Yoshikawa, Kohji; Umemura, Masayuki; Yoshida, Naoki
2013-01-10
We present a scheme for numerical simulations of collisionless self-gravitating systems which directly integrates the Vlasov-Poisson equations in six-dimensional phase space. Using the results from a suite of large-scale numerical simulations, we demonstrate that the present scheme can simulate collisionless self-gravitating systems properly. The integration scheme is based on the positive flux conservation method recently developed in plasma physics. We test the accuracy of our code by performing several test calculations, including the stability of King spheres, the gravitational instability, and the Landau damping. We show that the mass and the energy are accurately conserved for all the test cases we study. The results are in good agreement with linear theory predictions and/or analytic solutions. The distribution function keeps the property of positivity and remains non-oscillatory. The largest simulations are run on 64{sup 6} grids. The computation speed scales well with the number of processors, and thus our code performs efficiently on massively parallel supercomputers.
Flavored quantum Boltzmann equations
Cirigliano, Vincenzo; Lee, Christopher; Ramsey-Musolf, Michael J.; Tulin, Sean
2010-05-15
We derive from first principles, using nonequilibrium field theory, the quantum Boltzmann equations that describe the dynamics of flavor oscillations, collisions, and a time-dependent mass matrix in the early universe. Working to leading nontrivial order in ratios of relevant time scales, we study in detail a toy model for weak-scale baryogenesis: two scalar species that mix through a slowly varying time-dependent and CP-violating mass matrix, and interact with a thermal bath. This model clearly illustrates how the CP asymmetry arises through coherent flavor oscillations in a nontrivial background. We solve the Boltzmann equations numerically for the density matrices, investigating the impact of collisions in various regimes.
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.
Discrete Boltzmann equation for microfluidics.
Li, Baoming; Kwok, Daniel Y
2003-03-28
We propose a discrete Boltzmann model for microfluidics based on the Boltzmann equation with external forces using a single relaxation time collision model. Considering the electrostatic interactions in microfluidics systems, we introduce an equilibrium distribution function that differs from the Maxwell-Boltzmann distribution by an exponential factor to represent the action of an external force field. A statistical mechanical approach is applied to derive the equivalent external acceleration force exerting on the lattice particles based on a mean-field approximation, resulting from the electro-static potential energy and intermolecular potential energy between fluid-fluid and fluid-substrate interactions.
Lattice Boltzmann solver of Rossler equation
NASA Astrophysics Data System (ADS)
Yan, Guangwu; Ruan, Li
2000-06-01
We proposed a lattice Boltzmann model for the Rossler equation. Using a method of multiscales in the lattice Boltzmann model, we get the diffusion reaction as a special case. If the diffusion effect disappeared, we can obtain the lattice Boltzmann solution of the Rossler equation on the mesescopic scale. The numerical results show the method can be used to simulate Rossler equation.
Classical Boltzmann equation and high-temperature QED
NASA Astrophysics Data System (ADS)
Brandt, F. T.; Ferreira, R. B.; Thuorst, J. F.
2015-02-01
The equivalence between thermal field theory and the Boltzmann transport equation is investigated at higher orders in the context of quantum electrodynamics. We compare the contributions obtained from the collisionless transport equation with the high temperature limit of the one-loop thermal Green's function. Our approach employs the representation of the thermal Green's functions in terms of forward scattering amplitudes. The general structure of these amplitudes clearly indicates that the physics described by the leading high temperature limit of quantum electrodynamics can be obtained from the Boltzman transport equation. We also present some explicit examples of this interesting equivalence.
Approximation method for the kinetic Boltzmann equation
NASA Technical Reports Server (NTRS)
Shakhov, Y. M.
1972-01-01
The further development of a method for approximating the Boltzmann equation is considered and a case of pseudo-Maxwellian molecules is treated in detail. A method of approximating the collision frequency is discussed along with a method for approximating the moments of the Boltzmann collision integral. Since the return collisions integral and the collision frequency are expressed through the distribution function moments, use of the proposed methods make it possible to reduce the Boltzmann equation to a series of approximating equations.
Global existence proof for relativistic Boltzmann equation
Dudynski, M. ); Ekiel-Jezewska, M.L. )
1992-02-01
The existence and causality of solutions to the relativistic Boltzmann equation in L[sup 1] and in L[sub loc][sup 1] are proved. The solutions are shown to satisfy physically natural a priori bounds, time-independent in L[sup 1]. The results rely upon new techniques developed for the nonrelativistic Boltzmann equation by DiPerna and Lions.
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 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
NASA Astrophysics Data System (ADS)
Alexanian, Moorad
2014-08-01
The modeling of particle transport involves anomalous diffusion, ⟨x2(t) ⟩ ∝ tα 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.
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.
Entropic lattice Boltzmann model for Burgers's equation.
Boghosian, Bruce M; Love, Peter; Yepez, Jeffrey
2004-08-15
Entropic lattice Boltzmann models are discrete-velocity models of hydrodynamics that possess a Lyapunov function. This feature makes them useful as nonlinearly stable numerical methods for integrating hydrodynamic equations. Over the last few years, such models have been successfully developed for the Navier-Stokes equations in two and three dimensions, and have been proposed as a new category of subgrid model of turbulence. In the present work we develop an entropic lattice Boltzmann model for Burgers's equation in one spatial dimension. In addition to its pedagogical value as a simple example of such a model, our result is actually a very effective way to simulate Burgers's equation in one dimension. At moderate to high values of viscosity, we confirm that it exhibits no trace of instability. At very small values of viscosity, however, we report the existence of oscillations of bounded amplitude in the vicinity of the shock, where gradient scale lengths become comparable with the grid size. As the viscosity decreases, the amplitude at which these oscillations saturate tends to increase. This indicates that, in spite of their nonlinear stability, entropic lattice Boltzmann models may become inaccurate when the ratio of gradient scale length to grid spacing becomes too small. Similar inaccuracies may limit the utility of the entropic lattice Boltzmann paradigm as a subgrid model of Navier-Stokes turbulence.
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.
Consistent lattice Boltzmann equations for phase transitions.
Siebert, D N; Philippi, P C; Mattila, K K
2014-11-01
Unlike conventional computational fluid dynamics methods, the lattice Boltzmann method (LBM) describes the dynamic behavior of fluids in a mesoscopic scale based on discrete forms of kinetic equations. In this scale, complex macroscopic phenomena like the formation and collapse of interfaces can be naturally described as related to source terms incorporated into the kinetic equations. In this context, a novel athermal lattice Boltzmann scheme for the simulation of phase transition is proposed. The continuous kinetic model obtained from the Liouville equation using the mean-field interaction force approach is shown to be consistent with diffuse interface model using the Helmholtz free energy. Density profiles, interface thickness, and surface tension are analytically derived for a plane liquid-vapor interface. A discrete form of the kinetic equation is then obtained by applying the quadrature method based on prescribed abscissas together with a third-order scheme for the discretization of the streaming or advection term in the Boltzmann equation. Spatial derivatives in the source terms are approximated with high-order schemes. The numerical validation of the method is performed by measuring the speed of sound as well as by retrieving the coexistence curve and the interface density profiles. The appearance of spurious currents near the interface is investigated. The simulations are performed with the equations of state of Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, and Carnahan-Starling.
Consistent lattice Boltzmann equations for phase transitions
NASA Astrophysics Data System (ADS)
Siebert, D. N.; Philippi, P. C.; Mattila, K. K.
2014-11-01
Unlike conventional computational fluid dynamics methods, the lattice Boltzmann method (LBM) describes the dynamic behavior of fluids in a mesoscopic scale based on discrete forms of kinetic equations. In this scale, complex macroscopic phenomena like the formation and collapse of interfaces can be naturally described as related to source terms incorporated into the kinetic equations. In this context, a novel athermal lattice Boltzmann scheme for the simulation of phase transition is proposed. The continuous kinetic model obtained from the Liouville equation using the mean-field interaction force approach is shown to be consistent with diffuse interface model using the Helmholtz free energy. Density profiles, interface thickness, and surface tension are analytically derived for a plane liquid-vapor interface. A discrete form of the kinetic equation is then obtained by applying the quadrature method based on prescribed abscissas together with a third-order scheme for the discretization of the streaming or advection term in the Boltzmann equation. Spatial derivatives in the source terms are approximated with high-order schemes. The numerical validation of the method is performed by measuring the speed of sound as well as by retrieving the coexistence curve and the interface density profiles. The appearance of spurious currents near the interface is investigated. The simulations are performed with the equations of state of Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, and Carnahan-Starling.
On the full Boltzmann equations for leptogenesis
Garayoa, J.; Pastor, S.; Pinto, T.; Rius, N.; Vives, O. E-mail: pastor@ific.uv.es E-mail: nuria@ific.uv.es
2009-09-01
We consider the full Boltzmann equations for standard and soft leptogenesis, instead of the usual integrated Boltzmann equations which assume kinetic equilibrium for all species. Decays and inverse decays may be inefficient for thermalising the heavy-(s)neutrino distribution function, leading to significant deviations from kinetic equilibrium. We analyse the impact of using the full kinetic equations in the case of a previously generated lepton asymmetry, and find that the washout of this initial asymmetry due to the interactions of the right-handed neutrino is larger than when calculated via the integrated equations. We also solve the full Boltzmann equations for soft leptogenesis, where the lepton asymmetry induced by the soft SUSY-breaking terms in sneutrino decays is a purely thermal effect, since at T = 0 the asymmetry in leptons cancels the one in sleptons. In this case, we obtain that in the weak washout regime (K ∼< 1) the final lepton asymmetry can change up to a factor four with respect to previous estimates.
Non-markovian boltzmann equation
Kremp, D.; Bonitz, M.; Kraeft, W.D.; Schlanges, M.
1997-08-01
A quantum kinetic equation for strongly interacting particles (generalized binary collision approximation, ladder or T-matrix approximation) is derived in the framework of the density operator technique. In contrast to conventional kinetic theory, which is valid on large time scales as compared to the collision (correlation) time only, our approach retains the full time dependencies, especially also on short time scales. This means retardation and memory effects resulting from the dynamics of binary correlations and initial correlations are included. Furthermore, the resulting kinetic equation conserves total energy (the sum of kinetic and potential energy). The second aspect of generalization is the inclusion of many-body effects, such as self-energy, i.e., renormalization of single-particle energies and damping. To this end we introduce an improved closure relation to the Bogolyubov{endash}Born{endash}Green{endash}Kirkwood{endash}Yvon hierarchy. Furthermore, in order to express the collision integrals in terms of familiar scattering quantities (Mo/ller operator, T-matrix), we generalize the methods of quantum scattering theory by the inclusion of medium effects. To illustrate the effects of memory and damping, the results of numerical simulations are presented. {copyright} 1997 Academic Press, Inc.
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.
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.
Boltzmann equations for neutrinos with flavor mixings
NASA Astrophysics Data System (ADS)
Yamada, Shoichi
2000-11-01
With a view of applications to the simulations of supernova explosions and protoneutron star cooling, we derive the Boltzmann equations for the neutrino transport with flavor mixing based on the real time formalism of the nonequilibrium field theory and the gradient expansion of the Green function. The relativistic kinematics is properly taken into account. The advection terms are derived in the mean field approximation for the neutrino self-energy while the collision terms are obtained in the Born approximation. The resulting equations take the familiar form of the Boltzmann equation with corrections due to mixing both in the advection part and in the collision part. These corrections are essentially the same as those derived by Sirera et al. for the advection terms and those by Raffelt et al. for the collision terms, respectively, though the formalism employed here is different from theirs. The derived equations will be easily implemented in numerical codes employed in the simulations of supernova explosions and protoneutron star cooling.
Fluctuating lattice Boltzmann method for the diffusion equation.
Wagner, Alexander J; Strand, Kyle
2016-09-01
We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method.
Electron Boltzmann equation in nonthermal plasmas
NASA Technical Reports Server (NTRS)
Kunc, J. A.; Soon, W. H.
1991-01-01
Numerical and analytical solutions of the electron Boltzmann equation for a two-temperature steady-state He plasma are examined in a broad range of conditions, i.e., atom temperature ranging from 5000 K to 20,000 K; electron temperature ranging from 10,000 K to 20,000 K; and atom density ranging from 10 to the 10th to 10 to the 18th per cu cm. The WKB analytical solution is shown to be satisfactory in most situations. Attention is also given to the deviation of the electron distribution from Maxwellian, and to the possibility of raising the tail of the distribution.
The Einstein-Boltzmann equations revisited
NASA Astrophysics Data System (ADS)
Nadkarni-Ghosh, Sharvari; Refregier, Alexandre
2017-10-01
The linear Einstein-Boltzmann (E-B) equations describe the evolution of perturbations in the universe and its numerical solutions play a central role in cosmology. We revisit this system of differential equations and present a detailed investigation of its mathematical properties. For this purpose, we focus on a simplified set of equations aimed at describing the broad features of the matter power spectrum. We first perform an eigenvalue analysis and study the onset of oscillations in the system signalled by the transition from real to complex eigenvalues. We then provide a stability criterion of different numerical schemes for this linear system and estimate the associated step size. We elucidate the stiffness property of the E-B system and show how it can be characterized in terms of the eigenvalues. While the parameters of the system are time dependent making it non-autonomous, we define an adiabatic regime where the parameters vary slowly enough for the system to be quasi-autonomous. We summarize the different regimes of the system for these different criteria as function of wavenumber k and scalefactor a. We also provide a compendium of analytic solutions for all perturbation variables in six limits on the k-a plane and express them explicitly in terms of initial conditions. These results are aimed to help the further development and testing of numerical cosmological Boltzmann solvers.
Boltzmann equation with double-well potentials
NASA Astrophysics Data System (ADS)
Chiacchiera, Silvia; Macrı, Tommaso; Trombettoni, Andrea
2016-10-01
We study the dynamics of an interacting classical gas trapped in a double-well potential at finite temperature. Two model potentials are considered: a cubic box with a square barrier in the middle, and a harmonic trap with a Gaussian barrier along one direction. The study is performed using the Boltzmann equation, solved numerically via the test-particle method. We introduce and discuss a simple analytical model that allows one to provide estimates of the relaxation time, which are compared with numerical results. Finally, we use our findings to make numerical and analytical predictions for the case of a fermionic mixture in the normal-fluid phase in a realistic double-well potential relevant for experiments with cold atoms.
Conservative form of Boltzmann's equation in general relativity
NASA Astrophysics Data System (ADS)
Shibata, Masaru; Nagakura, Hiroki; Sekiguchi, Yuichiro; Yamada, Shoichi
2014-04-01
We derive a conservative form of Boltzmann's equation in general relativity, which is concisely written. Several explicit forms of this equation are written for black-hole spacetime with several coordinate conditions in real spacetime and momentum-space coordinates.
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.
Efficient Asymptotic Preserving Deterministic methods for the Boltzmann Equation
2011-04-01
release, distribution unlimited 13. SUPPLEMENTARY NOTES See also ADA579248. Models and Computational Methods for Rarefied Flows (Modeles et methodes de...nonlinear collisional kinetic equation. The most well-known example is represented by the Boltzmann equation of rarefied gas dynamics (Cercignani, 1988...et al. (2010). Although the scope of our insights is wider, here we will focus mainly on the classical Boltzmann equation of rarefied gas dynamics
Boltzmann equations for a binary one-dimensional ideal gas.
Boozer, A D
2011-09-01
We consider a time-reversal invariant dynamical model of a binary ideal gas of N molecules in one spatial dimension. By making time-asymmetric assumptions about the behavior of the gas, we derive Boltzmann and anti-Boltzmann equations that describe the evolution of the single-molecule velocity distribution functions for an ensemble of such systems. We show that for a special class of initial states of the ensemble one can obtain an exact expression for the N-molecule velocity distribution function, and we use this expression to rigorously prove that the time-asymmetric assumptions needed to derive the Boltzmann and anti-Boltzmann equations hold in the limit of large N. Our results clarify some subtle issues regarding the origin of the time asymmetry of Boltzmann's H theorem.
Lattice Boltzmann equation method for the Cahn-Hilliard equation
NASA Astrophysics Data System (ADS)
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), 10.1016/j.jcp.2010.07.007]; 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.
Quantum linear Boltzmann equation with finite intercollision time
Diosi, Lajos
2009-12-15
Inconsistencies are pointed out in the usual quantum versions of the classical linear Boltzmann equation constructed for a quantized test particle in a gas. These are related to the incorrect formal treatment of momentum decoherence. We prove that ideal collisions with the molecules would result in complete momentum decoherence, the persistence of coherence is only due to the finite intercollision time. A corresponding quantum linear Boltzmann equation is proposed.
CMB spectral distortions as solutions to the Boltzmann equations
NASA Astrophysics Data System (ADS)
Ota, Atsuhisa
2017-01-01
We propose to re-interpret the cosmic microwave background spectral distortions as solutions to the Boltzmann equation. This approach makes it possible to solve the second order Boltzmann equation explicitly, with the spectral y distortion and the momentum independent second order temperature perturbation, while generation of μ distortion cannot be explained even at second order in this framework. We also extend our method to higher order Boltzmann equations systematically and find new type spectral distortions, assuming that the collision term is linear in the photon distribution functions, namely, in the Thomson scattering limit. As an example, we concretely construct solutions to the cubic order Boltzmann equation and show that the equations are closed with additional three parameters composed of a cubic order temperature perturbation and two cubic order spectral distortions. The linear Sunyaev-Zel'dovich effect whose momentum dependence is different from the usual y distortion is also discussed in the presence of the next leading order Kompaneets terms, and we show that higher order spectral distortions are also generated as a result of the diffusion process in a framework of higher order Boltzmann equations. The method may be applicable to a wider class of problems and has potential to give a general prescription to non-equilibrium physics.
NASA Astrophysics Data System (ADS)
Wang, Yahui; Yan, Liming; Ma, Yu
2017-06-01
Applications of the transient Boltzmann transport equation (BTE) have undergone much investigation, such as radiative heat transfer and neutron transport. This paper provides a lattice Boltzmann model to efficiently resolve the multidimensional transient BTE. For a higher angular resolution, enough transport directions are considered while the transient BTE in each direction is treated as a conservation law equation and solved independently. Both macroscopic equations recovered from a Chapman-Enskog expansion and simulated results of typical benchmark problems show not only the second-order accuracy but also the flexibility and applicability of the proposed lattice Boltzmann model. This approach may contribute a powerful technique for the parallel simulation of large-scale engineering and some alternative perspectives for solving the nonlinear transport problem further.
Wang, Yahui; Yan, Liming; Ma, Yu
2017-06-01
Applications of the transient Boltzmann transport equation (BTE) have undergone much investigation, such as radiative heat transfer and neutron transport. This paper provides a lattice Boltzmann model to efficiently resolve the multidimensional transient BTE. For a higher angular resolution, enough transport directions are considered while the transient BTE in each direction is treated as a conservation law equation and solved independently. Both macroscopic equations recovered from a Chapman-Enskog expansion and simulated results of typical benchmark problems show not only the second-order accuracy but also the flexibility and applicability of the proposed lattice Boltzmann model. This approach may contribute a powerful technique for the parallel simulation of large-scale engineering and some alternative perspectives for solving the nonlinear transport problem further.
Equations of state in a lattice Boltzmann model
NASA Astrophysics Data System (ADS)
Yuan, Peng; Schaefer, Laura
2006-04-01
In this paper we consider the incorporation of various equations of state into the single-component multiphase lattice Boltzmann model. Several cubic equations of state, including the van der Waals, Redlich-Kwong, and Peng-Robinson, as well as a noncubic equation of state (Carnahan-Starling), are incorporated into the lattice Boltzmann model. The details of phase separation in these nonideal single-component systems are presented by comparing the numerical simulation results in terms of density ratios, spurious currents, and temperature ranges. A comparison with a real fluid system, i.e., the properties of saturated water and steam, is also presented.
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; ...
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
Physical scales in the Wigner–Boltzmann equation
Nedjalkov, M.; Selberherr, S.; Ferry, D.K.; Vasileska, D.; Dollfus, P.; Querlioz, D.; Dimov, I.; Schwaha, P.
2013-01-01
The Wigner–Boltzmann equation provides the Wigner single particle theory with interactions with bosonic degrees of freedom associated with harmonic oscillators, such as phonons in solids. Quantum evolution is an interplay of two transport modes, corresponding to the common coherent particle-potential processes, or to the decoherence causing scattering due to the oscillators. Which evolution mode will dominate depends on the scales of the involved physical quantities. A dimensionless formulation of the Wigner–Boltzmann equation is obtained, where these scales appear as dimensionless strength parameters. A notion called scaling theorem is derived, linking the strength parameters to the coupling with the oscillators. It is shown that an increase of this coupling is equivalent to a reduction of both the strength of the electric potential, and the coherence length. Secondly, the existence of classes of physically different, but mathematically equivalent setups of the Wigner–Boltzmann evolution is demonstrated. PMID:23504194
Physical scales in the Wigner-Boltzmann equation.
Nedjalkov, M; Selberherr, S; Ferry, D K; Vasileska, D; Dollfus, P; Querlioz, D; Dimov, I; Schwaha, P
2013-01-01
The Wigner-Boltzmann equation provides the Wigner single particle theory with interactions with bosonic degrees of freedom associated with harmonic oscillators, such as phonons in solids. Quantum evolution is an interplay of two transport modes, corresponding to the common coherent particle-potential processes, or to the decoherence causing scattering due to the oscillators. Which evolution mode will dominate depends on the scales of the involved physical quantities. A dimensionless formulation of the Wigner-Boltzmann equation is obtained, where these scales appear as dimensionless strength parameters. A notion called scaling theorem is derived, linking the strength parameters to the coupling with the oscillators. It is shown that an increase of this coupling is equivalent to a reduction of both the strength of the electric potential, and the coherence length. Secondly, the existence of classes of physically different, but mathematically equivalent setups of the Wigner-Boltzmann evolution is demonstrated.
Lattice Boltzmann method for the Saint-Venant equations
NASA Astrophysics Data System (ADS)
Liu, Haifei; Wang, Hongda; Liu, Shu; Hu, Changwei; Ding, Yu; Zhang, Jie
2015-05-01
The Saint-Venant equations represent the hydrodynamic principles of unsteady flows in open channel network through a set of non-linear partial differential equations. In this paper, a new lattice Boltzmann approach to solving the one-dimensional Saint-Venant equations (LABSVE) is developed, demonstrating the variation of discharge and sectional area with external forces, such as bed slope and bed friction. Our research recovers the Saint-Venant equations through deducing the Chapman-Enskog expansion on the lattice Boltzmann equation, which is a mesoscopic technique, bridging the molecular movement and macroscopic physical variables. It is also a fully explicit process, providing simplicity for programming. The model is verified by three benchmark tests: (i) a one-dimensional subcritical gradient flow; (ii) a dam-break wave flow; (iii) a flood event on the Yongding River. The results showed the accuracy of the proposed method and its good applicability in solving Saint-Venant problems.
The Fluid Dynamic Limit of the Nonlinear Boltzmann Equation,
1980-02-01
dynamics is ’ . strongly nonlinear. Previously, Glikson [4] and Kaniel and Shinbrot [10 showed existence locally in time. Global existence of solutions... Glikson , A., On the existence of general solutions of the initial-value problem for the nonlinear Boltzmann equation with a cut-off, Arch. Rational
A fast iterative scheme for the linearized Boltzmann equation
NASA Astrophysics Data System (ADS)
Wu, Lei; Zhang, Jun; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.
2017-06-01
Iterative schemes to find steady-state solutions to the Boltzmann equation are efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator L into the form L = (L + Nδh) - Nδh, where δ is the gas rarefaction parameter, h is the velocity distribution function, and N is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when Nδ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference
The relativistic Boltzmann equation on a spherically symmetric gravitational field
NASA Astrophysics Data System (ADS)
Takou, Etienne; Ciake Ciake, Fidèle L.
2017-10-01
In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on the time, the position and the impulsion. We consider this equation on a spherically symmetric gravitational field spacetime. The collision kernel considered here is for the hard potentials case. We prove the existence of a unique global (in time) mild solution in a suitable weighted space.
Lattice Boltzmann model for the complex Ginzburg-Landau equation.
Zhang, Jianying; Yan, Guangwu
2010-06-01
A lattice Boltzmann model with complex distribution function for the complex Ginzburg-Landau equation (CGLE) is proposed. By using multiscale technique and the Chapman-Enskog expansion on complex variables, we obtain a series of complex partial differential equations. Then, complex equilibrium distribution function and its complex moments are obtained. Based on this model, the rotation and oscillation properties of stable spiral waves and the breaking-up behavior of unstable spiral waves in CGLE are investigated in detail.
Modeling adsorption with lattice Boltzmann equation.
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-06-03
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.
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.
Shock-wave structure using nonlinear model Boltzmann equations.
NASA Technical Reports Server (NTRS)
Segal, B. M.; Ferziger, J. H.
1972-01-01
The structure of strong plane shock waves in a perfect monatomic gas was studied using four nonlinear models of the Boltzmann equation. The models involved the use of a simplified collision operator with velocity-independent collision frequency, in place of the complicated Boltzmann collision operator. The models employed were the BGK and ellipsoidal models developed by earlier authors, and the polynomial and trimodal gain function models developed during the work. An exact set of moment equations was derived for the density, velocity, temperature, viscous stress, and heat flux within the shock. This set was reduced to a pair of coupled nonlinear integral equations and solved using specially adapted numerical techniques. A new and simple Gauss-Seidel iteration was developed during the work and found to be as efficient as the best earlier iteration methods.
Numerical methods for solving the Boltzmann equation (a review)
NASA Technical Reports Server (NTRS)
Limar, Y. F.; Shakhov, Y. M.; Shidlovskiy, V. P.
1972-01-01
The methods are reviewed which are utilized in principal attempts to obtain the numerical solution or modeling of the Boltzmann equation over a broad range of Knudsen numbers. The primary methods considered are the Monte Carlo and the discrete velocities methods. The conculsions drawn from the analysis include the following: (1) The Monte Carlo methods are not well suited in the area of small Knudsen numbers. (2) Among the Monte Carlo methods, the Bird method appears to be the most attractive, since it is more directly related to the Boltzmann equation. (3) The deterministic methods, which include the discrete ordinate technique, offer great possibilities but require exceedingly large computer times. (4) The use of approximating equations in combination with the discrete velocities method will possibly improve computation time and reduce the required memory volume.
Shock-wave structure using nonlinear model Boltzmann equations.
NASA Technical Reports Server (NTRS)
Segal, B. M.; Ferziger, J. H.
1972-01-01
The structure of strong plane shock waves in a perfect monatomic gas was studied using four nonlinear models of the Boltzmann equation. The models involved the use of a simplified collision operator with velocity-independent collision frequency, in place of the complicated Boltzmann collision operator. The models employed were the BGK and ellipsoidal models developed by earlier authors, and the polynomial and trimodal gain function models developed during the work. An exact set of moment equations was derived for the density, velocity, temperature, viscous stress, and heat flux within the shock. This set was reduced to a pair of coupled nonlinear integral equations and solved using specially adapted numerical techniques. A new and simple Gauss-Seidel iteration was developed during the work and found to be as efficient as the best earlier iteration methods.
On removal of charge singularity in Poisson-Boltzmann equation.
Cai, Qin; Wang, Jun; Zhao, Hong-Kai; Luo, Ray
2009-04-14
The Poisson-Boltzmann theory has become widely accepted in modeling electrostatic solvation interactions in biomolecular calculations. However the standard practice of atomic point charges in molecular mechanics force fields introduces singularity into the Poisson-Boltzmann equation. The finite-difference/finite-volume discretization approach to the Poisson-Boltzmann equation alleviates the numerical difficulty associated with the charge singularity but introduces discretization error into the electrostatic potential. Decomposition of the electrostatic potential has been explored to remove the charge singularity explicitly to achieve higher numerical accuracy in the solution of the electrostatic potential. In this study, we propose an efficient method to overcome the charge singularity problem. In our framework, two separate equations for two different potentials in two different regions are solved simultaneously, i.e., the reaction field potential in the solute region and the total potential in the solvent region. The proposed method can be readily implemented with typical finite-difference Poisson-Boltzmann solvers and return the singularity-free reaction field potential with a single run. Test runs on 42 small molecules and 4 large proteins show a very high agreement between the reaction field energies computed by the proposed method and those by the classical finite-difference Poisson-Boltzmann method. It is also interesting to note that the proposed method converges faster than the classical method, though additional time is needed to compute Coulombic potential on the dielectric boundary. The higher precision, accuracy, and efficiency of the proposed method will allow for more robust electrostatic calculations in molecular mechanics simulations of complex biomolecular systems.
