapse of simple initial conditions have been investigated for decades. We present the results of our attempts to describe the equilibria formed in N-body simulations using thermodynamically motivated models. Previous work has suggested that it is possible to define distribution functions for such systems that describe maximum entropy states. These distribution functions are used to create radial density and velocity distributions for comparison to those from simulations. A wide variety of N-body code conditions are used to reduce the chance that results are biased by numerical issues. We find that a subset of initial conditions studied lead to equilibria that can be accurately described by these models, and that direct calculation of the entropy shows maximum values being achieved.
Effects of electron inertia in collisionless magnetic reconnection
Andrés, Nahuel Gómez, Daniel; Martin, Luis; Dmitruk, Pablo
2014-07-15
We present a study of collisionless magnetic reconnection within the framework of full two-fluid MHD for a completely ionized hydrogen plasma, retaining the effects of the Hall current, electron pressure and electron inertia. We performed 2.5D simulations using a pseudo-spectral code with no dissipative effects. We check that the ideal invariants of the problem are conserved down to round-off errors. Our numerical results confirm that the change in the topology of the magnetic field lines is exclusively due to the presence of electron inertia. The computed reconnection rates remain a fair fraction of the Alfvén velocity, which therefore qualifies as fast reconnection.
Perpendicular diffusion of energetic particles in collisionless plasmas
NASA Astrophysics Data System (ADS)
Shalchi, A.
2015-01-01
A fundamental problem in plasma and astrophysics is the interaction between energetic particles and magnetized plasmas. In the current paper, we focus on particle diffusion across the guide magnetic field. It is shown that the perpendicular diffusion coefficient depends only on the parallel diffusion coefficient and the Kubo number. Therefore, one can find four asymptotic limits depending on the values of these two parameters. These regimes are the quasilinear limit, the Kadomtsev and Pogutse limit, the scaling of Rechester and Rosenbluth, and the scaling found by Zybin and Istomin. In the current article, we focus on the Rechester and Rosenbluth scenario because this was not discovered before in the context of collisionless plasmas. Examples and applications are discussed as well. We show that an energy independent ratio of perpendicular and parallel diffusion coefficients can be found and that this ratio can be very small but also close to unity. This is exactly what one observes in the solar wind.
Collisionless Plasma Modeling in an Arbitrary Potential Energy Distribution
NASA Technical Reports Server (NTRS)
Liemohn, M. W.; Khazanov, G. V.
1997-01-01
A new technique for calculating a collisionless plasma along a field line is presented. The primary feature of the new model is that it can handle an arbitrary (including nonmonotonic) potential energy distribution. This was one of the limiting constraints on the existing models in this class, and these constraints are generalized for an arbitrary potential energy composition. The formulation for relating current density to the field-aligned potential as well as formulas for density, temperature and energy flux calculations are presented for several distribution functions, ranging from a bi-Lorentzian with a loss cone to an isotropic Maxwellian. A comparison of these results with previous models shows that the formulation reduces.to the earlier models under similar assumptions.
Electron Force Balance in Steady Collisionless-Driven Reconnection
Li Bin; Horiuchi, Ritoku
2008-11-21
Steady collisionless-driven reconnection in an open system is investigated by means of full-particle simulations. A long thin electron current sheet extends towards the outflow direction when the system relaxes to a steady state. Although the pressure tensor term along the reconnection electric field contributes to the violation of the electron frozen-in condition, a new force balance in the inflow direction is realized between the Lorentz and electrostatic forces, which is quite different from that in Harris equilibrium. The strong electrostatic field is generated through the combined effect of the Hall term and a driving inflow. This new force balance is more evident in the three-dimensional case due to the growth of an instability along the reconnection electric field. It is also found that the normalized charge density is in proportion to the square of the electron Alfven velocity averaged over the electron dissipation region.
NASA Astrophysics Data System (ADS)
Ginzburg, S. L.; Dyachenko, V. F.; Orlov, Yu. N.; Fimin, N. N.; Chechetkin, V. M.