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.
Solving the homogeneous Boltzmann equation with arbitrary scattering kernel
NASA Astrophysics Data System (ADS)
Hohenegger, A.
2009-03-01
With applications in astroparticle physics in mind, we generalize a method for the solution of the nonlinear, space-homogeneous Boltzmann equation with an isotropic distribution function to arbitrary matrix elements. The method is based on the expansion of the scattering kernel in terms of two cosines of the “scattering angles.” The scattering functions used by previous authors in particle physics for matrix elements in the Fermi approximation are retrieved as lowest order results in this expansion. The method is designed for the unified treatment of reactive mixtures of particles obeying different scattering laws, including the quantum statistical terms for blocking or stimulated emission, in possibly large networks of Boltzmann equations. Although our notation is the relativistic one, as it is used in astroparticle physics, the results can also be applied in the classical case.
A method for direct numerical integration of the Boltzmann equation
NASA Technical Reports Server (NTRS)
Cheremisin, F. G.
1972-01-01
The principal difficulties in numerical solution of the Boltzmann equation are considered. The study is aimed at formulating a numerical solution in such a manner that it contains a minimum amount of excess information at the distribution function level. It is pointed out that the accurate calculation of the distribution function at each point in phase space requires a tremendous number of operations, due to the necessity of solving five-fold quadratures in the collision integral. This results in the operational memory of the digital computer being insufficient to store all the data on the distribution functions at the necessary points in phase space. An algorithm is constructed involving successive iterations of the Boltzmann equation which does not require storage of each step of the new distribution function.
Distributional Monte Carlo Methods for the Boltzmann Equation
2013-03-01
Examples of such violations arise in rarefied gas dynamics, hypersonic flows , and micro-scale flows . Additionally, there is an “equilibrium hypothesis...are rarefied flows and flows containing non-equilibrium phenomena. Applications of rarefied gas dynamics typically involve high-altitude flight and...1 1.1 Kinetic Theory and Rarefied Gas Dynamics . . . . . . . . . . . . . . . . . 3 1.2 Computational Methods for the Boltzmann equation
Distributional monte carlo methods for the boltzmann equation
NASA Astrophysics Data System (ADS)
Schrock, Christopher R.
Stochastic particle methods (SPMs) for the Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) technique, have gained popularity for the prediction of flows in which the assumptions behind the continuum equations of fluid mechanics break down; however, there are still a number of issues that make SPMs computationally challenging for practical use. In traditional SPMs, simulated particles may possess only a single velocity vector, even though they may represent an extremely large collection of actual particles. This limits the method to converge only in law to the Boltzmann solution. This document details the development of new SPMs that allow the velocity of each simulated particle to be distributed. This approach has been termed Distributional Monte Carlo (DMC). A technique is described which applies kernel density estimation to Nanbu's DSMC algorithm. It is then proven that the method converges not just in law, but also in solution for Linfinity(R 3) solutions of the space homogeneous Boltzmann equation. This provides for direct evaluation of the velocity density function. The derivation of a general Distributional Monte Carlo method is given which treats collision interactions between simulated particles as a relaxation problem. The framework is proven to converge in law to the solution of the space homogeneous Boltzmann equation, as well as in solution for Linfinity(R3) solutions. An approach based on the BGK simplification is presented which computes collision outcomes deterministically. Each technique is applied to the well-studied Bobylev-Krook-Wu solution as a numerical test case. Accuracy and variance of the solutions are examined as functions of various simulation parameters. Significantly improved accuracy and reduced variance are observed in the normalized moments for the Distributional Monte Carlo technique employing discrete BGK collision modeling.
Hypersonic Shock Wave Computations Using the Generalized Boltzmann Equation
NASA Astrophysics Data System (ADS)
Agarwal, Ramesh; Chen, Rui; Cheremisin, Felix G.
2006-11-01
Hypersonic shock structure in diatomic gases is computed by solving the Generalized Boltzmann Equation (GBE), where the internal and translational degrees of freedom are considered in the framework of quantum and classical mechanics respectively [1]. The computational framework available for the standard Boltzmann equation [2] is extended by including both the rotational and vibrational degrees of freedom in the GBE. There are two main difficulties encountered in computation of high Mach number flows of diatomic gases with internal degrees of freedom: (1) a large velocity domain is needed for accurate numerical description of the distribution function resulting in enormous computational effort in calculation of the collision integral, and (2) about 50 energy levels are needed for accurate representation of the rotational spectrum of the gas. Our methodology addresses these problems, and as a result the efficiency of calculations has increased by several orders of magnitude. The code has been validated by computing the shock structure in Nitrogen for Mach numbers up to 25 including the translational and rotational degrees of freedom. [1] Beylich, A., ``An Interlaced System for Nitrogen Gas,'' Proc. of CECAM Workshop, ENS de Lyon, France, 2000. [2] Cheremisin, F., ``Solution of the Boltzmann Kinetic Equation for High Speed Flows of a Rarefied Gas,'' Proc. of the 24th Int. Symp. on Rarefied Gas Dynamics, Bari, Italy, 2004.
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.
Numerical simulation for the Gross-Pitaevskii equation based on the lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Wang, Huimin
2017-09-01
A lattice Boltzmann model for the Gross-Pitaevskii equation is proposed in this paper. Some numerical tests for one- and two-dimensional Gross-Pitaevskii equation have been conducted. The waves of the Gross-Pitaevskii equation are simulated. Numerical results show that the lattice Boltzmann method is an effective method for the wave of the Gross-Pitaevskii equation.
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.
A lattice Boltzmann model for the Burgers-Fisher equation.
Zhang, Jianying; Yan, Guangwu
2010-06-01
A lattice Boltzmann model is developed for the one- and two-dimensional Burgers-Fisher equation based on the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. In order to obtain the two-dimensional Burgers-Fisher equation, vector sigma(j) has been used. And in order to overcome the drawbacks of "error rebound," a new assumption of additional distribution is presented, where two additional terms, in first order and second order separately, are used. Comparisons with the results obtained by other methods reveal that the numerical solutions obtained by the proposed method converge to exact solutions. The model under new assumption gives better results than that with second order assumption.
TAP equation for non-negative Boltzmann machine
NASA Astrophysics Data System (ADS)
Yasuda, Muneki; Tanaka, Kazuyuki
2012-01-01
Mean-field methods for spin systems are frequently used in not only statistical physics but also information sciences. We focus on the Plefka expansion method for spin systems with two-body interactions. The Plefka expansion is a useful perturbative expansion of the Gibbs free energy, and it can systematically provide the naive mean-field approximation, the Thouless-Anderson-Palmer (TAP) equation and higher-order approximations. In the first part of this paper, using the linear response relation, we derive a recurrence formula for perturbative coefficients in the Plefka expansion. Our recurrence formula enables us to systematically derive general order coefficients. In the latter part of the paper, we apply our recurrence formula to the non-negative Boltzmann machine in which all spin variables are constrained to have non-negative real values, and we obtain the TAP equation for this model. We verify the performance of our TAP equation by conducting some numerical experiments.
Kadomtsev−Petviashvili equation for a flow of highly nonisothermal collisionless plasma
Movsesyants, Yu. B.; Rukhadze, A. A.; Tyuryukanov, P. M.
2016-01-15
It is shown that the equations of two-fluid electrodynamics for a cold ions flow and Boltzmann electrons in the vicinity of the ion-sound point can be reduced to the Kadomtsev−Petviashvili equation. Examples of two-dimensional equilibria with pole singularities obtained by exactly solving the equations are presented. An exact self-similar solution describing a two-dimensional transonic flow and having no pole singularities is found.
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.
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.
Boundary conditions for the Boltzmann equation for rough walls
NASA Astrophysics Data System (ADS)
Brull, Stéphane; Charrier, Pierre
2014-12-01
In some applications, rarefied gases have to considered in a domain whose boundary presents some nanoscale roughness. That is why, we have considered (Brull,2014) a new derivation of boundary conditions for the Boltzmann equation, where the wall present some nanoscale roughness. In this paper, the interaction between the gas and the wall is represented by a kinetic equation defined in a surface layer at the scale of the nanometer close to the wall. The boundary conditions are obtained from a formal asymptotic expansion and are describded by a scattering kernel satisfying classical properties (non-negativeness, normalization, reciprocity). Finally, we present some numerical simulations of scattering diagrams showing the importance of the consideration of roughness for small scales in the model.
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)
Lattice Boltzmann equation method for multiple immiscible continuum fluids.
Spencer, T J; Halliday, I; Care, C M
2010-12-01
This paper generalizes the two-component algorithm of Sec. , extending it, in Sec. , to describe N>2 mutually immiscible fluids in the isothermal continuum regime. Each fluid has an independent interfacial tension. While retaining all its computational advantages, we remove entirely the empiricism associated with contact behavior in our previous multiple immiscible fluid models [M. M. Dupin, Phys. Rev. E 73, 055701(R) (2006); Med. Eng. Phys. 28, 13 (2006)] while solidifying the physical foundations. Moreover, the model relies upon a fluid-fluid segregation which is simpler, computationally faster, more free of artifacts (i.e., the interfacial microcurrent), and upon an interface-inducing force distribution which is analytic. The method is completely symmetric between any numbers of immiscible fluids and stable over a wide range of directly input interfacial tension. We present data on the steady-state properties of multiple interface model, which are in good agreement with theory [R. E. Johnson and S. S. Sadhal, Annu. Rev. Fluid Mech. 17, 289 (1985)], specifically on the shapes of multidrop systems. Section is an analysis of the kinetic and continuum-scale descriptions of the underlying two-component lattice Boltzmann model for immiscible fluids, extendable to more than two immiscible fluids. This extension requires (i) the use of a more local kinetic equation perturbation which is (ii) free from a reliance on measured interfacial curvature. It should be noted that viewed simply as a two-component method, the continuum algorithm is inferior to our previous methods, reported by Lishchuk [Phys. Rev. E 67, 036701 (2003)] and Halliday [Phys. Rev. E 76, 026708 (2007)]. Greater stability and parameter range is achieved in multiple drop simulations by using the forced multi-relaxation-time lattice Boltzmann method developed, along with (for completeness) a forced exactly incompressible Bhatnagar-Gross-Krook lattice Boltzmann model, in the Appendix. These appended schemes
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.
Solutions of Boltzmann Equation for Simulation of Particle Distributions in Plasmas
NASA Astrophysics Data System (ADS)
Hammond, Jason
2014-10-01
We are investigating the time evolution of the electron and excited state distribution functions. To accomplish this, we solve the time dependent Boltzmann equation to overcome some typical limitations of modeling high pressure plasmas using Monte Carlo methods. Here we focus on the numerical approach to solving the time dependent Boltzmann equation using a multi-term approximation of the electron distribution function. We also compare Boltzmann results for electron distribution evolution against multiple plasma simulations using experimental collisional cross-section data.
Acoustic equations of state for simple lattice Boltzmann velocity sets.
Viggen, Erlend Magnus
2014-07-01
The lattice Boltzmann (LB) method typically uses an isothermal equation of state. This is not sufficient to simulate a number of acoustic phenomena where the equation of state cannot be approximated as linear and constant. However, it is possible to implement variable equations of state by altering the LB equilibrium distribution. For simple velocity sets with velocity components ξ(iα)∈(-1,0,1) for all i, these equilibria necessarily cause error terms in the momentum equation. These error terms are shown to be either correctable or negligible at the cost of further weakening the compressibility. For the D1Q3 velocity set, such an equilibrium distribution is found and shown to be unique. Its sound propagation properties are found for both forced and free waves, with some generality beyond D1Q3. Finally, this equilibrium distribution is applied to a nonlinear acoustics simulation where both mechanisms of nonlinearity are simulated with good results. This represents an improvement on previous such simulations and proves that the compressibility of the method is still sufficiently strong even for nonlinear acoustics.
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.
A stochastic Galerkin method for the Boltzmann equation with uncertainty
Hu, Jingwei; Jin, Shi
2016-06-15
We develop a stochastic Galerkin method for the Boltzmann equation with uncertainty. The method is based on the generalized polynomial chaos (gPC) approximation in the stochastic Galerkin framework, and can handle random inputs from collision kernel, initial data or boundary data. We show that a simple singular value decomposition of gPC related coefficients combined with the fast Fourier-spectral method (in velocity space) allows one to compute the high-dimensional collision operator very efficiently. In the spatially homogeneous case, we first prove that the analytical solution preserves the regularity of the initial data in the random space, and then use it to establish the spectral accuracy of the proposed stochastic Galerkin method. Several numerical examples are presented to illustrate the validity of the proposed scheme.
Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation
NASA Astrophysics Data System (ADS)
Dou, Nicholas G.; Minnich, Austin J.
2016-01-01
Materials that possess low density, low thermal conductivity, and high stiffness are desirable for engineering applications, but most materials cannot realize these properties simultaneously due to the coupling between them. Nanotrusses, which consist of hollow nanoscale beams architected into a periodic truss structure, can potentially break these couplings due to their lattice architecture and nanoscale features. In this work, we study heat conduction in the exact nanotruss geometry by solving the frequency-dependent Boltzmann transport equation using a variance-reduced Monte Carlo algorithm. We show that their thermal conductivity can be described with only two parameters, solid fraction and wall thickness. Our simulations predict that nanotrusses can realize unique combinations of mechanical and thermal properties that are challenging to achieve in typical materials.
Boltzmann equation for the electron gas of a nondegenerate plasma
NASA Technical Reports Server (NTRS)
Gould, R. J.
1974-01-01
The collision terms in the Boltzmann equation associated with various processes are derived. For processes having a Fokker-Planck (F-P) limit, the associated F-P operator is derived by means of physical arguments to determine the form of the operator; its multiplying constant is fixed by calculating the total energy exchange rate and comparing with the rate computed by other means. In this manner, the F-P operator is derived for electron-ion scattering, electron-electron scattering in the high-velocity limit, electron-atom elastic scattering, Compton scattering, and the high-velocity limit for inelastic scattering. Other processes considered are bremsstrahlung, radiative-recombination, photoionization, collisional ionization of atoms, and suprathermal-particle ionization of atoms.
Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation
Dou, Nicholas G.; Minnich, Austin J.
2016-01-04
Materials that possess low density, low thermal conductivity, and high stiffness are desirable for engineering applications, but most materials cannot realize these properties simultaneously due to the coupling between them. Nanotrusses, which consist of hollow nanoscale beams architected into a periodic truss structure, can potentially break these couplings due to their lattice architecture and nanoscale features. In this work, we study heat conduction in the exact nanotruss geometry by solving the frequency-dependent Boltzmann transport equation using a variance-reduced Monte Carlo algorithm. We show that their thermal conductivity can be described with only two parameters, solid fraction and wall thickness. Our simulations predict that nanotrusses can realize unique combinations of mechanical and thermal properties that are challenging to achieve in typical materials.
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.
Lattice Boltzmann equation method for multiple immiscible continuum fluids
NASA Astrophysics Data System (ADS)
Spencer, T. J.; Halliday, I.; Care, C. M.
2010-12-01
This paper generalizes the two-component algorithm of Sec. , extending it, in Sec. , to describe N>2 mutually immiscible fluids in the isothermal continuum regime. Each fluid has an independent interfacial tension. While retaining all its computational advantages, we remove entirely the empiricism associated with contact behavior in our previous multiple immiscible fluid models [M. M. Dupin , Phys. Rev. E 73, 055701(R) (2006)10.1103/PhysRevE.73.055701; Med. Eng. Phys. 28, 13 (2006)10.1016/j.medengphy.2005.04.015] while solidifying the physical foundations. Moreover, the model relies upon a fluid-fluid segregation which is simpler, computationally faster, more free of artifacts (i.e., the interfacial microcurrent), and upon an interface-inducing force distribution which is analytic. The method is completely symmetric between any numbers of immiscible fluids and stable over a wide range of directly input interfacial tension. We present data on the steady-state properties of multiple interface model, which are in good agreement with theory [R. E. Johnson and S. S. Sadhal, Annu. Rev. Fluid Mech. 17, 289 (1985)10.1146/annurev.fl.17.010185.001445], specifically on the shapes of multidrop systems. Section is an analysis of the kinetic and continuum-scale descriptions of the underlying two-component lattice Boltzmann model for immiscible fluids, extendable to more than two immiscible fluids. This extension requires (i) the use of a more local kinetic equation perturbation which is (ii) free from a reliance on measured interfacial curvature. It should be noted that viewed simply as a two-component method, the continuum algorithm is inferior to our previous methods, reported by Lishchuk [Phys. Rev. E 67, 036701 (2003)]10.1103/PhysRevE.76.036701 and Halliday [Phys. Rev. E 76, 026708 (2007)]10.1103/PhysRevE.76.026708. Greater stability and parameter range is achieved in multiple drop simulations by using the forced multi-relaxation-time lattice Boltzmann method developed
Analytical solutions of the Poisson-Boltzmann equation: biological applications
NASA Astrophysics Data System (ADS)
Fenley, Andrew; Gordon, John; Onufriev, Alexey
2006-03-01
Electrostatic interactions are a key factor for determining many properties of bio-molecules. The ability to compute the electrostatic potential generated by a molecule is often essential in understanding the mechanism behind its biological function such as catalytic activity, ligand binding, and macromolecular association. We propose an approximate analytical solution to the (linearized) Poisson-Boltzmann (PB) equation that is suitable for computing electrostatic potential around realistic biomolecules. The approximation is tested against the numerical solutions of the PB equation on a test set of 600 representative structures including proteins, DNA, and macromolecular complexes. The approach allows one to generate, with the power of a desktop PC, electrostatic potential maps of virtually any molecule of interest, from single proteins to large protein complexes such as viral capsids. The new approach is orders of magnitude less computationally intense than its numerical counterpart, yet is almost equal in accuracy. When studying very large molecular systems, our method is a practical and inexpensive way of computing bio- molecular potential at atomic resolution. We demonstrate the usefullnes of the new approach by exploring the details of electrostatic potentials generated by two of such systems: the nucleosome core particle (25,000 atoms) and tobacco ring spot virus (500,000 atoms). Biologically relevant insights are generated.
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.
High-order regularization in lattice-Boltzmann equations
NASA Astrophysics Data System (ADS)
Mattila, Keijo K.; Philippi, Paulo C.; Hegele, Luiz A.
2017-04-01
A lattice-Boltzmann equation (LBE) is the discrete counterpart of a continuous kinetic model. It can be derived using a Hermite polynomial expansion for the velocity distribution function. Since LBEs are characterized by discrete, finite representations of the microscopic velocity space, the expansion must be truncated and the appropriate order of truncation depends on the hydrodynamic problem under investigation. Here we consider a particular truncation where the non-equilibrium distribution is expanded on a par with the equilibrium distribution, except that the diffusive parts of high-order non-equilibrium moments are filtered, i.e., only the corresponding advective parts are retained after a given rank. The decomposition of moments into diffusive and advective parts is based directly on analytical relations between Hermite polynomial tensors. The resulting, refined regularization procedure leads to recurrence relations where high-order non-equilibrium moments are expressed in terms of low-order ones. The procedure is appealing in the sense that stability can be enhanced without local variation of transport parameters, like viscosity, or without tuning the simulation parameters based on embedded optimization steps. The improved stability properties are here demonstrated using the perturbed double periodic shear layer flow and the Sod shock tube problem as benchmark cases.
L 1 and BV-type stability of the inelastic Boltzmann equation near vacuum
NASA Astrophysics Data System (ADS)
Wu, Zhigang
2010-03-01
The L 1 and BV-type stability to mild solutions of the inelastic Boltzmann equation is given in this paper. The result is an extension of the stability of the classical solution of the elastic Boltzmann equation proved in Ha (Arch. Ration. Mech. Anal. 173:25-42, 2004 [16]). The observation relies on the energy loss of the inelastic Boltzmann equation. This is a continuity work of Alonso (Indiana Univ. Math. J. [1]), where the author obtained the global existence of a mild solution for the inelastic Boltzmann equation. The proof is based on the mollification method and constructing some functionals as the one in Chae and Ha (Contin. Mech. Thermodyn. 17(7):511-524, 2006 [9]).
Global Well-Posedness in Spatially Critical Besov Space for the Boltzmann Equation
NASA Astrophysics Data System (ADS)
Duan, Renjun; Liu, Shuangqian; Xu, Jiang
2016-05-01
The unique global strong solution in the Chemin-Lerner type space to the Cauchy problem on the Boltzmann equation for hard potentials is constructed in a perturbation framework. Such a solution space is of critical regularity with respect to the spatial variable, and it can capture the intrinsic properties of the Boltzmann equation. For the proof of global well-posedness, we develop some new estimates on the nonlinear collision term through the Littlewood-Paley theory.
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.
Shock wave structure using nonlinear model Boltzmann equations
NASA Technical Reports Server (NTRS)
Segal, Ben Maurice
1971-01-01
The structure of a strong plane shock wave in a monatomic rarefied perfect gas is one of the simplest problems able to be posed in kinetic theory, and one of the hardest to solve. Its simplicity lies in the absence of solid boundaries, geometrical complications, or internal molecular energy. Its difficulty arises from the great departure of the gas from equilibrium within the shock, which invalidates many of the techniques used successfully elsewhere in kinetic theory. In addition to this theoretical challenge, the modern development of ballistics and hypersonic flight has helped to stimulate extensive theoretical and experimental interest in the shock problem. The experimenters in turn have encountered great difficulties on account of the very small physical dimensions of shocks. In fact, until very recently indeed, any close comparisons of theoretical and experimental shock structure results have been rather unprofitable due to the inadequacies of both theory and experiment. During the last few years this situation has been appreciably improved by development of the Monte Carlo method. This allows idealized 'experiments' to be performed on large computers instead of in wind tunnels, using a known intermolecular force law. The most developed of these methods has been shown to be equivalent theoretically to the Boltzmann equation and to give results which agree extremely closely with measurements of high accuracy. Thus Monte Carlo results not only form the soundest basis for our present theoretical knowledge of shock wave structure, but, for purposes of developing other theories, can also be considered a very valuable experimental resource. However, such results remain very expensive to obtain. In this thesis we develop more economical kinetic theory methods for the approximate prediction of shock structure, and compare our results with those of the Monte Carlo method.
Recovering Navier-Stokes Equations from Asymptotic Limits of the Boltzmann Gas Mixture Equation
NASA Astrophysics Data System (ADS)
Bianca, Carlo; Dogbe, Christian
2016-05-01
This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles. Specifically the hydrodynamics limit is performed by employing different time and space scalings. The paper shows that, depending on the magnitude of the parameters which define the scaling, the macroscopic quantities (number density, mean velocity and local temperature) are solutions of the acoustic equation, the linear incompressible Euler equation and the incompressible Navier-Stokes equation. The derivation is formally tackled by the recent moment method proposed by [C. Bardos, et al., J. Stat. Phys. 63 (1991) 323] and the results generalize the analysis performed in [C. Bianca, et al., Commun. Nonlinear Sci. Numer. Simulat. 29 (2015) 240].
Solitons of the Kadomtsev-Petviashvili equation based on lattice Boltzmann model
NASA Astrophysics Data System (ADS)
Wang, Huimin
2017-01-01
In this paper, a lattice Boltzmann model for the Kadomtsev-Petviashvili equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales are obtained. Due to the asymmetry in x direction and y direction of the equation, the moments of the equilibrium distribution function are selected are asymmetric. The numerical results demonstrate the lattice Boltzmann method is an effective method to simulate the solitons of the Kadomtsev-Petviashvili equation.
Numerical method based on the lattice Boltzmann model for the Fisher equation.
Yan, Guangwu; Zhang, Jianying; Dong, Yinfeng
2008-06-01
In this paper, a lattice Boltzmann model for the Fisher equation is proposed. First, the Chapman-Enskog expansion and the multiscale time expansion are used to describe higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. Second, the modified partial differential equation of the Fisher equation with the higher-order truncation error is obtained. Third, comparison between numerical results of the lattice Boltzmann models and exact solution is given. The numerical results agree well with the classical ones.
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.
Collisionless relaxation in galactic dynamics and the evolution of long-range order.
NASA Astrophysics Data System (ADS)
Kandrup, H. E.
This paper presents a critical assessment of certain aspects of collisionless galactic dynamics, focusing on the interpretation and limitations of the collisionless Boltzmann equation and the physical mechanisms associated with collisionless, or near-collisionsless, relaxation. Numerical and theoretical arguments are presented to motivate the idea that the evolution of a system far from equilibrium should be interpreted as involving nonlinear gravitational Landau damping, a picture which implies a greater overall coherence and remembrance of initial conditions than is implicit in the conventional paradigm of violent relaxation.
Chapman-Enskog analysis for finite-volume formulation of lattice Boltzmann equation
NASA Astrophysics Data System (ADS)
Patil, D. V.
2013-06-01
The classical Chapman-Enskog expansion is performed for the recently proposed finite-volume formulation of lattice Boltzmann equation (LBE) method [D.V. Patil, K.N. Lakshmisha, Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J. Comput. Phys. 228 (2009) 5262-5279]. First, a modified partial differential equation is derived from a numerical approximation of the discrete Boltzmann equation. Then, the multi-scale, small parameter expansion is followed to recover the continuity and the Navier-Stokes (NS) equations with additional error terms. The expression for apparent value of the kinematic viscosity is derived for finite-volume formulation under certain assumptions. The attenuation of a shear wave, Taylor-Green vortex flow and driven channel flow are studied to analyze the apparent viscosity relation.
Structural interactions in ionic liquids linked to higher-order Poisson-Boltzmann equations.
Blossey, R; Maggs, A C; Podgornik, R
2017-06-01
We present a derivation of generalized Poisson-Boltzmann equations starting from classical theories of binary fluid mixtures, employing an approach based on the Legendre transform as recently applied to the case of local descriptions of the fluid free energy. Under specific symmetry assumptions, and in the linearized regime, the Poisson-Boltzmann equation reduces to a phenomenological equation introduced by Bazant et al. [Phys. Rev. Lett. 106, 046102 (2011)]PRLTAO0031-900710.1103/PhysRevLett.106.046102, whereby the structuring near the surface is determined by bulk coefficients.
NASA Astrophysics Data System (ADS)
Ilyin, O. V.
2016-02-01
We consider the one-dimensional integro-differential Boltzmann equation for Maxwell particles with inelastic collisions. We show that the equation has a five-dimensional algebra of point symmetries for all dissipation parameter values and obtain an optimal system of one-dimensional subalgebras and classes of invariant solutions.
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.
The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential
NASA Astrophysics Data System (ADS)
Liu, Shuangqian; Yang, Xiongfeng
2017-01-01
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.
Solution of the Boltzmann kinetic equation for the relaxation of a gas mixture
NASA Technical Reports Server (NTRS)
Rykov, V. A.; Chukanova, T. I.
1972-01-01
The temporal behavior is considered of a quiescent mixture of gases of different temperatures with spatially uniform distribution. The process of heating a cold gas by a hot gas is treated on the basis of the Boltzmann kinetic equation. The mixture is assumed to be composed of absolutely hard smooth spheres, and the initial distribution functions for each gas is taken to the Maxwellian. With such a choice of initial distribution functions, it is shown that the solution of the Boltzmann kinetic equation depends only on the velocity modulus and the time.
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.
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.
Entropic Lattice Boltzmann Model for Burger’s Equation
2004-05-28
invariant multi- speed entropic lattice Boltzmann models. Physica D. (In the press.) (doi:10.1016/j.physd. 2004.01.018.) Chen, H., Kandasamy , S ., Orszag, S ...61102F 6. AUTHOR( S ) 5d. PROJECT NUMBER B. M. Boghosian*, P. Love*, and J. Yepez 230 So. TASK NUMBER 0T f. WORK UNIT NUMBER Bi 7. PERFORMING...ORGANIZATION NAME( S ) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT Air Force Research Laboratory/VSBYA NUMBER 29 Randolph Road Hanscom AFB MA 01731-3010
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)
Global classical solution to the inelastic Boltzmann equation with potential force
NASA Astrophysics Data System (ADS)
Wang, Xiaolong; Jiang, Zhenglu
2017-08-01
In this paper, motivated by Guo [Commun. Math. Phys. 218(2), 293-313 (2001)], we construct the global classical solution to the inelastic Boltzmann equation with potential force by using the standard contraction mapping theorem. Moreover, the dispersion estimates of the solution are obtained.
Quantum position diffusion and its implications for the quantum linear Boltzmann equation
Kamleitner, I.; Cresser, J.
2010-01-15
We derive a quantum linear Boltzmann equation from first principles to describe collisional friction, diffusion, and decoherence in a unified framework. In doing so, we discover that the previously celebrated quantum contribution to position diffusion is not a true physical process, but rather an artifact of the use of a coarse-grained time scale necessary to derive Markovian dynamics.
Ayik, S. Joint Inst. for Heavy Ion Research, Oak Ridge, TN ); Ivanov, Y.B.; Russkikh, V.N.; Noerenberg, W. )
1993-01-01
A reduction of the relativistic Boltzmann-Langevin Equation (BLE), to a stochastic two-fluid model is presented, and transport coefficients associated with fluid dynamical variables are extracted. The approach is applied to investigate equilibration in a counter-streaming nuclear system.
NASA Astrophysics Data System (ADS)
Ender, I. A.; Bakaleinikov, L. A.; Flegontova, E. Yu.; Gerasimenko, A. B.
2017-08-01
We have proposed an algorithm for the sequential construction of nonisotropic matrix elements of the collision integral, which are required to solve the nonlinear Boltzmann equation using the moments method. The starting elements of the matrix are isotropic and assumed to be known. The algorithm can be used for an arbitrary law of interactions for any ratio of the masses of colliding particles.
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.
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.
Nanoscale Heat Transfer using Phonon Boltzmann Transport Equation
2009-10-01
Fourier diffusive equation ( FDE ). The equation can be derived using a conservation law of energy and Fourier’s linear approximation of heat flux...using a temperature gradient. The FDE is a parabolic equation reflecting a diffusive nature of heat transport. An underlying assumption is that...the heat is effectively transferred between localized regions through sufficient scattering events of phonons within a medium. Therefore, the FDE
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.
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.
Kataoka, Takeshi; Tsutahara, Michihisa
2004-03-01
We have developed a lattice Boltzmann model for the compressible Navier-Stokes equations with a flexible specific-heat ratio. Several numerical results are presented, and they agree well with the corresponding solutions of the Navier-Stokes equations. In addition, an explicit finite-difference scheme is proposed for the numerical calculation that can make a stable calculation with a large Courant number.
Electric-field conditions for Landauer and Boltzmann-Drude conductance equations
NASA Astrophysics Data System (ADS)
Fenton, E. W.
1992-08-01
It is shown explicitly in a unified theory of conductance, for bulk metals and mesoscopic systems, that a Landauer type of conductance equation is compatible with a spatially localized continuous-q-spectrum electric field that is unidirectional, but not with a homogeneous q=0 field. The reverse field condition holds for the Boltzmann-Drude conductance equation for an inhomogeneous bulk metal that has no inelastic scattering. A Feynman-diagram form of Green-function theory shows explicitly the virtual processes and repeated quantum scattering from a single object that occur with Feynman path integrals. The distinction between repeated scattering of current and repeated one-electron scattering is important. For a mesoscopic system, infinite conduction would occur if scattering were to be exactly zero-there is no necessity for postulated contact potentials between lead wires and thermal reservoirs. This is because just in this translationally invariant case a q=0 electric field must occur, and for this the Landauer equation must be replaced by the Boltzmann-Drude equation with zero scattering. In contrast to the strong frequency dependence of the Boltzmann-Drude equation, it is shown that no frequency dependence of the conductance occurs in the Landauer type of equation for frequencies much smaller than the inverse of the electron transit time across the electric-field region.
The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium
NASA Astrophysics Data System (ADS)
Briant, Marc; Daus, Esther S.
2016-12-01
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^∞} 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.
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.
Fokker-Planck-Boltzmann equation for dissipative particle dynamics
NASA Astrophysics Data System (ADS)
Marsh, C. A.; Backx, G.; Ernst, M. H.
1997-05-01
The algorithm for Dissipative Particle Dynamics (DPD), as modified by Español and Warren, is used as a starting point for proving an H-theorem for the free energy and deriving hydrodynamic equations. Equilibrium and transport properties of the DPD fluid are explicitly calculated in terms of the system parameters for the continuous time version of the model.
NASA Astrophysics Data System (ADS)
Barbaroux, Jean-Marie; Hundertmark, Dirk; Ried, Tobias; Vugalter, Semjon
2017-08-01
It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties to the fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties to the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in {L^1_2(R^d)\\cap L log L(R^d)}, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.
NASA Astrophysics Data System (ADS)
Briant, Marc; Einav, Amit
2016-06-01
The Boltzmann-Nordheim equation is a modification of the Boltzmann equation, based on physical considerations, that describes the dynamics of the distribution of particles in a quantum gas composed of bosons or fermions. In this work we investigate the Cauchy theory of the spatially homogeneous Boltzmann-Nordheim equation for bosons, in dimension d≥slant 3. We show existence and uniqueness locally in time for any initial data in L^∞ (1+| v| ^s) with finite mass and energy, for a suitable s, as well as the instantaneous creation of moments of all order.
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.
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.
Well-posedness and Scattering for the Boltzmann Equations: Soft Potential with Cut-off
NASA Astrophysics Data System (ADS)
He, Lingbing; Jiang, Jin-Cheng
2017-07-01
We prove the global existence of the unique mild solution for the Cauchy problem of the cut-off Boltzmann equation for soft potential model γ =2-N with initial data small in L^N_{x,v} where N=2,3 is the dimension. The proof relies on the existing inhomogeneous Strichartz estimates for the kinetic equation by Ovcharov (SIAM J Math Anal 43(3):1282-1310, 2011) and convolution-like estimates for the gain term of the Boltzmann collision operator by Alonso et al. (Commun Math Phys 298:293-322, 2010). The global dynamics of the solution is also characterized by showing that the small global solution scatters with respect to the kinetic transport operator in L^N_{x,v}. Also the connection between function spaces and cut-off soft potential model -N<γ <2-N is characterized in the local well-posedness result for the Cauchy problem with large initial data.
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.
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.
Global classical solutions of the Boltzmann equation with long-range interactions.
Gressman, Philip T; Strain, Robert M
2010-03-30
This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential r(-(p-1)) with p > 2, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.
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.
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.
A Fokker-Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases
NASA Astrophysics Data System (ADS)
Mathiaud, J.; Mieussens, L.
2017-09-01
We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar-Gross-Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.
A Fokker-Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases
NASA Astrophysics Data System (ADS)
Mathiaud, J.; Mieussens, L.
2017-07-01
We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar-Gross-Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.
The Relativistic Boltzmann Equation on Bianchi Type I Space Time for Hard Potentials
NASA Astrophysics Data System (ADS)
Noutchegueme, Norbert; Takou, Etienne; Tchuengue, E. Kamdem
2017-08-01
In this paper, we consider the Cauchy problem for the spatially homogeneous relativistic Boltzmann equation with small initial data. The collision kernel considered here is for a hard potentials case. The background space-time in which the study is done is the Bianchi type I space-time. Under certain conditions made on the scattering kernel and on the metric, a uniqueness global (in time) solution is obtained in a suitable weighted functional space.
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.
Deterministic numerical solutions of the Boltzmann equation using the fast spectral method
NASA Astrophysics Data System (ADS)
Wu, Lei; White, Craig; Scanlon, Thomas J.; Reese, Jason M.; Zhang, Yonghao
2013-10-01
The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard-Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev-Krook-Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss-Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard-Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases
Fokker-Planck equation for Boltzmann-type and active particles: transfer probability approach.
Trigger, S A
2003-04-01
A Fokker-Planck equation with velocity-dependent coefficients is considered for various isotropic systems on the basis of probability transition (PT) approach. This method provides a self-consistent and universal description of friction and diffusion for Brownian particles. Renormalization of the friction coefficient is shown to occur for two-dimensional and three-dimensional cases, due to the tensorial character of diffusion. The specific forms of PT are calculated for Boltzmann-type and absorption-type collisions (the latter are typical in dusty plasmas and some other systems). The validity of the Einstein's relation for Boltzmann-type collisions is analyzed for the velocity-dependent friction and diffusion coefficients. For Boltzmann-type collisions in the region of very high grain velocity as well as it is always for non-Boltzmann collisions, such as, absorption collisions, the Einstein relation is violated, although some other relations (determined by the structure of PT) can exist. The generalized friction force is investigated in dusty plasmas in the framework of the PT approach. The relation among this force, the negative collecting friction force, and scattering and collecting drag forces is established. The concept of probability transition is used to describe motion of active particles in an ambient medium. On basis of the physical arguments, the PT for a simple model of the active particle is constructed and the coefficients of the relevant Fokker-Planck equation are found. The stationary solution of this equation is typical for the simplest self-organized molecular machines.
Bouchard, Hugo; Bielajew, Alex
2015-07-07
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.
Theory of the lattice Boltzmann equation: symmetry properties of discrete velocity sets.
Rubinstein, Robert; Luo, Li-Shi
2008-03-01
The lattice Boltzmann equation replaces continuous particle velocity space by a finite set; the velocity distribution function then varies over a finite-dimensional vector space instead of over an infinite-dimensional function space. The number of linearly independent moments of the distribution function in a lattice Boltzmann model cannot exceed the number of velocities; finite dimensionality therefore necessarily induces linear dependences among the moments that do not exist in a continuous theory. Given a finite velocity set, it is important to know which moments are free of these dependences. Elementary group theory is applied to the solution of this problem. It is found that decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group makes it straightforward to uncover linear dependences among the moments. The construction of some standard two- and three-dimensional models is reviewed from this viewpoint, and procedures for constructing higher-dimensional models are suggested.
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.
From Newton's Law to the Linear Boltzmann Equation Without Cut-Off
NASA Astrophysics Data System (ADS)
Ayi, Nathalie
2017-03-01
We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.
NASA Astrophysics Data System (ADS)
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.
From Newton's Law to the Linear Boltzmann Equation Without Cut-Off
NASA Astrophysics Data System (ADS)
Ayi, Nathalie
2017-01-01
We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.
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.
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.
Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers
NASA Astrophysics Data System (ADS)
Abramov, Rafail V.
2017-10-01
In the current work we construct a multimolecule random process which leads to the Boltzmann equation in the appropriate limit, and which is different from the deterministic real gas dynamics process. We approximate the statistical difference between the two processes via a suitable diffusion process, which is obtained in the multiscale homogenization limit. The resulting Boltzmann equation acquires a new spatially diffusive term, which subsequently manifests in the corresponding fluid dynamics equations. We test the Navier-Stokes and Grad closures of the diffusive fluid dynamics equations in the numerical experiments with the Couette flow for argon and nitrogen, and compare the results with the corresponding Direct Simulation Monte Carlo (DSMC) computations. We discover that the full-fledged Knudsen velocity boundary layers develop with all tested closures when the viscosity and diffusivity are appropriately scaled in the vicinity of the walls. Additionally, we find that the component of the heat flux parallel to the direction of the flow is comparable in magnitude to its transversal component near the walls, and that the nonequilibrium Grad closure approximates this parallel heat flux with good accuracy.
Ibrahim, R. S.; El-Kalaawy, O. H.
2006-10-15
The relativistic nonlinear self-consistent equations for a collisionless cold plasma with stationary ions [R. S. Ibrahim, IMA J. Appl. Math. 68, 523 (2003)] are extended to 3 and 3+1 dimensions. The resulting system of equations is reduced to the sine-Poisson equation. The truncated Painleve expansion and reduction of the partial differential equation to a quadrature problem (RQ method) are described and applied to obtain the traveling wave solutions of the sine-Poisson equation for stationary and nonstationary equations in 3 and 3+1 dimensions describing the charge-density equilibrium configuration model.
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).
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
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
Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff
NASA Astrophysics Data System (ADS)
Morimoto, Yoshinori; Wang, Shuaikun; Yang, Tong
2016-12-01
A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.
Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath
NASA Astrophysics Data System (ADS)
Cañizo, José A.; Lods, Bertrand
2016-05-01
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient α \\in (0,1) ) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which α is close to 1, and the rate of convergence is explicit and depends solely on the spectral gap of the elastic linear collision operator.
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.
NASA Astrophysics Data System (ADS)
Chvala, Frantisek
Subjected to an external electromagnetic field, a rare two-component spatially homogeneous gas consisting of charged and neutral particles is considered. The velocity distribution of the neutral particles being assumed known, the mixture is characterized by the velocity distribution of the charged particles, which is determined as a mild solution of the Boltzmann kinetic equation. Relying upon functional-analytic properties of the collision term, existence and uniqueness of the mild solution are established in some Lebesgue-type function spaces involving exponential weights.
Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation
NASA Astrophysics Data System (ADS)
Khurana, Saheba; Thachuk, Mark
2016-03-01
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.
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.
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.
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 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.
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.
NASA Astrophysics Data System (ADS)
Wang, Huimin
2017-01-01
In this paper, a new lattice Boltzmann model for the Korteweg-de Vries (KdV) equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales and several higher- order moments of equilibrium distribution functions are obtained. In order to make the scheme obey the three conservation laws of the KdV equation, two equilibrium distribution functions are used and a correlation between the first conservation law and the second conservation law is constructed. In numerical examples, the numerical results of the KdV equation obtained by this scheme are compared with those results obtained by the previous lattice Boltzmann model. Numerical experiments demonstrate this scheme can be used to reduce the truncation error of the lattice Boltzmann scheme and preserve the three conservation laws.
NASA Astrophysics Data System (ADS)
Kuang, Yangyu; Tang, Huazhong
2017-06-01
This paper extends the model reduction method by the operator projection to the one-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order globally hyperbolic moment system is built on our careful study of two families of the complicate Grad type orthogonal polynomials depending on a parameter. We derive their recurrence relations, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros and coefficient matrices in the recurrence formulas. Some properties of the moment system are also proved. They include the eigenvalues and their bound as well as eigenvectors, hyperbolicity, characteristic fields, linear stability, and Lorentz covariance. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. The results show that the solutions of our hyperbolic moment system converge to the solution of the special relativistic Boltzmann equation as the order of the hyperbolic moment system increases.
Harris, Robert C; Boschitsch, Alexander H; Fenley, Marcia O
2017-08-08
Many researchers compute surface maps of the electrostatic potential (φ) with the Poisson-Boltzmann (PB) equation to relate the structural information obtained from X-ray and NMR experiments to biomolecular functions. Here we demonstrate that the usual method of obtaining these surface maps of φ, by interpolating from neighboring grid points on the solution grid generated by a PB solver, generates large errors because of the large discontinuity in the dielectric constant (and thus in the normal derivative of φ) at the surface. The Cartesian Poisson-Boltzmann solver contains several features that reduce the numerical noise in surface maps of φ: First, CPB introduces additional mesh points at the Cartesian grid/surface intersections where the PB equation is solved. This procedure ensures that the solution for interior mesh points only references nodes on the interior or on the surfaces; similarly for exterior points. Second, for added points on the surface, a second order least-squares reconstruction (LSR) is implemented that analytically incorporates the discontinuities at the surface. LSR is used both during the solution phase to compute φ at the surface and during postprocessing to obtain φ, induced charges, and ionic pressures. Third, it uses an adaptive grid where the finest grid cells are located near the molecular surface.
Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation.
Harris, Robert C; Boschitsch, Alexander H; Fenley, Marcia O
2014-02-21
Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.
Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation
Harris, Robert C.; Boschitsch, Alexander H.; Fenley, Marcia O.
2014-01-01
Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel. PMID:24559370
Nanoscale roughness effect on Maxwell-like boundary conditions for the Boltzmann equation
Brull, S. Charrier, P. Mieussens, L.
2016-08-15
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.
Thermal transport at the nanoscale: A Fourier's law vs. phonon Boltzmann equation study
NASA Astrophysics Data System (ADS)
Kaiser, J.; Feng, T.; Maassen, J.; Wang, X.; Ruan, X.; Lundstrom, M.
2017-01-01
Steady-state thermal transport in nanostructures with dimensions comparable to the phonon mean-free-path is examined. Both the case of contacts at different temperatures with no internal heat generation and contacts at the same temperature with internal heat generation are considered. Fourier's law results are compared to finite volume method solutions of the phonon Boltzmann equation in the gray approximation. When the boundary conditions are properly specified, results obtained using Fourier's law without modifying the bulk thermal conductivity are in essentially exact quantitative agreement with the phonon Boltzmann equation in the ballistic and diffusive limits. The errors between these two limits are examined in this paper. For the four cases examined, the error in the apparent thermal conductivity as deduced from a correct application of Fourier's law is less than 6%. We also find that the Fourier's law results presented here are nearly identical to those obtained from a widely used ballistic-diffusive approach but analytically much simpler. Although limited to steady-state conditions with spatial variations in one dimension and to a gray model of phonon transport, the results show that Fourier's law can be used for linear transport from the diffusive to the ballistic limit. The results also contribute to an understanding of how heat transport at the nanoscale can be understood in terms of the conceptual framework that has been established for electron transport at the nanoscale.
NASA Astrophysics Data System (ADS)
Huang, Juntao; Hu, Zexi; Yong, Wen-An
2016-04-01
In this paper, we present a kind of second-order curved boundary treatments for the lattice Boltzmann method solving two-dimensional convection-diffusion equations with general nonlinear Robin boundary conditions. The key idea is to derive approximate boundary values or normal derivatives on computational boundaries, with second-order accuracy, by using the prescribed boundary condition. Once the approximate information is known, the second-order bounce-back schemes can be perfectly adopted. Our boundary treatments are validated with a number of numerical examples. The results show the utility of our boundary treatments and very well support our theoretical predications on the second-order accuracy thereof. The idea is quite universal. It can be directly generalized to 3-dimensional problems, multiple-relaxation-time models, and the Navier-Stokes 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.
Lid-driven cavity flow using a discrete velocity method for solving the Boltzmann equation
NASA Astrophysics Data System (ADS)
Sekaran, Aarthi; Varghese, Philip; Estes, Samuel; Goldstein, David
2016-11-01
We extend the discrete velocity method for solving the Boltzmann equation previously used for one-dimensional problems to two spatial dimensions. The collision integral is computed using collisions between velocity classes selected randomly using a Monte Carlo method. Arbitrary post-collision velocities are mapped back onto the grid using a projection scheme which conserves mass, momentum, and energy. In addition, a variance reduction scheme is implemented to decrease noise and further reduce computational effort. The convection part of the equation is computed using first order upwind finite differences. We apply this discrete velocity scheme to the 2D lid-driven square cavity flow problem with Ar as the fluid medium and explore the effect of the additional flexibility available in this quasi-particle based stochastic method on the accuracy and noise level in the solutions obtained.
Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential
Sharipov, Felix Bertoldo, Guilherme
2009-05-20
A numerical procedure to solve the linearized Boltzmann equation with an arbitrary intermolecular potential by the discrete velocity method is elaborated. The equation is written in terms of the kernel, which contains the differential cross section and represents a singularity. As an example, the Lennard-Jones potential is used and the corresponding differential cross section is calculated and tabulated. Then, the kernel is calculated so that to overcome its singularity. Once, the kernel is known and stored it can be used for many kinds of gas flows. In order to test the method, the transport coefficients, i.e. thermal conductivity and viscosity for all noble gases, are calculated and compared with those obtained by the variational method using the Sonine polynomials expansion. The fine agreement between the results obtained by the two different methods shows the feasibility of application of the proposed technique to calculate rarefied gas flows over the whole range of the Knudsen number.
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.
AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation.
Koehl, Patrice; Delarue, Marc
2010-02-14
The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE
Sun, HongGuang; Meerschaert, Mark M.; Zhang, Yong; Zhu, Jianting; Chen, Wen
2013-01-01
The traditional Richards’ equation implies that the wetting front in unsaturated soil follows Boltzmann scaling, with travel distance growing as the square root of time. This study proposes a fractal Richards’ equation (FRE), replacing the integer-order time derivative of water content by a fractal derivative, using a power law ruler in time. FRE solutions exhibit anomalous non-Boltzmann scaling, attributed to the fractal nature of heterogeneous media. Several applications are presented, fitting the FRE to water content curves from previous literature. PMID:23794783
Generalized Kinetic Description of Steady-State Collisionless Plasmas
NASA Technical Reports Server (NTRS)
Khazanov, G. V.; Liemohn, M. W.; Krivorutsky, E. N.
1997-01-01
We present a general solution to the collisionless Boltzmann (Vlasov) equation for a free-flowing plasma along a magnetic field line using Liouville's theorem, allowing for an arbitrary potential structure including non-monotonicities. The constraints of the existing collisionless kinetic transport models are explored, and the need for a more general approach to the problem of self- consistent potential energy calculations is described. Then a technique that handles an arbitrary potential energy distribution along the field line is presented and discussed. For precipitation of magnetospherically trapped hot plasma, this model yields moment calculations that vary by up to a factor of two for various potential energy structures with the same total potential drop. The differences are much greater for the high-latitude outflow scenario, giving order of magnitude variations depending on the shape of the potential energy distribution.
Generalized Kinetic Description of Steady-State Collisionless Plasmas
NASA Technical Reports Server (NTRS)
Khazanov, G. V.; Liemohn, M. W.; Krivorutsky, E. N.
1997-01-01
We present a general solution to the collisionless Boltzmann (Vlasov) equation for a free-flowing plasma along a magnetic field line using Liouville's theorem, allowing for an arbitrary potential structure including non-monotonicities. The constraints of the existing collisionless kinetic transport models are explored, and the need for a more general approach to the problem of self- consistent potential energy calculations is described. Then a technique that handles an arbitrary potential energy distribution along the field line is presented and discussed. For precipitation of magnetospherically trapped hot plasma, this model yields moment calculations that vary by up to a factor of two for various potential energy structures with the same total potential drop. The differences are much greater for the high-latitude outflow scenario, giving order of magnitude variations depending on the shape of the potential energy distribution.
Botello-Smith, Wesley M; Luo, Ray
2015-10-26
Continuum solvent models have been widely used in biomolecular modeling applications. Recently much attention has been given to inclusion of implicit membranes 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 multigrid 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.
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.
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.
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.
NASA Astrophysics Data System (ADS)
Arnab, Sarkar; Manjeet, Singh
2017-02-01
We report spectroscopic studies on plasma electron number density of laser-induced plasma produced by ns-Nd:YAG laser light pulses on an aluminum sample in air at atmospheric pressure. The effect of different laser energy and the effect of different laser wavelengths were compared. The experimentally observed line profiles of neutral aluminum have been used to extract the excitation temperature using the Boltzmann plot method, whereas the electron number density has been determined from the Stark broadened as well as using the Saha-Boltzmann equation (SBE). Each approach was also carried out by using the Al emission line and Mg emission lines. It was observed that the SBE method generated a little higher electron number density value than the Stark broadening method, but within the experimental uncertainty range. Comparisons of N e determined by the two methods show the presence of a linear relation which is independent of laser energy or laser wavelength. These results show the applicability of the SBE method for N e determination, especially when the system does not have any pure emission lines whose electron impact factor is known. Also use of Mg lines gives superior results than Al lines.
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
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
NASA Astrophysics Data System (ADS)
Farrell, Patricio; Koprucki, Thomas; Fuhrmann, Jürgen
2017-10-01
We compare three thermodynamically consistent numerical fluxes known in the literature, appearing in a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics. Our discussion includes an extension of the Scharfetter-Gummel scheme to non-Boltzmann (e.g. Fermi-Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.
NASA Astrophysics Data System (ADS)
Hong, Liu; Yang, Zaibao; Zhu, Yi; Yong, Wen-An
2015-12-01
In this article, we propose a novel approach to construct macroscopic balance equations and constitutive equations describing various irreversible phenomena. It is based on the general principles of non-equilibrium thermodynamics and consists of four basic steps: picking suitable state variables, choosing a strictly concave entropy function, properly separating entropy fluxes and production rates, and determining a dissipation matrix. Our approach takes advantage of both extended irreversible thermodynamics and GENERIC formalisms and shows a direct correspondence with Levermore's moment-closure hierarchies for the Boltzmann equation. As a direct application, a new ten-moment model beyond the classical hierarchies is constructed and is shown to recover the Euler equations in the equilibrium state. These interesting results may put various macroscopic modeling approaches, starting from the general principles of non-equilibrium thermodynamics, on a solid microscopic foundation based on the Boltzmann equation.
NASA Astrophysics Data System (ADS)
Chiloyan, Vazrik; Zeng, Lingping; Huberman, Samuel; Maznev, Alexei A.; Nelson, Keith A.; Chen, Gang
2016-04-01
The phonon Boltzmann transport equation (BTE) is a powerful tool for studying nondiffusive thermal transport. Here, we develop a new universal variational approach to solving the BTE that enables extraction of phonon mean free path (MFP) distributions from experiments exploring nondiffusive transport. By utilizing the known Fourier heat conduction solution as a trial function, we present a direct approach to calculating the effective thermal conductivity from the BTE. We demonstrate this technique on the transient thermal grating experiment, which is a useful tool for studying nondiffusive thermal transport and probing the MFP distribution of materials. We obtain a closed form expression for a suppression function that is materials dependent, successfully addressing the nonuniversality of the suppression function used in the past, while providing a general approach to studying thermal properties in the nondiffusive regime.
Raman scattering in a two-dimensional electron gas: Boltzmann equation approach
NASA Astrophysics Data System (ADS)
Mishchenko, E. G.
1999-06-01
The inelastic light scattering in a two-dimensional electron gas is studied theoretically using the Boltzmann equation techniques. Electron-hole excitations produce the Raman spectrum essentially different from the one predicted for the 3D case. In the clean limit it has the form of a strong nonsymmetric resonance due to the square-root singularity at the electron-hole frequency ω=vk, while in the opposite dirty limit the usual Lorentzian shape of the cross section is reestablished. The effects of electromagnetic field are considered self-consistently, and the contribution from collective plasmon modes is found. It is shown that unlike 3D metals where plasmon excitations are unobservable (because of very large required transferred frequencies), the two-dimensional electron system gives rise to a low-frequency (ω~k1/2) plasmon peak. A measurement of the width of this peak can provide data on the magnitude of the electron-scattering rate.
Membrane potential and ion partitioning in an erythrocyte using the Poisson-Boltzmann equation.
Barbosa, Nathalia S V; Lima, Eduardo R A; Boström, Mathias; Tavares, Frederico W
2015-05-28
In virtually all mammal cells, we can observe a much higher concentration of potassium ions inside the cell and vice versa for sodium ions. Classical theories ignore the specific ion effects and the difference in the thermodynamic reference states between intracellular and extracellular environments. Usually, this differential ion partitioning across a cell membrane is attributed exclusively to the active ion transport. Our aim is to investigate how much the dispersion forces contribute to active ion pumps in an erythrocyte (red blood cell) as well as the correction of chemical potential reference states between intracellular and extracellular environments. The ionic partition and the membrane potential in an erythrocyte are analyzed by the modified Poisson-Boltzmann equation, considering nonelectrostatic interactions between ions and macromolecules. Results show that the nonelectrostatic potential calculated by Lifshitz theory has only a small influence with respect to the high concentration of K(+) in the intracellular environment in comparison with Na(+).
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.
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.
The electron Boltzmann equation in a plasma generated by fission fragments
NASA Technical Reports Server (NTRS)
Hassan, H. A.; Deese, J. E.
1976-01-01
A Boltzmann equation formulation is presented for the determination of the electron distribution function in a plasma generated by fission fragments. The formulation takes into consideration ambipolar diffusion, elastic and inelastic collisions, recombination and ionization, and allows for the fact that the primary electrons are not monoenergetic. Calculations for He in a tube coated with fissionable material show that, over a wide pressure and neutron flux range, the distribution function is non-Maxwellian, but the electrons are essentially thermal. Moreover, about a third of the energy of the primary electrons is transferred into the inelastic levels of He. This fraction of energy transfer is almost independent of pressure and neutron flux but increases sharply in the presence of a sustainer electric field.
Multiple-component lattice Boltzmann equation for fluid-filled vesicles in flow
NASA Astrophysics Data System (ADS)
Halliday, I.; Lishchuk, S. V.; Spencer, T. J.; Pontrelli, G.; Care, C. M.