2016-09-01
The evolution of a collisionless electron-proton plasma in the self-consistent approximation is investigated. The plasma is assumed to move initially as a whole in a vacuum with the Lorentz factor. The behavior of the dynamical system is analyzed by applying a three-dimensional model based on the Vlasov-Maxwell equations with allowance for retarded potentials. It is shown that the analysis of the solution to the problem is not valid in the "center-of-mass frame" of the plasmoid (since it cannot be correctly defined for a relativistic plasma interacting via an electromagnetic field) and the transition to a laboratory frame of reference is required. In the course of problem solving, a chaotic electromagnetic field is generated by the plasma particles. As a result, the particle distribution functions in the phase space change substantially and differ from their Maxwell-Juttner form. Computations show that the kinetic energies of the electron and proton components and the energy of the self-consistent electromagnetic field become identical. A tendency to the isotropization of the particle momentum distribution in the direction of the initial plasmoid motion is observed.
Cremaschini, Claudio; Tessarotto, Massimo
2011-11-15
A largely unsolved theoretical issue in controlled fusion research is the consistent kinetic treatment of slowly-time varying plasma states occurring in collisionless and magnetized axisymmetric plasmas. The phenomenology may include finite pressure anisotropies as well as strong toroidal and poloidal differential rotation, characteristic of Tokamak plasmas. Despite the fact that physical phenomena occurring in fusion plasmas depend fundamentally on the microscopic particle phase-space dynamics, their consistent kinetic treatment remains still essentially unchallenged to date. The goal of this paper is to address the problem within the framework of Vlasov-Maxwell description. The gyrokinetic treatment of charged particles dynamics is adopted for the construction of asymptotic solutions for the quasi-stationary species kinetic distribution functions. These are expressed in terms of the particle exact and adiabatic invariants. The theory relies on a perturbative approach, which permits to construct asymptotic analytical solutions of the Vlasov-Maxwell system. In this way, both diamagnetic and energy corrections are included consistently into the theory. In particular, by imposing suitable kinetic constraints, the existence of generalized bi-Maxwellian asymptotic kinetic equilibria is pointed out. The theory applies for toroidal rotation velocity of the order of the ion thermal speed. These solutions satisfy identically also the constraints imposed by the Maxwell equations, i.e., quasi-neutrality and Ampere's law. As a result, it is shown that, in the presence of nonuniform fluid and EM fields, these kinetic equilibria can sustain simultaneously toroidal differential rotation, quasi-stationary finite poloidal flows and temperature anisotropy.
Linear theory for fast collisionless magnetic reconnection in the lower-hybrid frequency range
NASA Astrophysics Data System (ADS)
Jovanović, D.; Shukla, P. K.
2005-05-01
A linear theory is presented for the interplay between the fast collisionless magnetic reconnection and the lower-hybrid waves that has been observed in recent computer simulations [J. F. Drake, M. Swisdak, C. Cattell et al., Science 299, 873 (2003)]. In plasma configurations with a strong guide field and anisotropic electron temperature, the electron dynamics is described within the framework of standard electron magnetohydrodynamic equations, accounting also for the effects of the electron polarization and ion motions in the presence of perpendicular electric fields. In the linear phase, we find two types of instabilities of a thin current sheet with steep edges, corresponding to its filamentation (or tearing) and bending. Using a surface-wave formalism for the perturbations whose wavelength is larger than the thickness of the current sheet, the corresponding growth rates are calculated as the contributions of singularities in the plasma dispersion function. These are governed by the electron inertia and the linear coupling of the reconnecting magnetic field with local plasma modes propagating in the perpendicular direction that are subject to the Buneman instability. The linear surface wave instability may be particularly important as a secondary instability, dissipating the thin current sheets that develop in the course of the fast reconnection in the shear-Alfvén and kinetic-Alfvén regimes, and providing the anomalous resistivity for the growth of magnetic islands beyond the shear-Alfvén and kinetic-Alfvén scales.
Hamiltonian derivation of a gyrofluid model for collisionless magnetic reconnection
NASA Astrophysics Data System (ADS)
Tassi, E.