2013-02-01
We document the derivation and implementation of extensions to a two-dimensional, multicomponent lattice Boltzmann equation model, with Laplace law interfacial tension. The extended model behaves in such a way that the boundary between its immiscible drop and embedding fluid components can be shown to describe a vesicle of constant volume bounded by a membrane with conserved length, specified interface compressibility, bending rigidity, preferred curvature, and interfacial tension. We describe how to apply this result to several, independent vesicles. The extended scheme is completely Eulerian, and it represents a two-way coupled vesicle membrane and flow within a single framework. Unlike previous methods, our approach dispenses entirely with the need explicitly to track the membrane, or boundary, and makes no use whatsoever of computationally expensive and intricate interface tracking and remeshing. Validation data are presented, which demonstrate the utility of the method in the simulation of the flow of high volume fraction suspensions of deformable objects.
dugksFoam: An open source OpenFOAM solver for the Boltzmann model equation
NASA Astrophysics Data System (ADS)
Zhu, Lianhua; Chen, Songze; Guo, Zhaoli
2017-04-01
A deterministic Boltzmann model equation solver called dugksFoam has been developed in the framework of the open source CFD toolbox OpenFOAM. The solver adopts the discrete unified gas kinetic scheme (Guo et al., 2015) with the Shakhov collision model. It has been validated by simulating several test cases covering different flow regimes including the one dimensional shock tube problem, a two dimensional thermal induced flow and the three dimensional lid-driven cavity flow. The solver features a parallel computing ability based on the velocity space decomposition, which is different from the physical space decomposition based approach provided by the OpenFOAM framework. The two decomposition approaches have been compared in both two and three dimensional cases. The parallel performance improves significantly using the newly implemented approach. A speed up by two orders of magnitudes has been observed using 256 cores on a small cluster.
On Weak Solutions to the Linear Boltzmann Equation with Inelastic Coulomb Collisions
Pettersson, Rolf
2011-05-20
This paper considers the time- and space-dependent linear Boltzmann equation with general boundary conditions in the case of inelastic (granular) collisions. First, in the (angular) cut-off case, mild L{sup 1}-solutions are constructed as limits of the iterate functions and boundedness of higher velocity moments are discussed in the case of inverse power collisions forces. Then the problem of the weak solutions, as weak limit of a sequence of mild solutions, is studied for a bounded body, in the case of very soft interactions (including the Coulomb case). Furthermore, strong convergence of weak solutions to the equilibrium, when time goes to infinity, is discussed, using a generalized H-theorem, together with a translation continuity property.
Multiple-component lattice Boltzmann equation for fluid-filled vesicles in flow.
Halliday, I; Lishchuk, S V; Spencer, T J; Pontrelli, G; Care, C M
2013-02-01
We document the derivation and implementation of extensions to a two-dimensional, multicomponent lattice Boltzmann equation model, with Laplace law interfacial tension. The extended model behaves in such a way that the boundary between its immiscible drop and embedding fluid components can be shown to describe a vesicle of constant volume bounded by a membrane with conserved length, specified interface compressibility, bending rigidity, preferred curvature, and interfacial tension. We describe how to apply this result to several, independent vesicles. The extended scheme is completely Eulerian, and it represents a two-way coupled vesicle membrane and flow within a single framework. Unlike previous methods, our approach dispenses entirely with the need explicitly to track the membrane, or boundary, and makes no use whatsoever of computationally expensive and intricate interface tracking and remeshing. Validation data are presented, which demonstrate the utility of the method in the simulation of the flow of high volume fraction suspensions of deformable objects.
High throughput solution of Boltzmann transport equation: phonons, thermal conductivity and beyond
NASA Astrophysics Data System (ADS)
Plata, Jose; Nath, Pinku; Usanmaz, Demet; Toher, Cormac; Fornari, Marco; Buongiorno Nardelli, Marco; Curtarolo, Stefano
Quantatively accurate predictions of the lattice thermal conductivity have important implications for key technologies ranging from thermoelectrics to thermal barrier coatings. Of the many approaches with varying computational costs and accuracy, which have been developed in the last years, the solution of the Boltzmann transport equation (BTE) is the only approach that guarantees accurate predictions of this property. We have implemented this methodology in the AFLOW high throughput materials science framework, which enables us to compute these anharmonic force constants and solve BTE to obtain the lattice thermal conductivity and related properties automatically in a single step. This technique can be combined with less expensive methodologies previously implemented in AFLOW to create an efficient and fast framework to accelerate the discovery of materials with interesting thermal properties.
Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium
Cai, W.; Lax, M.; Alfano, R. R.
2000-04-01
We consider an analytical solution of the time-dependent elastic Boltzmann transport equation in an infinite uniform isotropic medium with an arbitrary phase function. We obtain (1) the exact distribution in angle, (2) the exact first and second spatial cumulants at any angle, and (3) an approximate combined distribution in position and angle and a spatial distribution whose central position and half-width of spread are always exact. The resulting Gaussian distribution has a center that advances in time, and an ellipsoidal contour that grows and changes shape providing a clear picture of the time evolution of the particle migration from near ballistic, through snakelike and into the final diffusive regime. (c) 2000 The American Physical Society.
Numerical simulation of flow around rectangular cylinders using the Boltzmann equation
NASA Astrophysics Data System (ADS)
Rovenskaya, O. I.; Aristov, V. V.
2016-11-01
A two-dimensional unsteady flow past a rectangular cylinder has been investigated numerically using the Boltzmann equation. The effect of cylinder aspect ratio varying from 1 to 10 and the flow Reynolds number from 10 to 400 on flow pattern has been analyzed. Results are presented in terms of drag, lift and pressure coefficients and Strouhal number of vortex shedding. Flow visualization images in the wake of the cylinder are shown and discussed. It was found that the shape and size of the recirculation bubble downstream of the cylinder are strong functions of its aspect ratio. Drag, lift, pressure coefficients and Strouhal number strongly depend on Re in steady regime while a dependence becomes weaker in unsteady flow regime. The flow configuration over a cylinder also varies with the aspect ratio. In addition, the present predictions are compared with numerical and experimental results from other works and a good agreement is reached.
Simulation of nonequilibrium turbulent flows on the basis of the Boltzmann equation
NASA Astrophysics Data System (ADS)
Aristov, V. V.; Rovenskaya, O. I.
2012-11-01
The present paper is devoted to simulating the compressible turbulent flows at the new level. The principal importance of the kinetic approach for description of turbulence is discussed. An explicit-implicit numerical scheme in the framework of the direct approach for the Boltzmann kinetic equation is examined. Numerical investigation of supersonic jets and the supersonic turbulent separated flow in the vicinity of backward/forward-facing step with leeward-face angle equals to 8°, 25°, 45° and 90° and Ma ≈ 3 for a model 2D case is made. At lower Knudsen number values (starting from 0.001) instabilities in the jet flow is observed. The comparison with available experimental and numerical results is performed in terms of the pressure and skin friction distributions over the body surface and the distribution of heat-transfer coefficient. A good agreement with the experimental heat-transfer coefficient is observed.
NASA Astrophysics Data System (ADS)
Bazow, D.; Denicol, G. S.; Heinz, U.; Martinez, M.; Noronha, J.
2016-12-01
The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of nonhydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the nonhydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation. However, the latter probes additional high-momentum details unresolved by the relaxation time approximation. While the expansion of the FLRW spacetime is slow enough for the system to move towards (and not away from) local thermal equilibrium, it is not sufficiently slow for the system to actually ever reach complete local equilibrium. Equilibration is fastest in the relaxation time approximation, followed, in turn, by kinetic evolution with a linearized and a fully nonlinear Boltzmann collision term.
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.
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.
1989-12-01
Numerical Solution ....... ....................... 113 E.1 Gauss- Jordan .......................................... 113 E.2 L-U Decomposition...2.4 Numerical Solution of the Boltzmann Equation Four numerical were used to solve equation (39). These were: : Gauss- Jordan , L-U Decom- position...both the Gauss- Jordan and L-U decomposition methods. 2.5 Transport Coefficiente In order to provide the required input to the laser design program CO2OSC
Verschaeve, Joris C G
2011-06-13
By means of the continuity equation of the incompressible Navier-Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.
Ion strength limit of computed excess functions based on the linearized Poisson-Boltzmann equation.
Fraenkel, Dan
2015-12-05
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. © 2015 Wiley Periodicals, Inc.
Asymptotic analysis of lattice Boltzmann method to solveconvection-diffusion equation
NASA Astrophysics Data System (ADS)
Hou, Y.; Jiang, J.; Wu, J.
2016-12-01
Convection-diffusion (CDE) is widely used to describe the transport of solute in physics and hydrology. However, numerical dispersion and oscillation may appear especially when advection process dominates transport .In the past few years lattice Boltzmann method has emerged as an efficient numerical method to solve convection-diffusion equation. LBM can simultaneously eliminate numerical dispersion and oscillation both in advection-dominated problems and in dispersion-dominated problems. To solve CDE by LBM, the key step is to derive the CDE from the BGK lattice Boltzmann model. The usual derivation method in literature is Chapman-Enskog multi-scale expansion. Though Chapman-Enskog multi-scale expansion can derive CDE from LBM, the asymptotic behavior of the error has not been clearly studied. As illustrated in this paper, the error of LBM at beginning is larger than the error at later time.Therefore, the asymptotic analysis of LBM is important when LBM is used to solve CDE. In this paper we propose an intuitive method to derive DLBM, i.e. the diffusion coefficient of LBM. This diffusion coefficient is a function of time and clearly shows the asymptotic behavior of LBM. DLBM approaches to the diffusion coefficient derived by multi-scale expansion as simulation time increases. The difference between them is inversely proportional to the simulation time. Then, we used numerical examples to test our derivation. The numerical results show that the concentration simulated by LBM tends to analytical solution of CDE with a relative error which is approximately proportional to the inverse of time. This asymptotic behavior guarantees the accuracy of LBM when using it to solve CDE. Also, our derivation does not need the hypothesis of two scales of time (the basis for multi-scale expansion method) and enables us understand LBM in a more intuitive way.
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
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
Robson, R E; Winkler, R; Sigeneger, F
2002-05-01
The Boltzmann equation corresponding to a general "multiterm" representation of the phase space distribution function f(r,c,t) for charged particles in a gas in an electric field was reformulated entirely in terms of spherical tensors f(l)(m) some time ago, and numerous applications, including extension to time varying and crossed electric and magnetic fields, have followed. However, these applications have, by and large, been limited to the hydrodynamic conditions that prevail in swarm experiments and the full potential of the tensor formalism has thus never been realized. This paper resumes the discussion in the context of the more general nonhydrodynamic situation. Geometries for which a simple Legendre polynomial expansion suffices to represent f are discussed briefly, but the emphasis is upon cylindrical geometry, where such simplification does not arise. In particular, we consider an axisymmetric cylindrical column of weakly ionized plasma, and derive an infinite hierarchy of integrodifferential equations for the expansion coefficients of the phase space distribution function, valid for both electrons and ions, and for all types of binary interaction with neutral gas molecules.
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.
Asymptotic analysis of the Poisson-Boltzmann equation describing electrokinetics in porous media
NASA Astrophysics Data System (ADS)
Allaire, Grégoire; Dufrêche, Jean-François; Mikelić, Andro; Piatnitski, Andrey
2013-03-01
We consider the Poisson-Boltzmann equation in a periodic cell, representative of a porous medium. It is a model for the electrostatic distribution of N chemical species diluted in a liquid at rest, occupying the pore space with charged solid boundaries. We study the asymptotic behaviour of its solution depending on a parameter β, which is the square of the ratio between a characteristic pore length and the Debye length. For small β we identify the limit problem which is still a nonlinear Poisson equation involving only one species with maximal valence, opposite to the average of the given surface charge density. This result justifies the Donnan effect, observing that the ions for which the charge is that of the solid phase are expelled from the pores. For large β we prove that the solution behaves like a boundary layer near the pore walls and is constant far away in the bulk. Our analysis is valid for Neumann boundary conditions (namely for imposed surface charge densities) and establishes rigorously that solid interfaces are uncoupled from the bulk fluid so that simplified additive theories, such as the popular Derjaguin, Landau, Verwey and Overbeek approach, can be used. We show that the asymptotic behaviour is completely different in the case of Dirichlet boundary conditions (namely for imposed surface potential).
NASA Astrophysics Data System (ADS)
Collins, Kimberlee C.; Maznev, Alexei A.; Tian, Zhiting; Esfarjani, Keivan; Nelson, Keith A.; Chen, Gang
2013-09-01
The relaxation of an one-dimensional transient thermal grating (TTG) in a medium with phonon-mediated thermal transport is analyzed within the framework of the Boltzmann transport equation (BTE), with the goal of extracting phonon mean free path (MFP) information from TTG measurements of non-diffusive phonon transport. Both gray-medium (constant MFP) and spectrally dependent MFP models are considered. In the gray-medium approximation, an analytical solution is derived. For large TTG periods compared to the MFP, the model yields an exponential decay of grating amplitude with time in agreement with Fourier's heat diffusion equation, and at shorter periods, phonon transport transitions to the ballistic regime, with the decay becoming strongly non-exponential. Spectral solutions are obtained for Si and PbSe at 300 K using phonon dispersion and lifetime data from density functional theory calculations. The spectral decay behaviors are compared to several approximate models: a single MFP solution, a frequency-integrated gray-medium model, and a "two-fluid" BTE solution. We investigate the utility of using the approximate models for the reconstruction of phonon MFP distributions from non-diffusive TTG measurements.
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.
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.
NASA Astrophysics Data System (ADS)
Khajepor, Sorush; Chen, Baixin
2016-01-01
A method is developed to analytically and consistently implement cubic equations of state into the recently proposed multipseudopotential interaction (MPI) scheme in the class of two-phase lattice Boltzmann (LB) models [S. Khajepor, J. Wen, and B. Chen, Phys. Rev. E 91, 023301 (2015)], 10.1103/PhysRevE.91.023301. An MPI forcing term is applied to reduce the constraints on the mathematical shape of the thermodynamically consistent pseudopotentials; this allows the parameters of the MPI forces to be determined analytically without the need of curve fitting or trial and error methods. Attraction and repulsion parts of equations of state (EOSs), representing underlying molecular interactions, are modeled by individual pseudopotentials. Four EOSs, van der Waals, Carnahan-Starling, Peng-Robinson, and Soave-Redlich-Kwong, are investigated and the results show that the developed MPI-LB system can satisfactorily recover the thermodynamic states of interest. The phase interface is predicted analytically and controlled via EOS parameters independently and its effect on the vapor-liquid equilibrium system is studied. The scheme is highly stable to very high density ratios and the accuracy of the results can be enhanced by increasing the interface resolution. The MPI drop is evaluated with regard to surface tension, spurious velocities, isotropy, dynamic behavior, and the stability dependence on the relaxation time.
Khajepor, Sorush; Chen, Baixin
2016-01-01
A method is developed to analytically and consistently implement cubic equations of state into the recently proposed multipseudopotential interaction (MPI) scheme in the class of two-phase lattice Boltzmann (LB) models [S. Khajepor, J. Wen, and B. Chen, Phys. Rev. E 91, 023301 (2015)]10.1103/PhysRevE.91.023301. An MPI forcing term is applied to reduce the constraints on the mathematical shape of the thermodynamically consistent pseudopotentials; this allows the parameters of the MPI forces to be determined analytically without the need of curve fitting or trial and error methods. Attraction and repulsion parts of equations of state (EOSs), representing underlying molecular interactions, are modeled by individual pseudopotentials. Four EOSs, van der Waals, Carnahan-Starling, Peng-Robinson, and Soave-Redlich-Kwong, are investigated and the results show that the developed MPI-LB system can satisfactorily recover the thermodynamic states of interest. The phase interface is predicted analytically and controlled via EOS parameters independently and its effect on the vapor-liquid equilibrium system is studied. The scheme is highly stable to very high density ratios and the accuracy of the results can be enhanced by increasing the interface resolution. The MPI drop is evaluated with regard to surface tension, spurious velocities, isotropy, dynamic behavior, and the stability dependence on the relaxation time.
A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations
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
NASA Astrophysics Data System (ADS)
Suga, Shinsuke
2010-08-01
An accurate and unconditionally stable explicit finite difference scheme for 1D diffusion equations is derived from the lattice Boltzmann method with rest particles. The system of the lattice Boltzmann equations for the distribution of the number of the fictitious particles is rewritten as a four-level explicit finite difference equation for the concentration of the diffused matter with two parameters. The consistency analysis of the four-level scheme shows that the two parameters which appear in the scheme, the relaxation parameter and the amount of rest particles, can be determined such that the scheme has the truncation error of fourth order. Numerical experiments demonstrate the fourth-order rate of convergence for various combinations of model parameters.
Yoshimura, Kazuyoshi; Kuwabara, Sinzi
2011-05-20
Relaxation phenomena in the binary gas-mixture with different temperature and different velocities are discussed on the basis of two Boltzmann equations. The Hermite expansion method, extended by H.Grad to multidimensional space, is applied to express distribution functions and the Galerkin method is used to solve two Boltzmann equations. Thus, a system of differential equations for the expansion coefficients is obtained. The time development of the system is calculated numerically.
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
New insights into self-heating in double-gate transistors by solving Boltzmann transport equations
Thu Trang Nghiêm, T.; Saint-Martin, J.; Dollfus, P.
2014-08-21
Electro-thermal effects become one of the most critical issues for continuing the downscaling of electron devices. To study this problem, a new efficient self-consistent electron-phonon transport model has been developed. Our model of phonon Boltzmann transport equation (pBTE) includes the decay of optical phonons into acoustic modes and a generation term given by electron-Monte Carlo simulation. The solution of pBTE uses an analytic phonon dispersion and the relaxation time approximation for acoustic and optical phonons. This coupled simulation is applied to investigate the self-heating effects in a 20 nm-long double gate MOSFET. The temperature profile per mode and the comparison between Fourier temperature and the effective temperature are discussed. Some significant differences occur mainly in the hot spot region. It is shown that under the influence of self-heating effects, the potential profile is modified and both the drain current and the electron ballisticity are reduced because of enhanced electron-phonon scattering rates.
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
Shrinkage of bubbles and drops in the lattice Boltzmann equation method for nonideal gases
NASA Astrophysics Data System (ADS)
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.
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.
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.
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.
Analysis of force treatment in the pseudopotential lattice Boltzmann equation method
NASA Astrophysics Data System (ADS)
Zheng, Lin; Zhai, Qinglan; Zheng, Song
2017-04-01
In this paper, different force treatments are analyzed in detail for a pseudopotential lattice Boltzmann equation (LBE), and the contribution of third-order error terms to pressure tensor with a force scheme is analyzed by a higher-order Chapman-Enskog expansion technique. From the theoretical analysis, the performance of the original force treatment of Shan-Chen (SC), Ladd, Guo et al., and the exact difference method (EDM) are ɛLadd<ɛGuo<ɛEDM≤ɛSC with the relaxation time τ ≥1 , while ɛLadd<ɛGuo<ɛSC<ɛEDM with τ <1 ; here ɛ is a parameter related to the mechanical stability and the subscripts are the corresponding force scheme. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to show that the predictions of modified force treatment of the pseudopotential LBE are all in good agreement with the analytical solution and other predictions.
pK(A) in proteins solving the Poisson-Boltzmann equation with finite elements.
Sakalli, Ilkay; Knapp, Ernst-Walter
2015-11-05
Knowledge on pK(A) values is an eminent factor to understand the function of proteins in living systems. We present a novel approach demonstrating that the finite element (FE) method of solving the linearized Poisson-Boltzmann equation (lPBE) can successfully be used to compute pK(A) values in proteins with high accuracy as a possible replacement to finite difference (FD) method. For this purpose, we implemented the software molecular Finite Element Solver (mFES) in the framework of the Karlsberg+ program to compute pK(A) values. This work focuses on a comparison between pK(A) computations obtained with the well-established FD method and with the new developed FE method mFES, solving the lPBE using protein crystal structures without conformational changes. Accurate and coarse model systems are set up with mFES using a similar number of unknowns compared with the FD method. Our FE method delivers results for computations of pK(A) values and interaction energies of titratable groups, which are comparable in accuracy. We introduce different thermodynamic cycles to evaluate pK(A) values and we show for the FE method how different parameters influence the accuracy of computed pK(A) values.
NASA Astrophysics Data System (ADS)
Halliday, I.; Lishchuk, S. V.; Spencer, T. J.; Burgin, K.; Schenkel, T.
2017-10-01
We describe, analyse and reduce micro-current effects in one class of lattice Boltzmann equation simulation method describing immiscible fluids within the continuum approximation, due to Lishchuk et al. (2003). This model's micro-current flow field and associated density adjustment, when considered in the linear, low-Reynolds number regime, may be decomposed into independent, superposable contributions arising from various error terms in its immersed boundary force. Error force contributions which are rotational (solenoidal) are mainly responsible for the micro-current (corresponding density adjustment). Rotationally anisotropic error terms arise from numerical derivatives and from the sampling of the interface-supporting force. They may be removed, either by eliminating the causal error force or by negating it. It is found to be straightforward to design more effective stencils with significantly improved performance. Practically, the micro-current activity arising in Lishchuk's method is reduced by approximately three quarters by using an appropriate stencil and approximately by an order of magnitude when the effects of sampling are removed.
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.
Hurtado, Pablo I
2005-10-01
We investigate the nonequilibrium behavior of a one-dimensional binary fluid on the basis of Boltzmann equation, using an infinitely strong shock wave as probe. Density, velocity, and temperature profiles are obtained as a function of the mixture mass ratio mu. We show that temperature overshoots near the shock layer, and that heavy particles are denser, slower, and cooler than light particles in the strong nonequilibrium region around the shock. The shock width omega(mu), which characterizes the size of this region, decreases as omega(mu) approximately mu(1/3) for mu-->0. In this limit, two very different length scales control the fluid structure, with heavy particles equilibrating much faster than light ones. Hydrodynamic fields relax exponentially toward equilibrium: phi(chi) approximately exp[-chi/lambda]. The scale separation is also apparent here, with two typical scales, lambda1 and lambda2, such that lambda1 approximately mu(1/2 as mu-->0, while lambda2, which is the slow scale controlling the fluid's asymptotic relaxation, increases to a constant value in this limit. These results are discussed in light of recent numerical studies on the nonequilibrium behavior of similar one-dimensional binary fluids.
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
Interface-capturing lattice Boltzmann equation model for two-phase flows
NASA Astrophysics Data System (ADS)
Lou, Qin; Guo, Zhaoli
2015-01-01
In this work, an interface-capturing lattice Boltzmann equation (LBE) model is proposed for two-phase flows. In the model, a Lax-Wendroff propagation scheme and a properly chosen equilibrium distribution function are employed. The Lax-Wendroff scheme is used to provide an adjustable Courant-Friedrichs-Lewy (CFL) number, and the equilibrium distribution is presented to remove the dependence of the relaxation time on the CFL number. As a result, the interface can be captured accurately by decreasing the CFL number. A theoretical expression is derived for the chemical potential gradient by solving the LBE directly for a two-phase system with a flat interface. The result shows that the gradient of the chemical potential is proportional to the square of the CFL number, which explains why the proposed model is able to capture the interface naturally with a small CFL number, and why large interface error exists in the standard LBE model. Numerical tests, including a one-dimensional flat interface problem, a two-dimensional circular droplet problem, and a three-dimensional spherical droplet problem, demonstrate that the proposed LBE model performs well and can capture a sharp interface with a suitable CFL number.
Interface-capturing lattice Boltzmann equation model for two-phase flows.
Lou, Qin; Guo, Zhaoli
2015-01-01
In this work, an interface-capturing lattice Boltzmann equation (LBE) model is proposed for two-phase flows. In the model, a Lax-Wendroff propagation scheme and a properly chosen equilibrium distribution function are employed. The Lax-Wendroff scheme is used to provide an adjustable Courant-Friedrichs-Lewy (CFL) number, and the equilibrium distribution is presented to remove the dependence of the relaxation time on the CFL number. As a result, the interface can be captured accurately by decreasing the CFL number. A theoretical expression is derived for the chemical potential gradient by solving the LBE directly for a two-phase system with a flat interface. The result shows that the gradient of the chemical potential is proportional to the square of the CFL number, which explains why the proposed model is able to capture the interface naturally with a small CFL number, and why large interface error exists in the standard LBE model. Numerical tests, including a one-dimensional flat interface problem, a two-dimensional circular droplet problem, and a three-dimensional spherical droplet problem, demonstrate that the proposed LBE model performs well and can capture a sharp interface with a suitable CFL number.
NASA Astrophysics Data System (ADS)
Wang, Shyh-Wei; Guo, Shuang-Fa
1998-01-01
New techniques for more accurate and efficient simulation of ion implantations by a stepwise numerical integration of the Boltzmann transport equation (BTE) have been developed in this work. Instead of using uniform energy grid, a non-uniform grid is employed to construct the momentum distribution matrix. A more accurate simulation result is obtained for heavy ions implanted into silicon. In the same time, rather than utilizing the conventional Lindhard, Nielsen and Schoitt (LNS) approximation, an exact evaluation of the integrals involving the nuclear differential scattering cross-section (dσn=2πp dp) is proposed. The impact parameter p as a function of ion energy E and scattering angle φ is obtained by solving the magic formula iteratively and an interpolation techniques is devised during the simulation process. The simulation time using exact evaluation is about 3.5 times faster than that using the Littmark and Ziegler (LZ) spline fitted cross-section function for phosphorus implantation into silicon.
Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions.
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)JCTPAH0021-999110.1016/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.
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.
A new splitting scheme to the discrete Boltzmann equation for non-ideal gases on non-uniform meshes
NASA Astrophysics Data System (ADS)
Patel, Saumil; Lee, Taehun
2016-12-01
We present a novel numerical procedure for solving the discrete Boltzmann equations (DBE) on non-uniform meshes. Our scheme is based on the Strang splitting method where we seek to investigate two-phase flow applications. In this note, we investigate the onset of parasitic currents which arise in many computational two-phase algorithms. To the best of our knowledge, the results presented in this work show, for the first time, a spectral element discontinuous Galerkin (SEDG) discretization of a discrete Boltzmann equation which successfully eliminates parasitic currents on non-uniform meshes. With the hope that this technique can be used for applications in complex geometries, calculations are performed on non-uniform mesh distributions by using high-order (spectral), body-fitting quadrilateral elements. Validation and verification of our work is carried out by comparing results against the classical 2D Young-Laplace law problem for a static drop.
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
Solving the Vlasov equation in general relativity
NASA Astrophysics Data System (ADS)
Rasio, Frederic A.; Shapiro, Stuart L.; Teukolsky, Saul A.
1989-09-01
A new numerical method for determining the dynamical evolution of a collisionless system in full general relativity is developed. The method exploits Liouville's theorem to determine the evolution of the distribution function of matter in phase space directly. The distribution function is governed by the collisionless Boltzmann (Vlasov) equation coupled to Einstein's equations for the gravitational field. The method accurately tracks the increasingly complicated fine-grained structure developed by the distribution function due to phase mixing. It can be used to study Newtonian as well as fully relativistic systems. Applications include violent relaxation, the stability of relativistic star clusters, and the collapse of unstable relativistic star clusters to black holes.
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)
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.
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
Wu, Zhen; Zhang, Xian; Zhou, Chunjiao; Pang, Jing-Lin; Zhang, Panyue
2017-02-22
Single-molecule aluminum salt AlCl3, medium polymerized polyaluminum chloride (PAC), and high polymerized polyaluminum chloride (HPAC) were prepared in a laboratory. The characteristics and coagulation properties of these prepared aluminum salts were investigated. The Langmuir, Freundlich, and Sips adsorption isotherms were first used to describe the adsorption neutralization process in coagulation, and the Boltzmann equation was used to fit the reaction kinetics of floc growth in flocculation. It was novel to find that the experimental data fitted well with the Sips and Boltzmann equation, and the significance of parameters in the equations was discussed simultaneously. Through the Sips equation, the adsorption neutralization reaction was proved to be spontaneous and the adsorption neutralization capacity was HPAC > PAC > AlCl3. Sips equation also indicated that the zeta potential of water samples would reach a limit with the increase of coagulant dosage, and the equilibrium zeta potential values were 30.25, 30.23, and 27.25 mV for AlCl3, PAC, and HPAC, respectively. The lower equilibrium zeta potential value of HPAC might be the reason why the water sample was not easy to achieve restabilization at a high coagulant dosage. Through the Boltzmann equation modeling, the maximum average floc size formed by AlCl3, PAC, and HPAC were 196.0, 188.0, and 203.6 μm, respectively, and the halfway time of reactions were 31.23, 17.08, and 9.55 min, respectively. The HPAC showed the strongest floc formation ability and the fastest floc growth rate in the flocculation process, which might be caused by the stronger adsorption and bridging functions of Alb and Alc contained in HPAC.
SU-E-T-270: Diffusion Synthetic Acceleration for Linear Boltzmann Transport Equation
Chen, G; Hong, X; Gao, H
2015-06-15
Purpose: Linear Boltzmann transport equation (LBTE) is as accurate as the Monte Carlo method (MC) for dose calculation in photon/particle therapy (LBTE is a deterministic and Eulerian formulation and MC is a statistical and Lagrangian description). An advantage of LBTE is that numerous acceleration techniques can be utilized for acceleration. This work is to explore the acceleration of LBTE via diffusion synthetic acceleration (DSA). Methods: For simplicity, two-dimensional, steady-state, and within-group LBTE is considered with two angular dimensions and two spatial dimensions. The discrete ordinate method is developed for solving this integro-differential equation. The angular variables are discretized using a level-symmetric quadrature set on the unit sphere. The spatial variables are discretized on the structured grid based on the diamond scheme. The source-iteration method (SI) is used to solve the discretized system.Since SI is slow in optically thick and highly scattering regime. DSA is developed to accelerate SI. The motivation for DSA is that diffusion equation (DE) is a good approximation of LBTE in the above regime. However, DE is much cheaper than LBTE computationally since DE only involves spatial variables. Thus, in each DSA iteration, DSA adds to the SI step a computationally-negligible DE step, i.e., to first solve DE with the SI residual as source term, and then compensate the SI solution with DE solution. Results: DSA was benchmarked and compared with SI. The difference between two methods was within 0.12% which verifies the accuracy of DSA, while DSA demonstrated the great advantage in speed, e.g., the reduction of iteration number to 6% and 4% respectively for cases with 100 and 1,000 scattering-absorption ratio that commonly occur in clinical dose calculation. Conclusion: DSA has been developed as one of many possible means for accelerating the numerical solver of LBTE for dose calculation. The authors were partially supported by the NSFC
Multigrid solution of the nonlinear Poisson-Boltzmann equation and calculation of titration curves.
Oberoi, H; Allewell, N M
1993-01-01
Although knowledge of the pKa values and charge states of individual residues is critical to understanding the role of electrostatic effects in protein structure and function, calculating these quantities is challenging because of the sensitivity of these parameters to the position and distribution of charges. Values for many different proteins which agree well with experimental results have been obtained with modified Tanford-Kirkwood theory in which the protein is modeled as a sphere (reviewed in Ref. 1); however, convergence is more difficult to achieve with finite difference methods, in which the protein is mapped onto a grid and derivatives of the potential function are calculated as differences between the values of the function at grid points (reviewed in Ref. 6). Multigrid methods, in which the size of the grid is varied from fine to coarse in several cycles, decrease computational time, increase rates of convergence, and improve agreement with experiment. Both the accuracy and computational advantage of the multigrid approach increase with grid size, because the time required to achieve a solution increases slowly with grid size. We have implemented a multigrid procedure for solving the nonlinear Poisson-Boltzmann equation, and, using lysozyme as a test case, compared calculations for several crystal forms, different refinement procedures, and different charge assignment schemes. The root mean square difference between calculated and experimental pKa values for the crystal structure which yields best agreement with experiment (1LZT) is 1.1 pH units, with the differences in calculated and experimental pK values being less than 0.6 pH units for 16 out of 21 residues. The calculated titration curves of several residues are biphasic. Images FIGURE 8 PMID:8369451
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.
Xie, Yang; Ying, Jinyong; Xie, Dexuan
2017-03-30
SMPBS (Size Modified Poisson-Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson-Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson-Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile-friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware-accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc.
Frydel, Derek
2011-06-21
We incorporate ion polarizabilities into the Poisson-Boltzmann equation by modifying the effective dielectric constant and the Boltzmann distribution of ions. The extent of the polarizability effects is controlled by two parameters, γ(1) and γ(2); γ(1) determines the polarization effects in a dilute system and γ(2) regulates the dependence of the polarizability effects on the concentration of ions. For a polarizable ion in an aqueous solution γ(1) ≈ 0.01 and the polarizability effects are negligible. The conditions where γ(1) and/or γ(2) are large and the polarizability is relevant involve the low dielectric constant media, high surface charge, and/or large ionic concentrations. © 2011 American Institute of Physics
NASA Astrophysics Data System (ADS)
He, Ping
2012-01-01
The long-standing puzzle surrounding the statistical mechanics of self-gravitating systems has not yet been solved successfully. We formulate a systematic theoretical framework of entropy-based statistical mechanics for spherically symmetric collisionless self-gravitating systems. We use an approach that is very different from that of the conventional statistical mechanics of short-range interaction systems. We demonstrate that the equilibrium states of self-gravitating systems consist of both mechanical and statistical equilibria, with the former characterized by a series of velocity-moment equations and the latter by statistical equilibrium equations, which should be derived from the entropy principle. The velocity-moment equations of all orders are derived from the steady-state collisionless Boltzmann equation. We point out that the ergodicity is invalid for the whole self-gravitating system, but it can be re-established locally. Based on the local ergodicity, using Fermi-Dirac-like statistics, with the non-degenerate condition and the spatial independence of the local microstates, we rederive the Boltzmann-Gibbs entropy. This is consistent with the validity of the collisionless Boltzmann equation, and should be the correct entropy form for collisionless self-gravitating systems. Apart from the usual constraints of mass and energy conservation, we demonstrate that the series of moment or virialization equations must be included as additional constraints on the entropy functional when performing the variational calculus; this is an extension to the original prescription by White & Narayan. Any possible velocity distribution can be produced by the statistical-mechanical approach that we have developed with the extended Boltzmann-Gibbs/White-Narayan statistics. Finally, we discuss the questions of negative specific heat and ensemble inequivalence for self-gravitating systems.
Lloyd, S A M; Ansbacher, W
2013-01-01
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. 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(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. 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 support the dose perturbations
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
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.
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
Boschitsch, Alexander H; Fenley, Marcia O
2007-04-15
The nonlinear Poisson-Boltzmann equation (PBE) has been successfully used for the prediction of numerous electrostatic properties of highly charged biopolyelectrolytes immersed in aqueous salt solutions. While numerous numerical solvers for the 3D PBE have been developed, the formulation of the outer boundary treatments used in these methods has only been loosely addressed, especially in the nonlinear case. The de facto standard in current nonlinear PBE implementations is to either set the potential at the outer boundaries to zero or estimate it using the (linear) Debye-Hückel (DH) approximation. However, an assessment of how these outer boundary treatments affect the overall solution accuracy does not appear to have been previously made. As will be demonstrated here, both approximations can, under certain conditions, produce completely erroneous estimates of the potential and energy salt dependencies. A related concern for calculations carried out on grids of finite extent (e.g., all current finite difference and finite element implementations) is the contribution to the energy and salt dependence from the exterior region outside the computational grid. This too is shown to be significant, especially at low salt concentration where essentially all of the contributions to the excess osmotic pressure and ion stress energies originate from this exterior region. In this paper the authors introduce a new outer boundary treatment that is valid for both the linear and nonlinear PBE. The authors also formulate energy corrections to account for contributions from outside the computational domain. Finally, the authors also consider the effects of general ion exclusion layers upon biomolecular electrostatics. It is shown that while these layers tend to increase the surface electrostatic potential, under physiological salt conditions and high net charges their effect on the excess osmotic pressure term, which is a measure of the salt dependence of the total electrostatic
NASA Astrophysics Data System (ADS)
Di, Shaoyan; Shen, Lei; Chang, Pengying; Zhao, Kai; Lu, Tiao; Du, Gang; Liu, Xiaoyan
2017-04-01
A deterministic time-dependent Boltzmann transport equation (BTE) solver is employed to carry out a comparison work among 10 nm double-gate n-type MOSFETs with channel materials of Si, In0.53Ga0.47As, and GaSb in different surface orientations. Results show that the GaSb device has the highest drive current, while scattering affects carrier transport in the Si device the most. The InGaAs device exhibits the highest injection velocity but suffers from the density of state (DOS) bottleneck seriously.
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.
Silalahi, Alexander R.J.; Boschitsch, Alexander H.; Harris, Robert C.; Fenley, Marcia O.
2011-01-01
The ion size-modified Poisson Boltzmann equation (SMPBE) is applied to the simple model problem of a low-dielectric spherical cavity containing a central charge, in an aqueous salt solution to investigate the finite ion size effect upon the electrostatic free energy and its sensitivity to changes in salt concentration. The SMPBE is shown to predict a very different electrostatic free energy than the nonlinear Poisson-Boltzmann equation (NLPBE) due to the additional entropic cost of placing ions in solution. Although the energy predictions of the SMPBE can be reproduced by fitting an appropriatelysized Stern layer, or ion-exclusion layer to the NLPBE calculations, the size of the Stern layer is difficult to estimate a priori. The SMPBE also produces a saturation layer when the central charge becomes sufficiently large. Ion-competition effects on various integrated quantities such the total number of ions predicted by the SMPBE are qualitatively similar to those given by the NLPBE and those found in available experimental results. PMID:22723750
Phase-field modeling by the method of lattice Boltzmann equations.
Fakhari, Abbas; Rahimian, Mohammad H
2010-03-01
In this paper, at first, a lattice Boltzmann method for binary fluids, which is applicable at low viscosity values, is developed. The presented scheme is extension of the free-energy-based approach to a multi-relaxation-time collision model. Various benchmark problems such as the well-known Laplace law for stationary bubbles and capillary-wave test are conducted for validation. As an appealing application, instability of a rising bubble in an enclosed duct is studied and irregular behavior of the bubble is observed at very high Reynolds numbers. In order to highlight its capability to simulate high Reynolds number flows, which is a challenge for many other models, a typical wobbling bubble in the turbulent regime is simulated successfully. Then, in the context of phase-field modeling, a lattice Boltzmann method is proposed for multiphase flows with a density contrast. Unlike most of the previous models based on the phase-field theory, the proposed scheme not only tolerates very low viscosity values but also emerges as a promising method for investigation of two-phase flow problems with moderate density ratios. In addition to comparison to the kinetic-based model, the proposed approach is further verified by judging against the theoretical solutions and experimental data. Various case studies including the rising bubble, droplet splashing on a wet surface, and falling droplet are conducted to show the versatility of the presented lattice Boltzmann model.
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
Hybrid discrete ordinates and characteristics method for solving the linear Boltzmann equation
NASA Astrophysics Data System (ADS)
Yi, Ce
With the ability of computer hardware and software increasing rapidly, deterministic methods to solve the linear Boltzmann equation (LBE) have attracted some attention for computational applications in both the nuclear engineering and medical physics fields. Among various deterministic methods, the discrete ordinates method (SN) and the method of characteristics (MOC) are two of the most widely used methods. The SN method is the traditional approach to solve the LBE for its stability and efficiency. While the MOC has some advantages in treating complicated geometries. However, in 3-D problems requiring a dense discretization grid in phase space (i.e., a large number of spatial meshes, directions, or energy groups), both methods could suffer from the need for large amounts of memory and computation time. In our study, we developed a new hybrid algorithm by combing the two methods into one code, TITAN. The hybrid approach is specifically designed for application to problems containing low scattering regions. A new serial 3-D time-independent transport code has been developed. Under the hybrid approach, the preferred method can be applied in different regions (blocks) within the same problem model. Since the characteristics method is numerically more efficient in low scattering media, the hybrid approach uses a block-oriented characteristics solver in low scattering regions, and a block-oriented SN solver in the remainder of the physical model. In the TITAN code, a physical problem model is divided into a number of coarse meshes (blocks) in Cartesian geometry. Either the characteristics solver or the SN solver can be chosen to solve the LBE within a coarse mesh. A coarse mesh can be filled with fine meshes or characteristic rays depending on the solver assigned to the coarse mesh. Furthermore, with its object-oriented programming paradigm and layered code structure, TITAN allows different individual spatial meshing schemes and angular quadrature sets for each coarse
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 Astrophysics Data System (ADS)
Stokes, Peter W.; Philippa, Bronson; Cocks, Daniel; White, Ronald D.
2016-03-01
We present a general phase-space kinetic model for charged-particle transport through combined localized and delocalized states, capable of describing scattering collisions, trapping, detrapping, and losses. The model is described by a generalized Boltzmann equation, for which an analytical solution is found in Fourier-Laplace space. The velocity of the center of mass and the diffusivity about it are determined analytically, together with the flux transport coefficients. Transient negative values of the free particle center-of-mass transport coefficients can be observed due to the trapping to, and detrapping from, localized states. A Chapman-Enskog-type perturbative solution technique is applied, confirming the analytical results and highlighting the emergence of a density gradient representation in the weak-gradient hydrodynamic regime. A generalized diffusion equation with a unique global time operator is shown to arise, reducing to the standard diffusion equation and a Caputo fractional diffusion equation in the normal and dispersive limits. A subordination transformation is used to solve the generalized diffusion equation by mapping from the solution of a corresponding standard diffusion equation.
Thermoelectric power calculation by the Boltzmann equation: Na(x)CoO(2).
Hamada, N; Imai, T; Funashima, H
2007-09-12
A full-potential augmented-plane-wave (FLAPW) band-structure calculation in the local density approximation (LDA) was carried out for hexagonal Na(x)CoO(2) (x = 0.45, 0.55, 0.66 and 0.75). The Seebeck tensor was estimated by the Boltzmann theory, assuming that the relaxation time is constant on the Fermi surface. The Seebeck tensor is extremely anisotropic; the c-axis Seebeck coefficient varies dramatically with the Na content. The calculation reproduces the experiment semiquantitatively.
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.
NASA Astrophysics Data System (ADS)
Grofulović, M.; Pinhão, N.; Alves, Ll; Guerra, V.; Loffhagen, D.; Korolov, I.; Vass, M.; Donkó, Z.
2016-09-01
The plasma-based CO2 conversion is a promising route for achieving the reduction of fossil fuel consumption and of CO2 emission. An accurate description of the electron kinetics by solving the electron Boltzmann equation (EBE) is necessary for this application. This work is dedicated to the inter-comparison between various calculation techniques of the EBE (two-term, multi-term and space gradients of the electron density) and the Monte-Carlo reference technique for the analysis of swarm parameters and their comparison with previously available and present experimental data. We adopt the complete set of electron-impact cross sections for CO2, to be published on the IST-LISBON database with LXCat. Results show that despite the fact that the IST-LISBON cross sections were derived to fit measured swarm parameters when used in a two-term expansion Boltzmann code, good agreement with the other solution and simulation techniques is generally obtained for the electron swarm parameters under consideration. Work partially supported by grant OTKA-K-105476 and by Portuguese FCT, under project UID/FIS/50010/2013 and grant PD/BD/105884/2014 (PD-F APPLAuSE).
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.
NASA Astrophysics Data System (ADS)
Beralso e Silva, Leandro; de Siqueira Pedra, Walter; Sodré, Laerte; Perico, Eder L. D.; Lima, Marcos
2017-09-01
The collapse of a collisionless self-gravitating system, with the fast achievement of a quasi-stationary state, is driven by violent relaxation, with a typical particle interacting with the time-changing collective potential. It is traditionally assumed that this evolution is governed by the Vlasov–Poisson equation, in which case entropy must be conserved. We run N-body simulations of isolated self-gravitating systems, using three simulation codes, NBODY-6 (direct summation without softening), NBODY-2 (direct summation with softening), and GADGET-2 (tree code with softening), for different numbers of particles and initial conditions. At each snapshot, we estimate the Shannon entropy of the distribution function with three different techniques: Kernel, Nearest Neighbor, and EnBiD. For all simulation codes and estimators, the entropy evolution converges to the same limit as N increases. During violent relaxation, the entropy has a fast increase followed by damping oscillations, indicating that violent relaxation must be described by a kinetic equation other than the Vlasov–Poisson equation, even for N as large as that of astronomical structures. This indicates that violent relaxation cannot be described by a time-reversible equation, shedding some light on the so-called “fundamental paradox of stellar dynamics.” The long-term evolution is well-described by the orbit-averaged Fokker–Planck model, with Coulomb logarithm values in the expected range 10{--}12. By means of NBODY-2, we also study the dependence of the two-body relaxation timescale on the softening length. The approach presented in the current work can potentially provide a general method for testing any kinetic equation intended to describe the macroscopic evolution of N-body systems.
NASA Astrophysics Data System (ADS)
Jungemann, C.; Pham, A. T.; Meinerzhagen, B.; Ringhofer, C.; Bollhöfer, M.
2006-07-01
The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real spaces by box integration. This ensures exact current continuity for the discrete equations. Spurious oscillations of the distribution function are suppressed by stabilization based on a maximum entropy dissipation principle avoiding the H transformation. The derived formulation can be used on arbitrary grids as long as box integration is possible. The approach works not only with analytical bands but also with full band structures in the case of holes. Results are presented for holes in bulk silicon based on a full band structure and electrons in a Si NPN bipolar junction transistor. The convergence of the spherical harmonics expansion is shown for a device, and it is found that the quasiballistic transport in nanoscale devices requires an expansion of considerably higher order than the usual first one. The stability of the discretization is demonstrated for a range of grid spacings in the real space and bias points which produce huge gradients in the electron density and electric field. It is shown that the resultant large linear system of equations can be solved in a memory efficient way by the numerically robust package ILUPACK.
Shestakov, A I; Milovich, J L; Noy, A
2000-12-27
The nonlinear Poisson-Boltzmann (PB) equation is solved using Pseudo Transient Continuation. The PB solver is constructed by modifying the nonlinear diffusion module of a 3D, massively parallel, unstructured-grid, finite element, radiation-hydrodynamics code. The solver also computes the electrostatic energy and evaluates the force on a user-specified contour. Either Dirichlet or mixed boundary conditions are allowed. The latter specifies surface charges, approximates far-field conditions, or linearizes conditions ''regulating'' the surface charge. The code may be run in either Cartesian, cylindrical, or spherical coordinates. The potential and force due to a conical probe interacting with a flat plate is computed and the result compared with direct force measurements by chemical force microscopy.
NASA Astrophysics Data System (ADS)
Rashid, Shahid
1990-07-01
The kinematically based concept of quasi-free electron states developed by Rashid (1988) is extended to obtain a coupling/decoupling technique capable of separating dynamical and kinematical factors. This method is then applied to a reduced form of the Boltzmann equation (assuming that the electrons are in quantum states with quasi-four-momentum, as in an intense laser field). The solution obtained is applicable to the relativistic case and consists of the product of (1) the initial distribution function and (2) a time-evolving part dependent on the differential cross section of the plasma-heating scattering process. The significance of the present analysis for laser-fusion studies is briefly indicated.
NASA Astrophysics Data System (ADS)
Cai, Huanqing; Ye, Qizheng
2010-04-01
Based on the model of the Wigner-Seitz cell, the surface potential of the spherical macroparticle (radius a) expands in terms of the monopole (q). A dipole (p) model is assumed for an anisotropic boundary condition of the nonlinear Poisson-Boltzmann equation. Using the finite element method implemented by the FlexPDE software, the potential distribution around the macroparticle is obtained for different ratios p/qa. The calculated results for the potential show that there is an attractive region in the vicinity of the macroparticle when |p/qa|>1.1, and noticeably there is a potential well behind the macroparticle when |p/qa| = 1.1, i.e., there exists both an attractive region and a repulsive region simultaneously. This means that the attractive interaction between macroparticles may arise from the anisotropic distribution of the surrounding plasmas, which well explains some experimental observations.
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.
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.
NASA Technical Reports Server (NTRS)
Zank, G. P.; Pauls, H. L.; Williams, L. L.; Hall, D. T.
1995-01-01
The importance of interstellar neutrals in understanding and modelling the global interaction of the solar wind with the local interstellar medium is becoming increasingly apparent. Unfortunately the self-consistent inclusion of a neutral interstellar component into time-dependent, dynamical models is formidably difficult due to the extremely large mean free paths associated with the neutrals and the creation of essentially different neutral distributions from different interaction regions of the solar wind and LISM. In full generality, one has to address the problem by treating the neutrals kinetically with the appropriate extinction and creation source terms. In this paper, a limited set of simulations will be presented in which the solar wind and interstellar plasma is described as a 2D fully compressible time-dependent fluid while the interstellar neutral distribution is derived by solving the appropriate Boltzmann equation directly.
NASA Astrophysics Data System (ADS)
Dekura, Haruhiko; Tsuchiya, Taku
2017-05-01
Lattice thermal conductivity κlat of MgO at high pressures P and temperatures T up to 150 GPa and 4000 K are determined using lattice dynamics calculations and the linearized phonon Boltzmann transport equation (BTE) beyond the relaxation time approximation (RTA) from first principles. It is found that the complete solution of the linearized BTE substantially corrects values of κlat calculated with the RTA by ˜30 % , from ˜42 to ˜54 W m-1K-1 under ambient conditions. The calculated values of κlat are in good agreement with those from the existing experiments. At conditions representative of the Earth's core-mantle boundary (P =136 GPa and T =3800 K ), κlat is predicted to be ˜32 and ˜40 W m-1K-1 by RTA and the full solution of BTE, respectively. We report a detailed comparison of our study with earlier theoretical studies.
Global well-posedness for the Fokker-Planck-Boltzmann equation in Besov-Chemin-Lerner type spaces
NASA Astrophysics Data System (ADS)
Liu, Zhengrong; Tang, Hao
2016-06-01
In this paper, motivated by [16], we use the Littlewood-Paley theory to establish some estimates on the nonlinear collision term, which enable us to investigate the Cauchy problem of the Fokker-Planck-Boltzmann equation. When the initial data is a small perturbation of the Maxwellian equilibrium state, under the Grad's angular cutoff assumption, the unique global solution for the hard potential case is obtained in the Besov-Chemin-Lerner type spaces C ([ 0 , ∞) ; L˜ξ 2 (B2,rs)) with 1 ≤ r ≤ 2 and s > 3 / 2 or s = 3 / 2 and r = 1. Besides, we also obtain the uniform stability of the dependence on the initial data.
Langevin Poisson-Boltzmann equation: point-like ions and water dipoles near a charged surface.
Gongadze, Ekaterina; van Rienen, Ursula; Kralj-Iglič, Veronika; Iglič, Aleš
2011-06-01
Water ordering near a charged membrane surface is important for many biological processes such as binding of ligands to a membrane or transport of ions across it. In this work, the mean-field Poisson-Boltzmann theory for point-like ions, describing an electrolyte solution in contact with a planar charged surface, is modified by including the orientational ordering of water. Water molecules are considered as Langevin dipoles, while the number density of water is assumed to be constant everywhere in the electrolyte solution. It is shown that the dielectric permittivity of an electrolyte close to a charged surface is decreased due to the increased orientational ordering of water dipoles. The dielectric permittivity close to the charged surface is additionally decreased due to the finite size of ions and dipoles.
SU-E-T-22: A Deterministic Solver of the Boltzmann-Fokker-Planck Equation for Dose Calculation
Hong, X; Gao, H; Paganetti, H
2015-06-15
Purpose: The Boltzmann-Fokker-Planck equation (BFPE) accurately models the migration of photons/charged particles in tissues. While the Monte Carlo (MC) method is popular for solving BFPE in a statistical manner, we aim to develop a deterministic BFPE solver based on various state-of-art numerical acceleration techniques for rapid and accurate dose calculation. Methods: Our BFPE solver is based on the structured grid that is maximally parallelizable, with the discretization in energy, angle and space, and its cross section coefficients are derived or directly imported from the Geant4 database. The physical processes that are taken into account are Compton scattering, photoelectric effect, pair production for photons, and elastic scattering, ionization and bremsstrahlung for charged particles.While the spatial discretization is based on the diamond scheme, the angular discretization synergizes finite element method (FEM) and spherical harmonics (SH). Thus, SH is used to globally expand the scattering kernel and FFM is used to locally discretize the angular sphere. As a Result, this hybrid method (FEM-SH) is both accurate in dealing with forward-peaking scattering via FEM, and efficient for multi-energy-group computation via SH. In addition, FEM-SH enables the analytical integration in energy variable of delta scattering kernel for elastic scattering with reduced truncation error from the numerical integration based on the classic SH-based multi-energy-group method. Results: The accuracy of the proposed BFPE solver was benchmarked against Geant4 for photon dose calculation. In particular, FEM-SH had improved accuracy compared to FEM, while both were within 2% of the results obtained with Geant4. Conclusion: A deterministic solver of the Boltzmann-Fokker-Planck equation is developed for dose calculation, and benchmarked against Geant4. Xiang Hong and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang
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. Copyright © 2013 Elsevier Ltd. All rights reserved.
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
Stable Equilibrium Based on Lévy Statistics:A Linear Boltzmann Equation Approach
NASA Astrophysics Data System (ADS)
Barkai, Eli
2004-06-01
To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, we consider stochastic collision models. The models are a generalization of the Rayleigh collision model, for a heavy one dimensional particle M interacting with ideal gas particles with a mass m<< M. Similar to previous approaches we assume elastic, uncorrelated, and impulsive collisions. We let the bath particle velocity distribution function to be of general form, namely we do not postulate a specific form of power-law equilibrium. We show, under certain conditions, that the velocity distribution function of the heavy particle is Lévy stable, the Maxwellian distribution being a special case. We demonstrate our results with numerical examples. The relation of the power law equilibrium obtained here to thermodynamics is discussed. In particular we compare between two models: a thermodynamic and an energy scaling approaches. These models yield insight into questions like the meaning of temperature for power law equilibrium, and into the issue of the universality of the equilibrium (i.e., is the width of the generalized Maxwellian distribution functions obtained here, independent of coupling constant to the bath).
On the Derivation of a High-Velocity Tail from the Boltzmann-Fokker-Planck Equation for Shear Flow
NASA Astrophysics Data System (ADS)
Acedo, L.; Santos, A.; Bobylev, A. V.
2002-12-01
Uniform shear flow is a paradigmatic example of a nonequilibrium fluid state exhibiting non-Newtonian behavior. It is characterized by uniform density and temperature and a linear velocity profile U x ( y)= ay, where a is the constant shear rate. In the case of a rarefied gas, all the relevant physical information is represented by the one-particle velocity distribution function f( r, v)= f( V), with V≡ v- U( r), which satisfies the standard nonlinear integro-differential Boltzmann equation. We have studied this state for a two-dimensional gas of Maxwell molecules with a collision rate K( θ)∝lim ∈→0 ∈ -2 δ( θ- ∈), where θ is the scattering angle, in which case the nonlinear Boltzmann collision operator reduces to a Fokker-Planck operator. We have found analytically that for shear rates larger than a certain threshold value a th≃0.3520 ν (where ν is an average collision frequency and a th/ ν is the real root of the cubic equation 64 x 3+16 x 2+12 x-9=0) the velocity distribution function exhibits an algebraic high-velocity tail of the form f( V; a)˜| V|-4- σ( a) Φ( ϕ; a), where ϕ≡tan V y / V x and the angular distribution function Φ( ϕ; a) is the solution of a modified Mathieu equation. The enforcement of the periodicity condition Φ( ϕ; a)= Φ( ϕ+ π; a) allows one to obtain the exponent σ( a) as a function of the shear rate. It diverges when a→ a th and tends to a minimum value σ min≃1.252 in the limit a→∞. As a consequence of this power-law decay for a> a th, all the velocity moments of a degree equal to or larger than 2+ σ( a) are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle ~ϕ( a), which rotates from ~ϕ=- π/4,3 π/4 when a→ a th to ~ϕ=0, π in the limit a→∞.
Biesheuvel, P. Maarten
2001-06-15
Simple solutions of the Poisson-Boltzmann (PB) equation for the electrostatic double-layer interaction of close, planar hydrophilic surfaces in water are evaluated. Four routes, being the weak overlap approximation, the Debye-Hückel linearization based on low electrostatic potentials, the Ettelaie-Buscall linearization based on small variations in the potential, and a new approach based on the fact that concentrations are virtually constant in the gap between close surfaces, are discussed. The Ettelaie-Buscall and constant-concentration approach become increasingly accurate for closer surfaces and are exact for touching surfaces, while the weak overlap approximation is exact for an isolated surface. The Debye-Hückel linearization is valid as long as potentials remain low, independent of separation. In contrast to the Ettelaie-Buscall approach and the weak overlap approximation, the Debye-Hückel linearization and constant-concentration approach can also be used for systems containing multivalent ions. Simulations in which the four approaches are compared with the PB equation for the constant-charge model, the constant-potential model, as being used in the DLVO theory, and the charge-regulation model are presented. Copyright 2001 Academic Press.
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.
Sixteen-moment approximation for a collisionless space plasma: Waves and instabilities
Kuznetsov, V. D.; Dzhalilov, N. S.