2014-11-01
We consider a simple electromagnetic gyrokinetic model for collisionless plasmas and show that it possesses a Hamiltonian structure. Subsequently, from this model we derive a two-moment gyrofluid model by means of a procedure which guarantees that the resulting gyrofluid model is also Hamiltonian. The first step in the derivation consists of imposing a generic fluid closure in the Poisson bracket of the gyrokinetic model, after expressing such bracket in terms of the gyrofluid moments. The constraint of the Jacobi identity, which every Poisson bracket has to satisfy, selects then what closures can lead to a Hamiltonian gyrofluid system. For the case at hand, it turns out that the only closures (not involving integro/differential operators or an explicit dependence on the spatial coordinates) that lead to a valid Poisson bracket are those for which the second order parallel moment, independently for each species, is proportional to the zero order moment. In particular, if one chooses an isothermal closure based on the equilibrium temperatures and derives accordingly the Hamiltonian of the system from the Hamiltonian of the parent gyrokinetic model, one recovers a known Hamiltonian gyrofluid model for collisionless reconnection. The proposed procedure, in addition to yield a gyrofluid model which automatically conserves the total energy, provides also, through the resulting Poisson bracket, a way to derive further conservation laws of the gyrofluid model, associated with the so called Casimir invariants. We show that a relation exists between Casimir invariants of the gyrofluid model and those of the gyrokinetic parent model. The application of such Hamiltonian derivation procedure to this two-moment gyrofluid model is a first step toward its application to more realistic, higher-order fluid or gyrofluid models for tokamaks. It also extends to the electromagnetic gyrokinetic case, recent applications of the same procedure to Vlasov and drift- kinetic systems.
Collisionless Reconnection with Weak Slow Shocks Under Anisotropic MHD Approximation
NASA Astrophysics Data System (ADS)
Hirabayashi, K.; Hoshino, M.
2014-12-01
Magnetic reconnection accompanied by a pair of slow-mode shock waves, known as Petschek's theory, has been widely studied as an efficient mechanism to convert magnetically stored energy to thermal and/or kinetic energy in plasmas. Satellite observations in the Earth's magnetotail, on the other hand, report that the detection of slow shocks is rare compared with the theory. As an important step to bridge the gap between the observational fact and the Petschek-type reconnection, we performed one- and two- dimensional collisionless magnetohydrodynamic (MHD) simulations of magnetic reconnection paying special attention to the effect of temperature anisotropy. In high-beta plasmas such as a plasma sheet in the magnetotail, it is expected that even weak temperature anisotropy can greatly modify the dynamics. We demonstrate that the slow shocks do exist in the reconnection layer even under the anisotropic temperature. The resultant shocks, however, are weaker than those in isotropic MHD in terms of plasma compression. In addition, the amount of magnetic energy released across the shock is extremely small, that is, the shock is no longer switch-off type. In spite of the weakness of the shocks, the reconnection rates measured by the inflow velocities are kept at the same level as the isotropic cases. Once the slow shock forms, the downstream plasma is heated in highly anisotropic manner, and the firehose-sense anisotropy affects the wave structure in the system. In particular, it is remarkable that the sequential order of propagation of slow shocks and rotational discontinuities reverses depending upon the magnitude of a superposed guide field. Our result is consistent with the rareness of the slow shock detection in the magnetotail, and implies that shocks do not necessarily play an important role. Furthermore, a variety of wave structure of a reconnection layer shown here will help interpretation of observational data in collisionless reconnection.
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.
Oxygen and nitrogen vibration in the thermosphere. [Boltzmann distribution discrepancy
NASA Technical Reports Server (NTRS)
Walker, J. C. G.
1973-01-01
Analysis of the departure of oxygen and nitrogen molecules from the Boltzmann distribution in the thermosphere. It is concluded that the daytime production rates are too low to cause departures from the Boltzmann distribution at altitudes below about 300 km for vibrational levels containing a significant fraction of total population. It is also pointed out that diffusion cannot perturb significantly the Boltzmann distribution at altitudes below about 370 km.
Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows
NASA Astrophysics Data System (ADS)
Liu, Haihu; Valocchi, Albert J.; Zhang, Yonghao; Kang, Qinjun
2013-01-01
A phase-field-based hybrid model that combines the lattice Boltzmann method with the finite difference method is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios. Using a phase field methodology, an interfacial force formula is analytically derived to model the interfacial tension force and the Marangoni stress. We present an improved lattice Boltzmann equation (LBE) method to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations. The LBE method allows not only use of variable mobility in the CHE, but also simulation of multiphase flows with high density ratio because a stable discretization scheme is used for calculating the derivative terms in forcing terms. An additional convection-diffusion equation is solved by the finite difference method for spatial discretization and the Runge-Kutta method for time marching to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. The model is first validated against analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of a three-dimensional deformable droplet and bubble at various Marangoni numbers and density ratios, and satisfactory agreement is obtained between numerical results and theoretical predictions.