2009-11-15
A study is carried out of waves and instabilities in an anisotropic collisionless plasma. In a strongly magnetized plasma, the velocity distributions along and across the magnetic field lines are different, which results in anisotropy of the total pressure and gives rise to an anisotropic heat flux. The fluid description of the plasma is based on the 16-moment integral transport equations, which are integral equations obtained from the Boltzmann-Vlasov kinetic equation. For small incompressible perturbations in a homogeneous plasma, the general dispersion relation implies that there are not only firehose modes, but also three additional modes, and that all four wave modes interact with each other if a heat flux is present. Heat fluxes do not change the properties of conventional firehose modes. The conditions for the onset of instabilities are investigated as functions of the parameters of the problems. Qualitative estimates for conditions typical of the solar corona are presented.
Alexander, F.J.; Garcia, A.L.; Alder, B.J.
1994-10-01
The direct simulation Monte Carlo method is modified with a post-collision displacement in order to obtain the hard sphere equation of state. This leads to consistent thermodynamic and transport properties in the low density regime. At higher densities, when the enhanced collision rate according to kinetic theory is introduced, the exact hard sphere equation of state is recovered. and the transport coefficients are comparable to those of the Enskog theory. The computational advantages of this scheme over hard sphere molecular dynamics are that it is significantly faster at low and moderate densities and that it is readily parallelizable.
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
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.
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.
Hu, Kainan; Zhang, Hongwu; Geng, Shaojuan
2016-10-01
A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e., the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion, so it is easy to construct new schemes of higher-order accuracy. To validate the proposed scheme, a one-dimensional shock tube simulation is performed. The numerical results agree with the analytical solutions very well.
Proton dose approximation in arbitrary convex geometry. [via Boltzmann transport equation
NASA Technical Reports Server (NTRS)
Wilson, J. W.; Khandelwal, G. S.
1974-01-01
An expansion is derived for the solution to the transport equation in two dimensions subject to boundary conditions given for an arbitrary convex region. Questions of high-energy transport are considered along with the properties of the dose response function. The expansion of the solution of the transport equation is presented in terms of a parameter which measures the lateral dispersion of an unidirectional beam. This parameter is usually small and the expansion is expected to converge rapidly. The dominant term in the expansion is related to fluence-to-dose conversion factors in a semiinfinite slab for normal incidence. A convenient parameterization of the conversion factors is provided along with numerical examples.
Proton dose approximation in arbitrary convex geometry. [via Boltzmann transport equation
NASA Technical Reports Server (NTRS)
Wilson, J. W.; Khandelwal, G. S.
1974-01-01
An expansion is derived for the solution to the transport equation in two dimensions subject to boundary conditions given for an arbitrary convex region. Questions of high-energy transport are considered along with the properties of the dose response function. The expansion of the solution of the transport equation is presented in terms of a parameter which measures the lateral dispersion of an unidirectional beam. This parameter is usually small and the expansion is expected to converge rapidly. The dominant term in the expansion is related to fluence-to-dose conversion factors in a semiinfinite slab for normal incidence. A convenient parameterization of the conversion factors is provided along with numerical examples.
NASA Astrophysics Data System (ADS)
Samian, R. S.; Abbassi, A.; Ghazanfarian, J.
2013-09-01
The thermal performance of two-dimensional (2D) field-effect transistors (FET) is investigated frequently by solving the Fourier heat diffusion law and the Boltzmann transport equation (BTE). With the introduction of the new generation of 3D FETs in which their thickness is less than the phonon mean-free-path it is necessary to carefully simulate the thermal performance of such devices. This paper numerically integrates the BTE in common 2D transistors including planar single layer and Silicon-On-Insulator (SOI) transistor, and the new generation of 3D transistors including FinFET and Tri-Gate devices. In order to decrease the directional dependency of results in 3D simulations; the Legendre equal-weight (PN-EW) quadrature set has been employed. It is found that if similar switching time is assumed for 3D and 2D FETs while the new generation of 3D FETs has less net energy consumption, they have higher hot-spot temperature. The results show continuous heat flux distribution normal to the silicon/oxide interface while the temperature jump is seen at the interface in double layer transistors.
NASA Astrophysics Data System (ADS)
Santos, A.; Ernst, M.
2003-07-01
The exact nonequilibrium steady-state solution of the nonlinear Boltzmann equation for a driven inelastic Maxwell model was obtained by Ben-Naim and Krapivsky [Phys. Rev. E 61, R5 (2000)] in the form of an infinite product for the Fourier transform of distribution function f(c). In this paper we have inverted the Fourier transform to express f(c) in the form of an infinite series of exponentially decaying terms. The dominant high-energy tail is exponential, f(c)≃A0 exp(-a|c|), where a≡2/(1-α2) and amplitude A0 is given in terms of a converging sum. This is explicitly shown in the totally inelastic limit (α→0) and in the quasielastic limit (α→1). In the latter case, the distribution is dominated by a Maxwellian for a very wide range of velocities, but a crossover from a Maxwellian to an exponential high-energy tail exists for velocities |c-c0|˜1/(q) around a crossover velocity c0≃ln q-1/(q), where q≡(1-α)/2≪1. In this crossover region the distribution function is extremely small, ln f(c0)≃q-1 ln q.
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.
Dynamic structure factor of the normal Fermi gas from the collisionless to the hydrodynamic regime
Watabe, Shohei; Nikuni, Tetsuro
2010-09-15
The dynamic structure factor of a normal Fermi gas is investigated by using the moment method for the Boltzmann equation. We determine the spectral function at finite temperatures over the full range of crossover from the collisionless regime to the hydrodynamic regime. We find that the Brillouin peak in the dynamic structure factor exhibits a smooth crossover from zero to first sound as functions of temperature and interaction strength. The dynamic structure factor obtained using the moment method also exhibits a definite Rayleigh peak ({omega}{approx}0), which is a characteristic of the hydrodynamic regime. We compare the dynamic structure factor obtained by the moment method with that obtained from the hydrodynamic equations.
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.
Tencer, John; Carlberg, Kevin; Larsen, Marvin; ...
2017-06-17
Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat-transfer applications, a quasi-steady assumption is valid, thereby removing time dependence. The dependence on wavelength is often treated through a weighted sum of gray gases (WSGG) approach. The discrete ordinates method (DOM) is one of the most common methods for approximating the angular (i.e., directional) dependence. The DOM exactly solves for the radiative intensity for a finite numbermore » of discrete ordinate directions and computes approximations to integrals over the angular space using a quadrature rule; the chosen ordinate directions correspond to the nodes of this quadrature rule. This paper applies a projection-based model-reduction approach to make high-order quadrature computationally feasible for the DOM for purely absorbing applications. First, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model constructed from the reduced basis. Finally, this results in a much more accurate solution than might have been achieved using only the ordinate directions used to compute the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.« less
Fluctuating multicomponent lattice Boltzmann model.
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
NASA Astrophysics Data System (ADS)
Péraud, Jean-Philippe M.; Hadjiconstantinou, Nicolas G.
2016-01-01
We derive the continuum equations and boundary conditions governing phonon-mediated heat transfer in the limit of a small but finite mean-free path from the asymptotic solution of the linearized Boltzmann equation in the relaxation time approximation. Our approach uses the ratio of the mean-free path to the characteristic system length scale, also known as the Knudsen number, as the expansion parameter to study the effects of boundaries on the breakdown of the Fourier description. We show that, in the bulk, the traditional heat conduction equation using Fourier's law as a constitutive relation is valid at least up to second order in the Knudsen number for steady problems and first order for time-dependent problems. However, this description does not hold within distances on the order of a few mean-free paths from the boundary; this breakdown is a result of kinetic effects that are always present in the boundary vicinity and require solution of a Boltzmann boundary layer problem to be determined. Matching the inner, boundary layer solution to the outer, bulk solution yields boundary conditions for the Fourier description as well as additive corrections in the form of universal kinetic boundary layers; both are found to be proportional to the bulk-solution gradients at the boundary and parametrized by the material model and the phonon-boundary interaction model (Boltzmann boundary condition). Our derivation shows that the traditional no-jump boundary condition for prescribed temperature boundaries and the no-flux boundary condition for diffusely reflecting boundaries are appropriate only to zeroth order in the Knudsen number; at higher order, boundary conditions are of the jump type. We illustrate the utility of the asymptotic solution procedure by demonstrating that it can be used to predict the Kapitza resistance (and temperature jump) associated with an interface between two materials. All results are validated via comparisons with low-variance deviational Monte
Phase space simulation of collisionless stellar systems on the massively parallel processor
NASA Technical Reports Server (NTRS)
White, Richard L.
1987-01-01
A numerical technique for solving the collisionless Boltzmann equation describing the time evolution of a self gravitating fluid in phase space was implemented on the Massively Parallel Processor (MPP). The code performs calculations for a two dimensional phase space grid (with one space and one velocity dimension). Some results from calculations are presented. The execution speed of the code is comparable to the speed of a single processor of a Cray-XMP. Advantages and disadvantages of the MPP architecture for this type of problem are discussed. The nearest neighbor connectivity of the MPP array does not pose a significant obstacle. Future MPP-like machines should have much more local memory and easier access to staging memory and disks in order to be effective for this type of problem.
Khabibrakhmanov, I.K. ); Galeev, A.A.; Galinsky, V.L. )
1993-02-01
A collisionless parallel shock model is presented which is based on solitary-type solutions of the modified derivative nonlinear Schrodinger equation (MDNLS) for parallel Alfven waves. We generalize the standard derivative nonlinear Schrodinger equation in order to include the possible anisotropy of the plasma distribution function and higher-order Korteweg-de Vies type dispersion. Stationary solutions of MDNLS are discussed. The new mechanism, which can be called [open quote]adiabatic[close quote] of ion reflection from the magnetic mirror of the parallel shock structure is the natural and essential feature of the parallel shock that introduces the irreversible properties into the nonlinear wave structure and may significantly contribute to the plasma heating upstream as well as downstream of the shock. The anisotropic nature of [open quotes]adiabatic[close quotes] reflections leads to the asymmetric particle distribution in the upstream as well 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. The number of adiabaticaly reflected ions define the threshold conditions of the fire-hose and mirror type instabilities in the downstream and upstream regions and thus determine a parameter region in which the described laminar parallel shock structure can exist. The threshold conditions for the fire hose and mirror-type instabilities in the downstream and upstream regions of the shock are defined by the number of reflected particles and thus determine a parameter region in which the described laminar parallel shock structure can exist. 29 refs., 4 figs.
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.
Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro
2015-04-05
The generalized Born model in the Onufriev, Bashford, and Case (Onufriev et al., Proteins: Struct Funct Genet 2004, 55, 383) implementation has emerged as one of the best compromises between accuracy and speed of computation. For simulations of nucleic acids, however, a number of issues should be addressed: (1) the generalized Born model is based on a linear model and the linearization of the reference Poisson-Boltmann equation may be questioned for highly charged systems as nucleic acids; (2) although much attention has been given to potentials, solvation forces could be much less sensitive to linearization than the potentials; and (3) the accuracy of the Onufriev-Bashford-Case (OBC) model for nucleic acids depends on fine tuning of parameters. Here, we show that the linearization of the Poisson Boltzmann equation has mild effects on computed forces, and that with optimal choice of the OBC model parameters, solvation forces, essential for molecular dynamics simulations, agree well with those computed using the reference Poisson-Boltzmann model.
NASA Astrophysics Data System (ADS)
Volkas, Raymond R.; Wong, Yvonne Y. Y.
2000-11-01
We demonstrate that the relic neutrino asymmetry evolution equation derived from the quantum kinetic equations (QKE's) reduces to the Boltzmann limit that is dependent only on the instantaneous neutrino distribution functions, in the adiabatic limit in conjunction with sufficient damping. An original physical and/or geometrical interpretation of the adiabatic approximation is given, which serves as a convenient visual aid for understanding the sharply contrasting resonance behaviors exhibited by the neutrino ensemble in opposing collision regimes. We also present a classical analogue for the evolution of the difference in the να and νs distribution functions which, in the Boltzmann limit, is akin to the behavior of the generic reaction A⇌B with equal forward and reverse reaction rate constants. A new characteristic quantity, the matter and collision-affected mixing angle of the neutrino ensemble, is identified here for the first time. The role of collisions is revealed to be twofold: (i) to wipe out the inherent oscillations, and (ii) to equilibrate the να and νs distribution functions in the long run. Studies on non-adiabatic evolution and its possible relation to rapid oscillations in lepton number generation are also featured, with the introduction of an adiabaticity parameter for collision-affected oscillations.
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.
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.
Horsten, N. Baelmans, M.; Dekeyser, W.; Samaey, G.
2016-01-15
We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation.
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.
St Aubin, J; Keyvanloo, A; Fallone, B G
2016-01-01
The advent of magnetic resonance imaging (MRI) guided radiotherapy systems demands the incorporation of the magnetic field into dose calculation algorithms of treatment planning systems. This is due to the fact that the Lorentz force of the magnetic field perturbs the path of the relativistic electrons, hence altering the dose deposited by them. Building on the previous work, the authors have developed a discontinuous finite element space-angle treatment of the linear Boltzmann transport equation to accurately account for the effects of magnetic fields on radiotherapy doses. The authors present a detailed description of their new formalism and compare its accuracy to geant4 Monte Carlo calculations for magnetic fields parallel and perpendicular to the radiation beam at field strengths of 0.5 and 3 T for an inhomogeneous 3D slab geometry phantom comprising water, bone, and air or lung. The accuracy of the authors' new formalism was determined using a gamma analysis with a 2%/2 mm criterion. Greater than 98.9% of all points analyzed passed the 2%/2 mm gamma criterion for the field strengths and orientations tested. The authors have benchmarked their new formalism against Monte Carlo in a challenging radiation transport problem with a high density material (bone) directly adjacent to a very low density material (dry air at STP) where the effects of the magnetic field dominate collisions. A discontinuous finite element space-angle approach has been proven to be an accurate method for solving the linear Boltzmann transport equation with magnetic fields for cases relevant to MRI guided radiotherapy. The authors have validated the accuracy of this novel technique against geant4, even in cases of strong magnetic field strengths and low density air.
St Aubin, J.; Keyvanloo, A.; Fallone, B. G.
2016-01-15
Purpose: The advent of magnetic resonance imaging (MRI) guided radiotherapy systems demands the incorporation of the magnetic field into dose calculation algorithms of treatment planning systems. This is due to the fact that the Lorentz force of the magnetic field perturbs the path of the relativistic electrons, hence altering the dose deposited by them. Building on the previous work, the authors have developed a discontinuous finite element space-angle treatment of the linear Boltzmann transport equation to accurately account for the effects of magnetic fields on radiotherapy doses. Methods: The authors present a detailed description of their new formalism and compare its accuracy to GEANT4 Monte Carlo calculations for magnetic fields parallel and perpendicular to the radiation beam at field strengths of 0.5 and 3 T for an inhomogeneous 3D slab geometry phantom comprising water, bone, and air or lung. The accuracy of the authors’ new formalism was determined using a gamma analysis with a 2%/2 mm criterion. Results: Greater than 98.9% of all points analyzed passed the 2%/2 mm gamma criterion for the field strengths and orientations tested. The authors have benchmarked their new formalism against Monte Carlo in a challenging radiation transport problem with a high density material (bone) directly adjacent to a very low density material (dry air at STP) where the effects of the magnetic field dominate collisions. Conclusions: A discontinuous finite element space-angle approach has been proven to be an accurate method for solving the linear Boltzmann transport equation with magnetic fields for cases relevant to MRI guided radiotherapy. The authors have validated the accuracy of this novel technique against GEANT4, even in cases of strong magnetic field strengths and low density air.
Collisional and collisionless expansion of Yukawa balls.
Piel, Alexander; Goree, John A
2013-12-01
The expansion of Yukawa balls is studied by means of molecular dynamics simulations of collisionless and collisional situations. High computation speed was achieved by using the parallel computing power of graphics processing units. When the radius of the Yukawa ball is large compared to the shielding length, the expansion process starts with the blow-off of the outermost layer. A rarefactive wave subsequently propagates radially inward at the speed of longitudinal phonons. This mechanism is fundamentally different from Coulomb explosions, which employ a self-similar expansion of the entire system. In the collisionless limit, the outer layers carry away most of the available energy. The simulations are compared with analytical estimates. In the collisional case, the expansion process can be described by a nonlinear diffusion equation that is a special case of the porous medium equation.
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.
Asinari, Pietro
2009-11-01
A finite difference lattice Boltzmann scheme for homogeneous mixture modeling, which recovers Maxwell-Stefan diffusion model in the continuum limit, without the restriction of the mixture-averaged diffusion approximation, was recently proposed [P. Asinari, Phys. Rev. E 77, 056706 (2008)]. The theoretical basis is the Bhatnagar-Gross-Krook-type kinetic model for gas mixtures [P. Andries, K. Aoki, and B. Perthame, J. Stat. Phys. 106, 993 (2002)]. In the present paper, the recovered macroscopic equations in the continuum limit are systematically investigated by varying the ratio between the characteristic diffusion speed and the characteristic barycentric speed. It comes out that the diffusion speed must be at least one order of magnitude (in terms of Knudsen number) smaller than the barycentric speed, in order to recover the Navier-Stokes equations for mixtures in the incompressible limit. Some further numerical tests are also reported. In particular, (1) the solvent and dilute test cases are considered, because they are limiting cases in which the Maxwell-Stefan model reduces automatically to Fickian cases. Moreover, (2) some tests based on the Stefan diffusion tube are reported for proving the complete capabilities of the proposed scheme in solving Maxwell-Stefan diffusion problems. The proposed scheme agrees well with the expected theoretical results.
Parametric lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Shim, Jae Wan
2017-06-01
The discretized equilibrium distributions of the lattice Boltzmann method are presented by using the coefficients of the Lagrange interpolating polynomials that pass through the points related to discrete velocities and using moments of the Maxwell-Boltzmann distribution. The ranges of flow velocity and temperature providing positive valued distributions vary with regulating discrete velocities as parameters. New isothermal and thermal compressible models are proposed for flows of the level of the isothermal and thermal compressible Navier-Stokes equations. Thermal compressible shock tube flows are simulated by only five on-lattice discrete velocities. Two-dimensional isothermal and thermal vortices provoked by the Kelvin-Helmholtz instability are simulated by the parametric models.
Wang, Peng; Wang, Lian-Ping; Guo, Zhaoli
2016-10-01
The main objective of this work is to perform a detailed comparison of the lattice Boltzmann equation (LBE) and the recently developed discrete unified gas-kinetic scheme (DUGKS) methods for direct numerical simulation (DNS) of the decaying homogeneous isotropic turbulence and the Kida vortex flow in a periodic box. The flow fields and key statistical quantities computed by both methods are compared with those from the pseudospectral method at both low and moderate Reynolds numbers. The results show that the LBE is more accurate and efficient than the DUGKS, but the latter has a superior numerical stability, particularly for high Reynolds number flows. In addition, we conclude that the DUGKS can adequately resolve the flow when the minimum spatial resolution parameter k_{max}η>3, where k_{max} is the maximum resolved wave number and η is the flow Kolmogorov length. This resolution requirement can be contrasted with the requirements of k_{max}η>1 for the pseudospectral method and k_{max}η>2 for the LBE. It should be emphasized that although more validations should be conducted before the DUGKS can be called a viable tool for DNS of turbulent flows, the present work contributes to the overall assessment of the DUGKS, and it provides a basis for further applications of DUGKS in studying the physics of turbulent flows.
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
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.
Shocks in collisionless plasmas
NASA Astrophysics Data System (ADS)
Parks, G. K.; Lee, E.; Fu, S. Y.; Lin, N.; Liu, Y.; Yang, Z. W.
2017-06-01
The Earth's bow shock is the best-known collisionless shock in space. Although much is known about the bow shock, the mechanisms of heating and thermalization processes still remain poorly understood. Collisionless shocks are different from ordinary fluid shocks, because a fraction of the incident solar wind is reflected from the bow shock and the transmitted particles are not immediately thermalized. The reflected particles interact with the incident solar wind producing waves and instabilities that can heat and accelerate particles to high energies. Some of the waves can grow to large amplitudes such as Short Large Amplitude Magnetic Structures. Other upstream nonlinear structures include hot flow anomalies and density holes. The upstream nonlinear structures subsequently convect Earthward with the SW and could impact the structure and dynamics of the bow shock. These observations have clearly indicated that the upstream dynamics are an integral part of the bow shock system. Although much has been learned about the behavior of Earth's bow shock dynamics from the existing data, many fundamental questions remain not answered. This article will review observations of ion dynamics of Earth's bow shock system, what we have learned from recent and past observations. We provide new perspectives from multi-spacecraft Cluster observations about the spatial and temporal variations including the fundamental shock heating, acceleration, and entropy generation processes.
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.
NASA Astrophysics Data System (ADS)
Hwang, Feng-Nan; Cai, Shang-Rong; Shao, Yun-Long; Wu, Jong-Shinn
2010-09-01
We investigate fully parallel Newton-Krylov-Schwarz (NKS) algorithms for solving the large sparse nonlinear systems of equations arising from the finite element discretization of the three-dimensional Poisson-Boltzmann equation (PBE), which is often used to describe the colloidal phenomena of an electric double layer around charged objects in colloidal and interfacial science. The NKS algorithm employs an inexact Newton method with backtracking (INB) as the nonlinear solver in conjunction with a Krylov subspace method as the linear solver for the corresponding Jacobian system. An overlapping Schwarz method as a preconditioner to accelerate the convergence of the linear solver. Two test cases including two isolated charged particles and two colloidal particles in a cylindrical pore are used as benchmark problems to validate the correctness of our parallel NKS-based PBE solver. In addition, a truly three-dimensional case, which models the interaction between two charged spherical particles within a rough charged micro-capillary, is simulated to demonstrate the applicability of our PBE solver to handle a problem with complex geometry. Finally, based on the result obtained from a PC cluster of parallel machines, we show numerically that NKS is quite suitable for the numerical simulation of interaction between colloidal particles, since NKS is robust in the sense that INB is able to converge within a small number of iterations regardless of the geometry, the mesh size, the number of processors. With help of an additive preconditioned Krylov subspace method NKS achieves parallel efficiency of 71% or better on up to a hundred processors for a 3D problem with 5 million unknowns.
Lee, Barry
2010-05-01
This paper presents a new multigrid method applied to the most common Sn discretizations (Petrov-Galerkin, diamond-differenced, corner-balanced, and discontinuous Galerkin) of the mono-energetic Boltzmann transport equation in the optically thick and thin regimes, and with strong anisotropic scattering. Unlike methods that use scalar DSA diffusion preconditioners for the source iteration, this multigrid method is applied directly to an integral equation for the scalar flux. Thus, unlike the former methods that apply a multigrid strategy to the scalar DSA diffusion operator, this method applies a multigrid strategy to the integral source iteration operator, which is an operator for 5 independent variables in spatial 3-d (3 in space and 2 in angle) and 4 independent variables in spatial 2-d (2 in space and 2 in angle). The core smoother of this multigrid method involves applications of the integral operator. Since the kernel of this integral operator involves the transport sweeps, applying this integral operator requires a transport sweep (an inversion of an upper triagular matrix) for each of the angles used. As the equation is in 5-space or 4-space, the multigrid approach in this paper coarsens in both angle and space, effecting efficient applications of the coarse integral operators. Although each V-cycle of this method is more expensive than a V-cycle for the DSA preconditioner, since the DSA equation does not have angular dependence, the overall computational efficiency is about the same for problems where DSA preconditioning {\\it is} effective. This new method also appears to be more robust over all parameter regimes than DSA approaches. Moreover, this new method is applicable to a variety of Sn spatial discretizations, to problems involving a combination of optically thick and thin regimes, and more importantly, to problems with anisotropic scattering cross-sections, all of which DSA approaches perform poorly or not applicable at all. This multigrid approach
Lou, James; Shih, Chun-Yu; Lee, Eric
2010-01-05
Diffusiophoresis in concentrated suspensions of spherical colloids with charge-regulated surface is investigated theoretically. The charge-regulated surface considered here is the generalization of conventional constant surface potential and constant surface charge density situations. Kuwabara's unit cell model is adopted to describe the system and a pseudospectral method based on Chebyshev polynomial is employed to solve the governing general electrokinetic equations. Excellent agreements with experimental data available in literature were obtained for the limiting case of constant surface potential and very dilute suspension. It is found, among other things, that in general the larger the number of dissociated functional groups on particle surface is, the higher the particle surface potential, hence the larger the magnitude of the particle mobility. The electric potential on particle surface depends on both the concentration of dissociated hydrogen ions and the concentration of electrolyte in the solution. The electric potential on particle surface turns out to be the dominant factor in the determination of the eventual particle diffusiophoretic mobility. Local maximum of diffusiophoretic mobility as a function of double layer thickness is observed. Its reason and influence is discussed. Corresponding behavior for the constant potential situation, however, may yield a monotonously increasing profile.
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.
A multi-term solution of the space–time Boltzmann equation for electrons in gases and liquids
NASA Astrophysics Data System (ADS)
Boyle, G. J.; Tattersall, W. J.; Cocks, D. G.; McEachran, R. P.; White, R. D.
2017-02-01
In this study we have developed a full multi-term space–time solution of Boltzmann’s equation for electron transport in gases and liquids. A Green’s function formalism is used that enables flexible adaptation to various experimental systems. The spatio-temporal evolution of electrons in liquids in the non-hydrodynamic regime is benchmarked for a model Percus–Yevick (PY) liquid against an independent Monte Carlo simulation, and then applied to liquid argon. The temporal evolution of Franck–Hertz oscillations in configuration and energy space are observed for the model liquid with large differences apparent when compared to the dilute gas case, for both the velocity distribution function components and the transport quantities. The packing density in the PY liquid is shown to influence both the magnitude and wavelength of Franck–Hertz oscillations of the steady-state Townsend (SST) simulation. Transport properties are calculated from the non-hydrodynamic theory in the long time limit under SST conditions which are benchmarked against hydrodynamic transport coefficients. Finally, the spatio-temporal relaxation of low-energy electrons in liquid argon was investigated, with striking differences evident in the spatio-temporal development of the velocity distribution function components between the uncorrelated gas and true liquid approximations, due largely to the presence of a Ramsauer minimum in the former and not in the latter.
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.
Altman, Michael D; Bardhan, Jaydeep P; White, Jacob K; Tidor, Bruce
2009-01-15
We present a boundary-element method (BEM) implementation for accurately solving problems in biomolecular electrostatics using the linearized Poisson-Boltzmann equation. Motivating this implementation is the desire to create a solver capable of precisely describing the geometries and topologies prevalent in continuum models of biological molecules. This implementation is enabled by the synthesis of four technologies developed or implemented specifically for this work. First, molecular and accessible surfaces used to describe dielectric and ion-exclusion boundaries were discretized with curved boundary elements that faithfully reproduce molecular geometries. Second, we avoided explicitly forming the dense BEM matrices and instead solved the linear systems with a preconditioned iterative method (GMRES), using a matrix compression algorithm (FFTSVD) to accelerate matrix-vector multiplication. Third, robust numerical integration methods were employed to accurately evaluate singular and near-singular integrals over the curved boundary elements. Fourth, we present a general boundary-integral approach capable of modeling an arbitrary number of embedded homogeneous dielectric regions with differing dielectric constants, possible salt treatment, and point charges. A comparison of the presented BEM implementation and standard finite-difference techniques demonstrates that for certain classes of electrostatic calculations, such as determining absolute electrostatic solvation and rigid-binding free energies, the improved convergence properties of the BEM approach can have a significant impact on computed energetics. We also demonstrate that the improved accuracy offered by the curved-element BEM is important when more sophisticated techniques, such as nonrigid-binding models, are used to compute the relative electrostatic effects of molecular modifications. In addition, we show that electrostatic calculations requiring multiple solves using the same molecular geometry, such as
Mikell, Justin K.; Mourtada, Firas
2010-09-15
Purpose: To evaluate the dose distributions of an {sup 192}Ir source (model VS2000) in homogeneous water geometry calculated using a deterministic grid-based Boltzmann transport equation solver (GBBS) in the commercial treatment planning system (TPS) (BRACHYVISION-ACUROS v8.8). Methods: Using percent dose differences (%{Delta}D), the GBBS (BV-ACUROS) was compared to the (1) published TG-43 data, (2) MCNPX Monte Carlo (MC) simulations of the {sup 192}Ir source centered in a 15 cm radius water sphere, and (3) TG-43 output from the TPS using vendor supplied (BV-TG43-vendor) and user extended (BV-TG43-extended) 2D anisotropy functions F(r,{theta}). BV-ACUROS assumes 1 mm of NiTi cable, while the TPS TG-43 algorithm uses data based on a 15 cm cable. MC models of various cable lengths were simulated. Results: The MC simulations resulted in >20% dose deviations along the cable for 1, 2, and 3 mm cable lengths relative to 15 cm. BV-ACUROS comparisons with BV-TG43-vendor and BV-TG43-extended yielded magnitude of differences, consistent with those seen in MC simulations. However, differences >20% extended further ({theta}{<=}10 deg.) when using the vendor supplied anisotropy function F{sub ven}(r,{theta}). These differences were also seen in comparisons of F(r,{theta}) derived from the TPS output. Conclusions: The results suggest that %{Delta}D near the cable region is larger than previously estimated. The spatial distribution of the dose deviation is highly dependent on the reference TG-43 data used to compare to GBBS. The differences observed, while important to realize, should not have an impact on clinical dosimetry in homogeneous water.
Fenley, Marcia O; Mascagni, Michael; McClain, James; Silalahi, Alexander R J; Simonov, Nikolai A
2010-01-01
Dielectric continuum or implicit solvent models provide a significant reduction in computational cost when accounting for the salt-mediated electrostatic interactions of biomolecules immersed in an ionic environment. These models, in which the solvent and ions are replaced by a dielectric continuum, seek to capture the average statistical effects of the ionic solvent, while the solute is treated at the atomic level of detail. For decades, the solution of the three-dimensional Poisson-Boltzmann equation (PBE), which has become a standard implicit-solvent tool for assessing electrostatic effects in biomolecular systems, has been based on various deterministic numerical methods. Some deterministic PBE algorithms have drawbacks, which include a lack of properly assessing their accuracy, geometrical difficulties caused by discretization, and for some problems their cost in both memory and computation time. Our original stochastic method resolves some of these difficulties by solving the PBE using the Monte Carlo method (MCM). This new approach to the PBE is capable of efficiently solving complex, multi-domain and salt-dependent problems in biomolecular continuum electrostatics to high precision. Here we improve upon our novel stochastic approach by simultaneouly computating of electrostatic potential and solvation free energies at different ionic concentrations through correlated Monte Carlo (MC) sampling. By using carefully constructed correlated random walks in our algorithm, we can actually compute the solution to a standard system including the linearized PBE (LPBE) at all salt concentrations of interest, simultaneously. This approach not only accelerates our MCPBE algorithm, but seems to have cost and accuracy advantages over deterministic methods as well. We verify the effectiveness of this technique by applying it to two common electrostatic computations: the electrostatic potential and polar solvation free energy for calcium binding proteins that are compared
NASA Astrophysics Data System (ADS)
Li, Wu
2015-08-01
We demonstrate the ab initio electrical transport calculation limited by electron-phonon coupling by using the full solution of the Boltzmann transport equation (BTE), which applies equally to metals and semiconductors. Numerical issues are emphasized in this work. We show that the simple linear interpolation of the electron-phonon coupling matrix elements from a relatively coarse grid to an extremely fine grid can ease the calculational burden, which makes the calculation feasible in practice. For the Brillouin zone (BZ) integration of the transition probabilities involving one δ function, the Gaussian smearing method with a physical choice of locally adaptive broadening parameters is employed. We validate the calculation in the cases of n -type Si and Al. The calculated conductivity and mobility are in good agreement with experiments. In the metal case we also demonstrate that the Gaussian smearing method with locally adaptive broadening parameters works excellently for the BZ integration with double δ functions involved in the Eliashberg spectral function and its transport variant. The simpler implementation is the advantage of the Gaussian smearing method over the tetrahedron method. The accuracy of the relaxation time approximation and the approximation made by Allen [Phys. Rev. B 17, 3725 (1978), 10.1103/PhysRevB.17.3725] has been examined by comparing with the exact solution of BTE. We also apply our method to n -type monolayer MoS2, for which a mobility of 150 cm2 v-1 s-1 is obtained at room temperature. Moreover, the mean free paths are less than 9 nm, indicating that in the presence of grain boundaries the mobilities should not be effectively affected if the grain boundary size is tens of nanometers or larger. The ab initio approach demonstrated in this paper can be directly applied to other materials without the need for any a priori knowledge about the electron-phonon scattering processes, and can be straightforwardly extended to study cases with
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.''
Nonlinear heavy-ion-acoustic waves in an adiabatic collisionless bi-ion plasma
NASA Astrophysics Data System (ADS)
Hossen, M. A.; Rahman, M. M.; Hossen, M. R.; Mamun, A. A.
2017-03-01
The basic properties of heavy-ion-acoustic (HIA) waves have been investigated in a collisionless plasma system which is supposed to be composed of nonthermal electrons, Boltzmann distributed light ions, and adiabatic positively charged inertial heavy ions. The Kortewg-de Vries and Burgers equations are derived in nonplanar (cylindrical and spherical) geometry by employing the standard reductive perturbation method for studying the basic features (viz. amplitude, phase speed, etc.) of HIA solitary and shock waves, which are associated with either positive or negative potential. It is found that the effects of nonplanar geometry, adiabaticity of positively charged inertial heavy ions, the presence of nonthermal (Cairns distributed) electrons, and number densities of the plasma components significantly modify the basic features of nonplanar HIA waves. It has been observed that the properties of solitary and shock waves associated with HIA waves in a nonplanar geometry differ from those in a planar geometry. The implications of our results may be helpful in understanding the electrostatic perturbations in various laboratory and astrophysical plasma environments.
Collisionless Shocks and Particle Acceleration.
NASA Astrophysics Data System (ADS)
Malkov, M.
2016-12-01
Collisionless shocks emerged in the 50s and 60s of the last century as an important branch of plasma physics and have remained ever since. New applications pose new challenges to our understanding of collisionless shock mechanisms. Particle acceleration in astrophysical settings, primarily studied concerning the putative origin of cosmic rays (CR) in supernova remnant (SNR) shocks, stands out with the collisionless shock mechanism being the key. Among recent laboratory applications, a laser-based tabletop proton accelerator is an affordable compact alternative to big synchrotron accelerators. The much-anticipated proof of cosmic ray (CR) acceleration in supernova remnants is hindered by our limited understanding of collisionless shock mechanisms. Over the last decade, dramatically improved observations were puzzling the theorists with unexpected discoveries. The difference between the helium/carbon and proton CR rigidity (momentum to charge ratio) spectra, seemingly inconsistent with the acceleration and propagation theories, and the perplexing positron excess in the 10-300 GeV range are just two recent examples. The latter is now also actively discussed in the particle physics and CR communities as a possible signature of decay or annihilation of hypothetical dark matter particles. By considering an initial (injection) phase of a diffusive shock acceleration mechanism, including particle reflection off the shock front - where an elemental similarity of particle dynamics does not apply - I will discuss recent suggestions of how to address the new data from the collisionless shock perspective. The backreaction of accelerated particles on the shock structure, its environment, and visibility across the electromagnetic spectrum from radio to gamma rays is another key aspect of collisionless shock that will be discussed.
Lattice Boltzmann approach for complex nonequilibrium flows.
Montessori, A; Prestininzi, P; La Rocca, M; Succi, S
2015-10-01
We present a lattice Boltzmann realization of Grad's extended hydrodynamic approach to nonequilibrium flows. This is achieved by using higher-order isotropic lattices coupled with a higher-order regularization procedure. The method is assessed for flow across parallel plates and three-dimensional flows in porous media, showing excellent agreement of the mass flow with analytical and numerical solutions of the Boltzmann equation across the full range of Knudsen numbers, from the hydrodynamic regime to ballistic motion.
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
Clamping in Boltzmann machines.
Livesey, M
1991-01-01
A certain assumption that appears in the proof of correctness of the standard Boltzmann machine learning procedure is investigated. The assumption, called the clamping assumption, concerns the behavior of a Boltzmann machine when some of its units are clamped to a fixed state. It is argued that the clamping assumption is essentially an assertion of the time reversibility of a certain Markov chain underlying the behavior of the Boltzmann machine. As such, the clamping assumption is generally false, though it is certainly true of the Boltzmann machines themselves. The author also considers how the concept of the Boltzmann machine may be generalized while retaining the validity of the clamping assumption.
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.
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.
Lattice Boltzmann model for simulating temperature-sensitive ferrofluids.
Niu, Xiao-Dong; Yamaguchi, Hiroshi; Yoshikawa, Keisuke
2009-04-01
In this paper, a lattice Boltzmann model for simulating temperature-sensitive ferrofluids is presented. The lattice Boltzmann equation for modeling the magnetic field is formulated using a scalar magnetic potential. Introducing a time derivative into the original elliptic equation for the scalar potential leads to an advection-diffusion equation, with an effective velocity determined by the temperature gradient. The time derivative is multiplied by an adjustable preconditioning parameter to ensure that the lattice Boltzmann solution remain close to a solution of the original elliptic equation for the scalar potential. To test the present lattice Boltzmann model, numerical simulations for the thermomagnetic nature convection of the ferrofluids in a cubic cavity are carried out. Good agreement between the obtained results and experimental data shows that the present lattice Boltzmann model is promising for studying temperature-sensitive ferrofluid flows.
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.
Hierarchical Boltzmann simulations and model error estimation
NASA Astrophysics Data System (ADS)
Torrilhon, Manuel; Sarna, Neeraj
2017-08-01
A hierarchical simulation approach for Boltzmann's equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics, but a subsequent refinement allows to successively improve the result to the complete Boltzmann result. We use Hermite discretization, or moment equations, for the steady linearized Boltzmann equation for a proof-of-concept of such a framework. All representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous Galerkin formulation. We demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains. The hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations. We present various model errors for a flow through a curved channel with obstacles.
Interaction of two collisionless shocks
NASA Technical Reports Server (NTRS)
Cargill, P. J.; Goodrich, C. C.; Papadopoulos, K.
1986-01-01
Kinetic simulations of the interaction between two collisionless shocks are presented. During the collision of two perpendicular shocks, the shock electromagnetic field structures pass through each other, while the previously shocked ions are kept separate by the electric field arising in the collision. When two supercritical shocks collide, a fraction of ions are accelerated up to an order of magnitude in energy by first being reflected at one shock, then interacting with the electric fields of the other shock.
NASA Astrophysics Data System (ADS)
Tripathy, Sushanta; Khuntia, Arvind; Tiwari, Swatantra Kumar; Sahoo, Raghunath
2017-05-01
In the continuation of our previous work, the transverse-momentum (pT) spectra and nuclear modification factor (R_{AA}) are derived using the relaxation time approximation of Boltzmann Transport Equation (BTE). The initial pT-distribution used to describe p + p collisions has been studied with the perturbative-Quantum Chromodynamics (pQCD) inspired power-law distribution, Hagedorn's empirical formula and with the Tsallis non-extensive statistical distribution. The non-extensive Tsallis distribution is observed to describe the complete range of the transverse-momentum spectra. The Boltzmann-Gibbs Blast Wave (BGBW) distribution is used as the equilibrium distribution in the present formalism, to describe the pT-distribution and nuclear modification factor in nucleus-nucleus collisions. The experimental data for Pb+Pb collisions at √{s_{NN}} = 2.76 TeV at the Large Hadron Collider at CERN have been analyzed for pions, kaons, protons, K^{\\ast0} and φ. It is observed that the present formalism while explaining the transverse-momentum spectra up to 5 GeV/ c, explains the nuclear modification factor very well up to 8 GeV/ c in pT for all these particles except for protons. R_{AA} is found to be independent of the degree of non-extensivity, q_{pp} after pT ˜ 8 GeV/ c.
Lattice Boltzmann model for compressible fluids
NASA Technical Reports Server (NTRS)
Alexander, F. J.; Chen, H.; Chen, S.; Doolen, G. D.
1992-01-01
A lattice Boltzmann model is derived which simulates compressible fluids. By choosing the parameters of the equilibrium distribution appropriately, the sound speed (which may be set arbitrarily low), bulk viscosity, and kinematic viscosity can be selected. This model simulates compressible flows and can include shocks. With a proper rescaling and zero-sound speed, this model simulates Burgers's equation. The viscosity determined by a Chapman-Enskog expansion compares well with that measured form simulations. The exact solutions of Burgers's equation on the unit circle are compared to solutions of lattice Boltzmann model finding reasonable agreement.
Suzuki, Hideyuki; Imura, Jun-ichi; Horio, Yoshihiko; Aihara, Kazuyuki
2013-01-01
The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented. PMID:23558425
NASA Astrophysics Data System (ADS)
Suzuki, Hideyuki; Imura, Jun-Ichi; Horio, Yoshihiko; Aihara, Kazuyuki
2013-04-01
The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented.
Suzuki, Hideyuki; Imura, Jun-ichi; Horio, Yoshihiko; Aihara, Kazuyuki
2013-01-01
The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented.
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.
Collisionless relaxation in beam-plasma systems
Backhaus, Ekaterina Yu.
2001-01-01
This thesis reports the results from the theoretical investigations, both numerical and analytical, of collisionless relaxation phenomena in beam-plasma systems. Many results of this work can also be applied to other lossless systems of plasma physics, beam physics and astrophysics. Different aspects of the physics of collisionless relaxation and its modeling are addressed. A new theoretical framework, named Coupled Moment Equations (CME), is derived and used in numerical and analytical studies of the relaxation of second order moments such as beam size and emittance oscillations. This technique extends the well-known envelope equation formalism, and it can be applied to general systems with nonlinear forces. It is based on a systematic moment expansion of the Vlasov equation. In contrast to the envelope equation, which is derived assuming constant rms beam emittance, the CME model allows the emittance to vary through coupling to higher order moments. The CME model is implemented in slab geometry in the absence of return currents. The CME simulation yields rms beam sizes, velocity spreads and emittances that are in good agreement with particle-in-cell (PIC) simulations for a wide range of system parameters. The mechanism of relaxation is also considered within the framework of the CME system. It is discovered that the rapid relaxation or beam size oscillations can be attributed to a resonant coupling between different modes of the system. A simple analytical estimate of the relaxation time is developed. The final state of the system reached after the relaxation is complete is investigated. New and accurate analytical results for the second order moments in the phase-mixed state are obtained. Unlike previous results, these connect the final values of the second order moments with the initial beam mismatch. These analytical estimates are in good agreement with the CME model and PIC simulations. Predictions for the final density and temperature are developed that show
Transport of parallel momentum by collisionless drift wave turbulence
Diamond, P. H.; McDevitt, C. J.; Guercan, Oe. D.; Hahm, T. S.; Naulin, V.
2008-01-15
This paper presents a novel, unified approach to the theory of turbulent transport of parallel momentum by collisionless drift waves. The physics of resonant and nonresonant off-diagonal contributions to the momentum flux is emphasized, and collisionless momentum exchange between waves and particles is accounted for. Two related momentum conservation theorems are derived. These relate the resonant particle momentum flux, the wave momentum flux, and the refractive force. A perturbative calculation, in the spirit of Chapman-Enskog theory, is used to obtain the wave momentum flux, which contributes significantly to the residual stress. A general equation for mean k{sub parallel} (
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'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 transport model for heavy-ion collisions
Ayik, S. |
1994-06-01
Heavy-ion collisions at intermediate energies exhibit catastrophic phenomena which requires descriptions based on stochastic transport models. First, the Boltzmann-Langevin model, which provides an example of such stochastic approaches, is briefly described. Then, a projection method for obtaining numerical solutions of the Boltzmann-Langevin equation is discussed. Finally, some applications of the model to heavy-ion collisions are presented.
Lattice Boltzmann Method for Two-Dimensional Unsteady Incompressible Flow
NASA Astrophysics Data System (ADS)
Mužík, Juraj
2016-12-01
A Lattice Boltzmann method is used to analyse incompressible fluid flow in a two-dimensional cavity and flow in the channel past cylindrical obstacle. The method solves the Boltzmann's transport equation using simple computational grid - lattice. With the proper choice of the collision operator, the Boltzmann's equation can be converted into incompressible Navier-Stokes equation. Lid-driven cavity benchmark case for various Reynolds numbers and flow past cylinder is presented in the article. The method produces stable solutions with results comparable to those in literature and is very easy to implement.
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
van der Weg, P B
2009-11-15
When the electrical contribution in the electrochemical potential of ionic species is reduced with a factor two from its traditional value, the ionic activity coefficients are closer to unity and need to account only for the short-range interactions at high concentrations. Such a change is needed to remove inconsistencies in the models and to comply with basic electrostatic principles. This will have serious implications, in many applications. For example, it will cause changes in many of the fundamental models that are used to explain measured data in the dilute range for the various disciplines that embrace classical electrochemistry. Examples are Debye-Hückel and Gouy-Chapman theories; Maxwell-Boltzmann distribution; Nernst theory; Donnan equilibrium, etc. These theories impact a wide range of observable phenomena such as activity coefficients of electrolytes, diffuse double layer capacitance, electrode potentials, membrane potentials, streaming potentials, electro-osmosis, flotation, sedimentation, corrosion, charged micellar behaviour, space-charge semiconductor behaviour, and electrical phenomena in biological tissue, e.g. membranes; cells; and nerves, etcetera.
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.
A time-reversal lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Vergnault, E.; Malaspinas, O.; Sagaut, P.
2011-09-01
In this paper we address the time-reversed simulation of viscous flows by the lattice Boltzmann method (LB). The theoretical derivation of the reversed LB from the Boltzmann equation is detailed, and the method implemented for weakly compressible flows using the D2Q9 scheme. The implementation of boundary conditions is also discussed. The accuracy and stability are illustrated by four test cases, namely the propagation of an acoustic wave in a medium at rest and in an uniform mean flow, the Taylor-Green vortex decay and the vortex pair-wall collision.
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.
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.
Magnetohydrodynamics for collisionless plasmas from the gyrokinetic perspective
Lee, W. W.
2016-07-15
The effort to obtain a set of MagnetoHydroDynamic (MHD) equations for a magnetized collisionless plasma was started nearly 60 years ago by Chew et al. [Proc. R. Soc. London, Ser. A 236(1204), 112–118 (1956)]. Many attempts have been made ever since. Here, we will show the derivation of a set of these equations from the gyrokinetic perspective, which we call it gyrokinetic MHD, and it is different from the conventional ideal MHD. However, this new set of equations still has conservation properties and, in the absence of fluctuations, recovers the usual MHD equilibrium. Furthermore, the resulting equations allow for the plasma pressure balance to be further modified by finite-Larmor-radius effects in regions with steep pressure gradients. The present work is an outgrowth of the paper on “Alfven Waves in Gyrokinetic Plasmas” by Lee and Qin [Phys. Plasmas 10, 3196 (2003)].
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
How good is the Lattice Boltzmann method?
NASA Astrophysics Data System (ADS)
Kocheemoolayil, Joseph; Barad, Michael; Kiris, Cetin
2016-11-01
Conflicting opinions exist in literature regarding how efficient the lattice Boltzmann method is relative to high-order finite difference approximations of the Navier-Stokes equations on Cartesian meshes, especially at high Mach numbers. We address the question from the pragmatic viewpoint of a practitioner. Dispersion, dissipation and aliasing errors of various lattice Boltzmann models are systematically quantified. The number of floating point operations and memory required for a desired accuracy level are carefully compared for the two numerical methods. Turbulent kinetic energy budgets for several standard test cases such as the decaying Taylor-Green vortex problem are used to evaluate how effective the stabilization mechanisms necessary for lattice Boltzmann method at high Reynolds numbers are. Detailed comments regarding the cyclomatic complexity of the underlying software, scalability of the underlying algorithm on state-of-the-art high-performance computing platforms and wall clock times and relative accuracy for selected simulations conducted using the two approaches are also made.
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
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
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.
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.
Boltzmann hierarchy for interacting neutrinos I: formalism
Oldengott, Isabel M.; Rampf, Cornelius; Wong, Yvonne Y.Y. E-mail: cornelius.rampf@port.ac.uk
2015-04-01
Starting from the collisional Boltzmann equation, we derive for the first time and from first principles the Boltzmann hierarchy for neutrinos including interactions with a scalar particle. Such interactions appear, for example, in majoron-like models of neutrino mass generation. We study two limits of the scalar mass: (i) An extremely massive scalar whose only role is to mediate an effective 4-fermion neutrino-neutrino interaction, and (ii) a massless scalar that can be produced in abundance and thus demands its own Boltzmann hierarchy. In contrast to, e.g., the first-order Boltzmann hierarchy for Thomson-scattering photons, our interacting neutrino/scalar Boltzmann hierarchies contain additional momentum-dependent collision terms arising from a non-negligible energy transfer in the neutrino-neutrino and neutrino-scalar interactions. This necessitates that we track each momentum mode of the phase space distributions individually, even if the particles were massless. Comparing our hierarchy with the commonly used (c{sub eff}{sup 2},c{sub vis}{sup 2})-parameterisation, we find no formal correspondence between the two approaches, which raises the question of whether the latter parameterisation even has an interpretation in terms of particle scattering. Lastly, although we have invoked majoron-like models as a motivation for our study, our treatment is in fact generally applicable to all scenarios in which the neutrino and/or other ultrarelativistic fermions interact with scalar particles.
a Study of One-Dimensional Nonlinear Hydromagnetic Waves and Collisionless Shocks.
NASA Astrophysics Data System (ADS)
Lyu, Ling-Hsiao
A variety of nonlinear hydromagnetic waves have been observed in the collisionless solar wind plasma. A comprehensive theoretical study of nonlinear hydromagnetic waves, including rotational discontinuities and collisionless shocks, is carried out in this thesis by means of both analytical solutions and numerical simulations. Nonlinear hydromagnetic waves are governed by the interplay of the dispersion process, the collisionless dissipation process and the nonlinear steepening process. The purpose of this thesis is to understand the nonlinear behavior of hydromagnetic waves in terms of these fundamental processes. It is shown that the rotational discontinuity structures observed in the solar wind and at the magnetopause are nonlinear Alfven wave solutions of the collisionless two-fluid plasma equations. In these nonlinear wave solutions, nonlinear steepening is self-consistently balanced by dispersion. Collisionless viscous dissipation is the dominant dissipation in high Mach number shocks, which converts the flow energy into thermal energy. Hybrid simulations show that the collisionless viscous dissipation can result from the reflection and pitch-angle scattering of incoming ions flowing through the magnetic structures in the shock transition region. Collisionless dissipations in hydromagnetic shocks is governed by the magnetic structures in the shock transition region. The dissipation in turn can modify the wave structures and balance the nonlinear steepening. However, such delicate balance of the dispersion, dissipation, and nonlinear steepening has been observed to break down momentarily in high Mach number quasi-parallel shocks. This leads to the so-called cyclic shock front reformation seen in the hybrid simulations. The shock front reformation can be explained in terms of momentary off-balance between the dispersion-dissipation on the one hand and the nonlinear steepening on the other hand. The off-balance occurs after a significant fraction of incoming ions
Linear collisionless Landau damping in Hilbert space
NASA Astrophysics Data System (ADS)
Zocco, Alessandro
2015-08-01
The equivalence between the Laplace transform (Landau, J. Phys. USSR 10 (1946), 25) and Hermite transform (Zocco and Schekochihin, Phys. Plasmas 18, 102309 (2011)) solutions of the linear collisionless Landau damping problem is proven.
Meshless lattice Boltzmann method for the simulation of fluid flows.
Musavi, S Hossein; Ashrafizaadeh, Mahmud
2015-02-01
A meshless lattice Boltzmann numerical method is proposed. The collision and streaming operators of the lattice Boltzmann equation are separated, as in the usual lattice Boltzmann models. While the purely local collision equation remains the same, we rewrite the streaming equation as a pure advection equation and discretize the resulting partial differential equation using the Lax-Wendroff scheme in time and the meshless local Petrov-Galerkin scheme based on augmented radial basis functions in space. The meshless feature of the proposed method makes it a more powerful lattice Boltzmann solver, especially for cases in which using meshes introduces significant numerical errors into the solution, or when improving the mesh quality is a complex and time-consuming process. Three well-known benchmark fluid flow problems, namely the plane Couette flow, the circular Couette flow, and the impulsively started cylinder flow, are simulated for the validation of the proposed method. Excellent agreement with analytical solutions or with previous experimental and numerical results in the literature is observed in all the simulations. Although the computational resources required for the meshless method per node are higher compared to that of the standard lattice Boltzmann method, it is shown that for cases in which the total number of nodes is significantly reduced, the present method actually outperforms the standard lattice Boltzmann method.
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.
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.
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.
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.
Lattice Boltzmann Stokesian dynamics.
Ding, E J
2015-11-01
Lattice Boltzmann Stokesian dynamics (LBSD) is presented for simulation of particle suspension in Stokes flows. This method is developed from Stokesian dynamics (SD) with resistance and mobility matrices calculated using the time-independent lattice Boltzmann algorithm (TILBA). TILBA is distinguished from the traditional lattice Boltzmann method (LBM) in that a background matrix is generated prior to the calculation. The background matrix, once generated, can be reused for calculations for different scenarios, thus the computational cost for each such subsequent calculation is significantly reduced. The LBSD inherits the merits of the SD where both near- and far-field interactions are considered. It also inherits the merits of the LBM that the computational cost is almost independent of the particle shape.
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.
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.
Maxwell iteration for the lattice Boltzmann method with diffusive scaling
NASA Astrophysics Data System (ADS)
Zhao, Weifeng; Yong, Wen-An
2017-03-01
In this work, we present an alternative derivation of the Navier-Stokes equations from Bhatnagar-Gross-Krook models of the lattice Boltzmann method with diffusive scaling. This derivation is based on the Maxwell iteration and can expose certain important features of the lattice Boltzmann solutions. Moreover, it will be seen to be much more straightforward and logically clearer than the existing approaches including the Chapman-Enskog expansion.
Maxwell iteration for the lattice Boltzmann method with diffusive scaling.
Zhao, Weifeng; Yong, Wen-An
2017-03-01
In this work, we present an alternative derivation of the Navier-Stokes equations from Bhatnagar-Gross-Krook models of the lattice Boltzmann method with diffusive scaling. This derivation is based on the Maxwell iteration and can expose certain important features of the lattice Boltzmann solutions. Moreover, it will be seen to be much more straightforward and logically clearer than the existing approaches including the Chapman-Enskog expansion.
Lattice Boltzmann method for electromagnetic wave propagation
NASA Astrophysics Data System (ADS)
Hanasoge, S. M.; Succi, S.; Orszag, S. A.
2011-10-01
We present a new Lattice Boltzmann (LB) formulation to solve the Maxwell equations for electromagnetic (EM) waves propagating in a heterogeneous medium. By using a pseudo-vector discrete Boltzmann distribution, the scheme is shown to reproduce the continuum Maxwell equations. The technique compares well with a pseudo-spectral method at solving for two-dimensional wave propagation in a heterogeneous medium, which by design contains substantial contrasts in the refractive index. The extension to three dimensions follows naturally and, owing to the recognized efficiency of LB schemes for parallel computation in irregular geometries, it gives a powerful method to numerically simulate a wide range of problems involving EM wave propagation in complex media.
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.
Nonlocal Boltzmann theory of plasma channels
NASA Astrophysics Data System (ADS)
Yu, S. S.; Melendez, R. E.
1983-01-01
The mathematical framework for the Lawrence Livermore National Lab. (LLNL) code NUTS is developed. This code is designed to study the evolution of an electron beam generated plasma channel at all pressures. The Boltzmann treatment of the secondary electrons presented include all inertial, nonlocal, electric and magnetic effects, as well as effects of atomic collisions. Field equations are advanced simultaneously and self-consistently with the evolving plasma currents.
Kinetic simulaitons of astrophysical collisionless shocks (Invited)
NASA Astrophysics Data System (ADS)
Spitkovsky, A.
2009-12-01
Nonthermal emission from a variety of astrophysical sources, including relativistic jets and supernova remnants, is often attributed to collisionless shocks. These shocks are inferred to accelerate particles and in some cases strongly amplify magnetic fields. How this happens remains to be clarified through both theory and observations. In this talk, I will present a summary of recent progress in kinetic modeling of collisionless shocks using particle-in-cell simulations. I will discuss the internal structure of relativistic and non-relativistic shocks, concentrating on the conditions necessary for particle acceleration. Large-scale shock simulations show ab-initio Fermi acceleration of particles from the thermal pool to power-law distributions and can set constraints on the shock acceleration efficiency and geometry. Other results that will be discussed include the amplification of magnetic fields by accelerated particles through streaming instabilities, and the electron-ion temperature equilibration in collisionless shocks.
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.
Laser-Driven Magnetized Collisionless Shocks
NASA Astrophysics Data System (ADS)
Schaeffer, Derek
2016-10-01
Collisionless shocks - supersonic plasma flows in which the interaction length scale is much shorter than the collisional mean free path - are common phenomena in space and astrophysical systems, including the solar wind, coronal mass ejections, supernovae remnants, and the jets of active galactic nuclei. These systems have been studied for decades, and in many the shocks are believed to efficiently accelerate particles to some of the highest observed energies. Only recently, however, have laser and diagnostic capabilities evolved sufficiently to allow the detailed study in the laboratory of the microphysics of collisionless shocks over a large parameter regime. We present experiments that demonstrate the formation of collisionless shocks utilizing the Phoenix laser laboratory and the LArge Plasma Device (LAPD) at UCLA. We also show recent observations of magnetized collisionless shocks on the Omega EP laser facility that extend the LAPD results to higher laser energy, background magnetic field, and ambient plasma density, and that may be relevant to recent experiments on strongly driven magnetic reconnection. Lastly, we discuss a new experimental regime for shocks with results from high-repetition (1 Hz), volumetric laser-driven measurements on the LAPD. These large parameter scales allow us to probe the formation physics of collisionless shocks over several Alfvénic Mach numbers (MA), from shock precursors (magnetosonic solitons with MA < 1) to subcritical (MA < 3) and supercritical (MA > 3) shocks. The results show that collisionless shocks can be generated using a laser-driven magnetic piston, and agree well with both 2D and 3D hybrid and PIC simulations. Additionally, using radiation-hydrodynamic modeling and measurements from multiple diagnostics, the different shock regimes are characterized with dimensionless formation parameters, allowing us to place disparate experiments in a common and predictive framework.
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.
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.
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.
Multiscale lattice Boltzmann schemes for low Mach number flows.
Filippova, Olga; Schwade, Bettina; Hänel, Dieter
2002-03-15
A low Mach number approximation (LMNA) of the Navier-Stokes equations is widely used in numerical methods for the simulation of low-speed thermal and athermal flows. The advanced lattice Boltzmann approach (Bhatnagar-Gross-Krook) for the solution of the LMNA equations is discussed and its performance is compared with the performance of the commercial CFD code FLUENT 5.
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.
New Description of Collisionless Dynamics in Beam-Plasma Systems
NASA Astrophysics Data System (ADS)
Backhaus, E. Yu.; Wurtele, J. S.
1999-11-01
The dynamics of collisionless relaxation has been an active topic of research in plasma and astrophysical systems for over thirty years. This is studied in the context of one-dimensional beam dynamics in overdense plasmas with slab geometry. A new description is presented in which the phase space distribution function is decomposed into a set of modes and the system is modeled as an initial value problem for the amplitudes of these modes. The model can also be considered as a natural extension of the well-celebrated envelope equation formalism to include the coupling to higher order moments. The predicted dynamics of the beam rms properties and the emittance growth during the violent relaxation are in a good agreement with the results of PIC simulations over a wide range of initial conditions. Predictions for beam density and temperature evolution contain main features observed in PIC simulations. Simple analytical estimates for the beam properties after phase-mixing agree with PIC simulation within 10% over a factor of 20 change in the initial mismatch of a gaussian beam. The method offers not only a fast numerical technique to describe the dynamics of the bulk beam properties but also offers new insights into the physics of collisionless relaxation: the resonant coupling between modes is shown to cause the fast relaxation of rms beam properties.
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
Noise source identification with the lattice Boltzmann method.
Vergnault, Etienne; Malaspinas, Orestis; Sagaut, Pierre
2013-03-01
In this paper the sound source identification problem is addressed with the use of the lattice Boltzmann method. To this aim, a time-reversed problem coupled to a complex differentiation method is used. In order to circumvent the inherent instability of the time-reversed lattice Boltzmann scheme, a method based on a split of the lattice Boltzmann equation into a mean and a perturbation component is used. Lattice Boltzmann method formulation around an arbitrary base flow is recalled and specific applications to acoustics are presented. The implementation of the noise source detection method for two-dimensional weakly compressible (low Mach number) flows is discussed, and the applicability of the method is demonstrated.
Boltzmann-type control of opinion consensus through leaders.
Albi, G; Pareschi, L; Zanella, M
2014-11-13
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.
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.
The fundamental and universal nature of Boltzmann`s constant
Biedenharn, L.C.; Solem, J.C.
1996-07-01
The nature of Boltzmann`s constant is very unclear in the physics literature. In the first part of this paper, on general considerations, the authors examine this situation in detail and demonstrate the conclusion that Boltzmann`s constant is indeed both fundamental and universal. As a consequence of their development they find there is an important implication of this work for the problem of the entropy of information. In the second part they discuss, Szilard`s famous construction showing in detail how his result is incompatible with the demonstrations in both parts 1 and 2.
Lattice Boltzmann model for traffic flow.
Meng, Jianping; Qian, Yuehong; Li, Xingli; Dai, Shiqiang
2008-03-01
Mesoscopic models for traffic flows are usually difficult to be employed because of the appearance of integro-differential terms in the models. In this work, a lattice Boltzmann model for traffic flow is introduced on the basis of the existing kinetics models by using the Bhatnagar-Gross-Krook-type approximation interaction term in the Boltzmann equation and discretizing it in time and phase space. The so-obtained model is simple while the relevant parameters are physically meaningful. Together with its discrete feature, the model can be easily used to investigate numerically the behavior of traffic flows. In consequence, the macroscopic dynamics of the model is derived using the Taylor and Chapman-Enskog expansions. For validating the model, numerical simulations are conducted under the periodic boundary conditions. It is found that the model could reasonably reproduce the fundamental diagram. Moreover, certain interesting physical phenomena can be captured by the model, such as the metastability and stop-and-go phenomena.
Investigation of the kinetic model equations.
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.
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.
Galilean-invariant lattice-Boltzmann models with H theorem
NASA Astrophysics Data System (ADS)
Boghosian, Bruce M.; Love, Peter J.; Coveney, Peter V.; Karlin, Iliya V.; Succi, Sauro; Yepez, Jeffrey
2003-08-01
We demonstrate that the requirement of Galilean invariance determines the choice of H function for a wide class of entropic lattice-Boltzmann models for the incompressible Navier-Stokes equations. The required H function has the form of the Burg entropy for D=2, and of a Tsallis entropy with q=1-(2/D) for D>2, where D is the number of spatial dimensions. We use this observation to construct a fully explicit, unconditionally stable, Galilean-invariant, lattice-Boltzmann model for the incompressible Navier-Stokes equations, for which attainable Reynolds number is limited only by grid resolution.
Collisionless Spectral Kinetic Simulation of Ideal Multipole Resonance Probe
NASA Astrophysics Data System (ADS)
Gong, Junbo; Wilczek, Sebastian; Szeremley, Daniel; Oberrath, Jens; Eremin, Denis; Dobrygin, Wladislaw; Schilling, Christian; Friedrichs, Michael; Brinkmann, Ralf Peter
2016-09-01
Active Plasma Resonance Spectroscopy denotes a class of industry-compatible plasma diagnostic methods which utilize the natural ability of plasmas to resonate on or near the electron plasma frequency ωpe. One particular realization of APRS with a high degree of geometric and electric symmetry is the Multipole Resonance Probe (MRP). The Ideal MRP(IMRP) is an even more symmetric idealization which is suited for theoretical investigations. In this work, a spectral kinetic scheme is presented to investigate the behavior of the IMRP in the low pressure regime. However, due to the velocity difference, electrons are treated as particles whereas ions are only considered as stationary background. In the scheme, the particle pusher integrates the equations of motion for the studied particles, the Poisson solver determines the electric field at each particle position. The proposed method overcomes the limitation of the cold plasma model and covers kinetic effects like collisionless damping.
Poisson–Boltzmann versus Size-Modified Poisson–Boltzmann Electrostatics Applied to Lipid Bilayers
2015-01-01
Mean-field methods, such as the Poisson–Boltzmann equation (PBE), are often used to calculate the electrostatic properties of molecular systems. In the past two decades, an enhancement of the PBE, the size-modified Poisson–Boltzmann equation (SMPBE), has been reported. Here, the PBE and the SMPBE are reevaluated for realistic molecular systems, namely, lipid bilayers, under eight different sets of input parameters. The SMPBE appears to reproduce the molecular dynamics simulation results better than the PBE only under specific parameter sets, but in general, it performs no better than the Stern layer correction of the PBE. These results emphasize the need for careful discussions of the accuracy of mean-field calculations on realistic systems with respect to the choice of parameters and call for reconsideration of the cost-efficiency and the significance of the current SMPBE formulation. PMID:25426875
Poisson-Boltzmann versus Size-Modified Poisson-Boltzmann Electrostatics Applied to Lipid Bilayers.
Wang, Nuo; Zhou, Shenggao; Kekenes-Huskey, Peter M; Li, Bo; McCammon, J Andrew
2014-12-26
Mean-field methods, such as the Poisson-Boltzmann equation (PBE), are often used to calculate the electrostatic properties of molecular systems. In the past two decades, an enhancement of the PBE, the size-modified Poisson-Boltzmann equation (SMPBE), has been reported. Here, the PBE and the SMPBE are reevaluated for realistic molecular systems, namely, lipid bilayers, under eight different sets of input parameters. The SMPBE appears to reproduce the molecular dynamics simulation results better than the PBE only under specific parameter sets, but in general, it performs no better than the Stern layer correction of the PBE. These results emphasize the need for careful discussions of the accuracy of mean-field calculations on realistic systems with respect to the choice of parameters and call for reconsideration of the cost-efficiency and the significance of the current SMPBE formulation.
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.
Electrostatic forces in the Poisson-Boltzmann systems.
Xiao, Li; Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2013-09-07
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.
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.
Study of astrophysical collisionless shocks at NIF
NASA Astrophysics Data System (ADS)
Park, Hye-Sook; Higginson, D. P.; Huntington, C. M.; Pollock, B. B.; Remington, B. A.; Rinderknecht, H.; Ross, J. S.; Ryutov, D. D.; Swadling, G. F.; Wilks, S. C.; Sakawa, Y.; Spitkovsky, A.; Petrasso, R.; Li, C. K.; Zylstra, A. B.; Lamb, D.; Tzeferacos, P.; Gregori, G.; Meinecke, J.; Manuel, M.; Froula, D.; Fiuza, F.
2016-10-01
High Mach number astrophysical plasmas can create collisionless shocks via plasma instabilities and turbulence that are responsible for magnetic field generations and cosmic ray acceleration. Recently, many laboratory experiments were successful to observe the Weibel instabilities and self-generated magnetic fields using high-power lasers that generated interpenetrating plasma flows. In order to create a fully formed shock, a series of NIF experiments have begun. The characteristics of flow interaction have been diagnosed by the neutrons and protons generated via beam-beam deuteron interactions, the x-ray emission from the hot plasmas and proton probe generated by imploding DHe3 capsules. This paper will present the latest results from the NIF collisionless shock experiments. Prepared by LLNL under Contract DE-AC52-07NA27344.
Finite-dimensional collisionless kinetic theory
NASA Astrophysics Data System (ADS)
Burby, J. W.
2017-03-01
A collisionless kinetic plasma model may often be cast as an infinite-dimensional noncanonical Hamiltonian system. I show that, when this is the case, the model can be discretized in space and particles while preserving its Hamiltonian structure, thereby producing a finite-dimensional Hamiltonian system that approximates the original kinetic model. I apply the general theory to two example systems: the relativistic Vlasov-Maxwell system with spin and a gyrokinetic Vlasov-Maxwell system.
Learning thermodynamics with Boltzmann machines
NASA Astrophysics Data System (ADS)
Torlai, Giacomo; Melko, Roger G.
2016-10-01
A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modeling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.
Stratified Simulations of Collisionless Accretion Disks
NASA Astrophysics Data System (ADS)
Hirabayashi, Kota; Hoshino, Masahiro
2017-06-01
This paper presents a series of stratified-shearing-box simulations of collisionless accretion disks in the recently developed framework of kinetic magnetohydrodynamics (MHD), which can handle finite non-gyrotropy of a pressure tensor. Although a fully kinetic simulation predicted a more efficient angular-momentum transport in collisionless disks than in the standard MHD regime, the enhanced transport has not been observed in past kinetic-MHD approaches to gyrotropic pressure anisotropy. For the purpose of investigating this missing link between the fully kinetic and MHD treatments, this paper explores the role of non-gyrotropic pressure and makes the first attempt to incorporate certain collisionless effects into disk-scale, stratified disk simulations. When the timescale of gyrotropization was longer than, or comparable to, the disk-rotation frequency of the orbit, we found that the finite non-gyrotropy selectively remaining in the vicinity of current sheets contributes to suppressing magnetic reconnection in the shearing-box system. This leads to increases both in the saturated amplitude of the MHD turbulence driven by magnetorotational instabilities and in the resultant efficiency of angular-momentum transport. Our results seem to favor the fast advection of magnetic fields toward the rotation axis of a central object, which is required to launch an ultra-relativistic jet from a black hole accretion system in, for example, a magnetically arrested disk state.
Ion-conserving Poisson-Boltzmann theory.
Sugioka, Hideyuki
2012-07-01
It is well known that the Poisson-Nernst-Planck (PNP) theory and the classical Gouy-Chapman theory are inconsistent at a high applied voltage. For solving this problem, we propose an ion-conserving Poisson-Boltzmann theory, which shows remarkable agreement with the numerical PNP solutions, even at a high applied voltage. In other words, we have found the exact analytical solutions for steady PNP equations; we believe that this finding greatly contributes to understanding surface science between solids and liquids.
The Poisson-Helmholtz-Boltzmann model.
Bohinc, K; Shrestha, A; May, S
2011-10-01
We present a mean-field model of a one-component electrolyte solution where the mobile ions interact not only via Coulomb interactions but also through a repulsive non-electrostatic Yukawa potential. Our choice of the Yukawa potential represents a simple model for solvent-mediated interactions between ions. We employ a local formulation of the mean-field free energy through the use of two auxiliary potentials, an electrostatic and a non-electrostatic potential. Functional minimization of the mean-field free energy leads to two coupled local differential equations, the Poisson-Boltzmann equation and the Helmholtz-Boltzmann equation. Their boundary conditions account for the sources of both the electrostatic and non-electrostatic interactions on the surface of all macroions that reside in the solution. We analyze a specific example, two like-charged planar surfaces with their mobile counterions forming the electrolyte solution. For this system we calculate the pressure between the two surfaces, and we analyze its dependence on the strength of the Yukawa potential and on the non-electrostatic interactions of the mobile ions with the planar macroion surfaces. In addition, we demonstrate that our mean-field model is consistent with the contact theorem, and we outline its generalization to arbitrary interaction potentials through the use of a Laplace transformation.
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.
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.
NASA Technical Reports Server (NTRS)
Tsuge, S.
1974-01-01
The Navier-Stokes equation expressed in terms of Klimontovich microscopic density is compared with the conventional one based on the Boltzmann formalism. Their difference is described by the two-point BBGKY equation. An implication is given that the averaging procedure from the Klimontovich to the Boltzmann formalism wipes out not only thermal agitation, but also a certain hydrodynamic turbulence. This is substantiated by agreement with measurements due to Schubauer and Klebanoff.
NASA Technical Reports Server (NTRS)
Tsuge, S.
1974-01-01
The Navier-Stokes equation expressed in terms of Klimontovich microscopic density is compared with the conventional one based on the Boltzmann formalism. Their difference is described by the two-point BBGKY equation. An implication is given that the averaging procedure from the Klimontovich to the Boltzmann formalism wipes out not only thermal agitation, but also a certain hydrodynamic turbulence. This is substantiated by agreement with measurements due to Schubauer and Klebanoff.
Perfect entropy functions of the Lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Karlin, I. V.; Ferrante, A.; Öttinger, H. C.
1999-07-01
In this letter, we derive entropy functions whose local equilibria are suitable to recover the Navier-Stokes equations in the framework of the Lattice Boltzmann method. For the two-dimensional nine-velocity lattice we demonstrate that such an entropy function is unique, and that the expansion of the corresponding local equilibrium is the well-known local equilibrium of Y. H. Qian et al. (Europhys. Lett., 17 (1992) 479). Based on the knowledge of entropy functions, we introduce a new version of the Lattice Boltzmann method with an H-theorem built in.
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.
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.
Information geometry of Boltzmann machines.
Amari, S; Kurata, K; Nagaoka, H
1992-01-01
A Boltzmann machine is a network of stochastic neurons. The set of all the Boltzmann machines with a fixed topology forms a geometric manifold of high dimension, where modifiable synaptic weights of connections play the role of a coordinate system to specify networks. A learning trajectory, for example, is a curve in this manifold. It is important to study the geometry of the neural manifold, rather than the behavior of a single network, in order to know the capabilities and limitations of neural networks of a fixed topology. Using the new theory of information geometry, a natural invariant Riemannian metric and a dual pair of affine connections on the Boltzmann neural network manifold are established. The meaning of geometrical structures is elucidated from the stochastic and the statistical point of view. This leads to a natural modification of the Boltzmann machine learning rule.
Lattice Boltzmann method for linear oscillatory noncontinuum flows
NASA Astrophysics Data System (ADS)
Shi, Yong; Yap, Ying Wan; Sader, John E.
2014-03-01
Oscillatory gas flows are commonly generated by micro- and nanoelectromechanical systems. Due to their small size and high operating frequencies, these devices often produce noncontinuum gas flows. Theoretical analysis of such flows requires solution of the unsteady Boltzmann equation, which can present a formidable challenge. In this article, we explore the applicability of the lattice Boltzmann (LB) method to such linear oscillatory noncontinuum flows; this method is derived from the linearized Boltzmann Bhatnagar-Gross-Krook (BGK) equation. We formulate four linearized LB models in the frequency domain, based on Gaussian-Hermite quadratures of different algebraic precision (AP). The performance of each model is assessed by comparison to high-accuracy numerical solutions to the linearized Boltzmann-BGK equation for oscillatory Couette flow. The numerical results demonstrate that high even-order LB models provide superior performance over the greatest noncontinuum range. Our results also highlight intrinsic deficiencies in the current LB framework, which is incapable of capturing noncontinuum behavior at high oscillation frequencies, regardless of quadrature AP and the Knudsen number.
Lattice Boltzmann method for linear oscillatory noncontinuum flows.
Shi, Yong; Yap, Ying Wan; Sader, John E
2014-03-01
Oscillatory gas flows are commonly generated by micro- and nanoelectromechanical systems. Due to their small size and high operating frequencies, these devices often produce noncontinuum gas flows. Theoretical analysis of such flows requires solution of the unsteady Boltzmann equation, which can present a formidable challenge. In this article, we explore the applicability of the lattice Boltzmann (LB) method to such linear oscillatory noncontinuum flows; this method is derived from the linearized Boltzmann Bhatnagar-Gross-Krook (BGK) equation. We formulate four linearized LB models in the frequency domain, based on Gaussian-Hermite quadratures of different algebraic precision (AP). The performance of each model is assessed by comparison to high-accuracy numerical solutions to the linearized Boltzmann-BGK equation for oscillatory Couette flow. The numerical results demonstrate that high even-order LB models provide superior performance over the greatest noncontinuum range. Our results also highlight intrinsic deficiencies in the current LB framework, which is incapable of capturing noncontinuum behavior at high oscillation frequencies, regardless of quadrature AP and the Knudsen number.
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
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.
Towards a parallel collisionless shock in LAPD
NASA Astrophysics Data System (ADS)
Weidl, M. S.; Heuer, P.; Schaeffer, D.; Dorst, R.; Winske, D.; Constantin, C.; Niemann, C.
2017-09-01
Using a high-energy laser to produce a super-Alfvénic carbon-ion beam in a strongly magnetized helium plasma, we expect to be able to observe the formation of a collisionless parallel shock inside the Large Plasma Device. We compare early magnetic-field measurements of the resonant right-hand instability with analytical predictions and find excellent agreement. Hybrid simulations show that the carbon ions couple to the background plasma and compress it, although so far the background ions are mainly accelerated perpendicular to the mean-field direction.
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.
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.
Caveat on the Boltzmann distribution function use in biology.
Sevcik, Carlos
2017-08-01
Sigmoid semilogarithmic functions with shape of Boltzmann equations, have become extremely popular to describe diverse biological situations. Part of the popularity is due to the easy availability of software which fits Boltzmann functions to data, without much knowledge of the fitting procedure or the statistical properties of the parameters derived from the procedure. The purpose of this paper is to explore the plasticity of the Boltzmann function to fit data, some aspects of the optimization procedure to fit the function to data and how to use this plastic function to differentiate the effect of treatment on data and to attest the statistical significance of treatment effect on the data. Copyright © 2017. Published by Elsevier Ltd.
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.
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.
Entropic Lattice Boltzmann Algorithms for Turbulence
NASA Astrophysics Data System (ADS)
Vahala, George; Yepez, Jeffrey; Soe, Min; Vahala, Linda; Keating, Brian; Carter, Jonathan
2007-11-01
For turbulent flows in non-trivial geometry, the scaling of CFD codes (now necessarily non-pseudo spectral) quickly saturate with the number of PEs. By projecting into a lattice kinetic phase space, the turbulent dynamics are simpler and much easier to solve since the underlying kinetic equation has only local algebraic nonlinearities in the macroscopic variables with simple linear kinetic advection. To achieve arbitrary high Reynolds number, a discrete H-theorem constraint is imposed on the collision operator resulting in an entropic lattice Boltzmann (ELB) algorithm that is unconditionally stable and scales almost perfectly with PE's on any supercomputer architecture. At this mesoscopic level, there are various kinetic lattices (ELB-27, ELB-19, ELB-15) which will recover the Navier-Stokes equation to leading order in the Chapman-Enskog asymptotics. We comment on the morphology of turbulence and its correlation to the rate of change of enstrophy as well as simulations on 1600^3 grids.
NASA Astrophysics Data System (ADS)
Nauenberg, Michael
2005-03-01
In 1916 Einstein published a remarkable paper entitled ``On the Quantum Theory of Radiation''ootnotetextA. Einstein ``On the Quantum theory of Radiation,'' Phys. Zeitschrift 18 (1917) 121. First printed in Mitteilungender Physikalischen Gesellschaft Zurich. No 18, 1916. Translated into English in Van der Waerden ``Sources of Quantum Mechanics'' (North Holland 1967) pp. 63-77. in which he obtained Planck's formula for black-body radiation by introducing a new statistical hypothesis for the emmision and absorption of electromagneic radiation based on discrete bundles of energy and momentum which are now called photons. Einstein radiation theory replaced Maxwell's classical theory by a stochastic process which, when properly interpreted, also gives well known statistics of massless particles with even spin.^2 This quantum distribution, however, was not discovered by Einstein but was communicated to him by Bose in 1924. Like Boltzmann's classical counterpart, Einstein's statistical theory leads to an irreversible approach to thermal equilibrium, but because this violates time reversal, Einstein theory can not be regarded as a fundamental theory of physical process.ootnotetextM. Nauenberg ``The evolution of radiation towards thermal equilibrium: A soluble model which illustrates the foundations of statistical mechanics,'' American Journal of Physics 72 (2004) 313 Apparently Einstein and his contemporaries were unaware of this problem, and even today this problem is ignored in contemporary discussions of Einstein's treatment of the black-body spectrum.
Park, H M; Lee, J S; Kim, T W
2007-11-15
In the analysis of electroosmotic flows, the internal electric potential is usually modeled by the Poisson-Boltzmann equation. The Poisson-Boltzmann equation is derived from the assumption of thermodynamic equilibrium where the ionic distributions are not affected by fluid flows. Although this is a reasonable assumption for steady electroosmotic flows through straight microchannels, there are some important cases where convective transport of ions has nontrivial effects. In these cases, it is necessary to adopt the Nernst-Planck equation instead of the Poisson-Boltzmann equation to model the internal electric field. In the present work, the predictions of the Nernst-Planck equation are compared with those of the Poisson-Boltzmann equation for electroosmotic flows in various microchannels where the convective transport of ions is not negligible.
Collisionless shock waves mediated by Weibel Instability
NASA Astrophysics Data System (ADS)
Naseri, Neda; Ruan, Panpan; Zhang, Xi; Khudik, Vladimir; Shvets, Gennady
2015-11-01
Relativistic collisionless shocks are common events in astrophysical environments. They are thought to be responsible for generating ultra-high energy particles via the Fermi acceleration mechanism. It has been conjectured that the formation of collisionless shocks is mediated by the Weibel instability that takes place when two initially cold, unmagnetized plasma shells counter-propagate into each other with relativistic drift velocities. Using a PIC code, VLPL, which is modified to suppress numerical Cherenkov instabilities, we study the shock formation and evolution for asymmetric colliding shells with different densities in their own proper reference frame. Plasma instabilities in the region between the shock and the precursor are also investigated using a moving-window simulation that advances the computational domain at the shock's speed. This method helps both to save computation time and avoid severe numerical Cherenkov instabilities, and it allows us to study the shock evolution in a longer time period. Project is supported by US DOE grants DE-FG02-04ER41321 and DE-FG02-07ER54945.
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.
Reformation and Microinstabilities at Perpendicular Collisionless Shocks
NASA Astrophysics Data System (ADS)
Umeda, T.; Kidani, Y.; Matsukiyo, S.; Yamazaki, R.
2014-12-01
Large-scale two-dimensional (2D) full particle-in-cell (PIC) simulations are carried out for studying the relationship between the dynamics of a perpendicular shock and microinstabilities generated at the shock foot. The structure and dynamics of collisionless shocks are generally determined by Alfven Mach number and plasma beta, while microinstabilities at the shock foot are controlled by the ratio of the upstream bulk velocity to the electron thermal velocity and the ratio of the plasma-to-cyclotron frequency. With a fixed Alfven Mach number and plasma beta, the ratio of the upstream bulk velocity to the electron thermal velocity is given as a function of the ion-to-electron mass ratio. The present 2D full PIC simulations with a relatively low Alfven Mach number (MA ˜ 6) show that the modified two-stream instability is dominant with higher ion-to-electron mass ratios. It is also confirmed that waves propagating downstream are more enhanced at the shock foot near the shock ramp as the mass ratio becomes higher. The result suggests that these waves play a role in the modification of the dynamics of collisionless shocks through the interaction with shock front ripples.
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.
Entropic lattice Boltzmann representations required to recover Navier-Stokes flows.
Keating, Brian; Vahala, George; Yepez, Jeffrey; Soe, Min; Vahala, Linda
2007-03-01
There are two disparate formulations of the entropic lattice Boltzmann scheme: one of these theories revolves around the analog of the discrete Boltzmann H function of standard extensive statistical mechanics, while the other revolves around the nonextensive Tsallis entropy. It is shown here that it is the nonenforcement of the pressure tensor moment constraints that lead to extremizations of entropy resulting in Tsallis-like forms. However, with the imposition of the pressure tensor moment constraint, as is fundamentally necessary for the recovery of the Navier-Stokes equations, it is proved that the entropy function must be of the discrete Boltzmann form. Three-dimensional simulations are performed which illustrate some of the differences between standard lattice Boltzmann and entropic lattice Boltzmann schemes, as well as the role played by the number of phase-space velocities used in the discretization.
Analysis of multifragmentation in a Boltzmann-Langevin approach
Zhang, F.; Suraud, E.
1995-06-01
By using the Boltzmann-Langevin equation, which incorporates dynamical fluctuations beyond usual transport theories, we simulate the {sup 40}Ca+{sup 40}Ca reaction system at different beam energies 20, 60, and 90 MeV/nucleon for different impact parameters. Dynamical fluctuations become larger and larger with increasing bombarding energy and the system can reach densities corresponding to the unstable region of the nuclear matter equation of state at energies above 60 MeV/nucleon. By coupling the Boltzmann-Langevin equation with a coalescence model in the late stages of the reaction, we obtain the distribution of the intermediate mass fragments in each event. From the correlation analysis of these fragments, we recover some trends of recent multifragmentation data. A critical behavior analysis is also provided.
Conservative deterministic spectral Boltzmann solver near the grazing collisions limit
NASA Astrophysics Data System (ADS)
Haack, Jeffrey R.; Gamba, Irene M.
2012-11-01
We present new results building on the conservative deterministic spectral method for the space homogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. Within this framework we have extended the formulation to the case of more general case of collision operators with anisotropic scattering mechanisms, which requires a new formulation of the convolution weights. We also derive the grazing collisions limit for the method, and show that it is consistent with the Fokker-Planck-Landau equations as the grazing collisions parameter goes to zero.
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, Guo-Wei
2011-05-21
The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external
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
Lattice-Boltzmann simulation of laser interaction with weakly ionized helium plasmas
Li Huayu; Ki, Hyungson
2010-07-15
This paper presents a lattice Boltzmann method for laser interaction with weakly ionized plasmas considering electron impact ionization and three-body recombination. To simulate with physical properties of plasmas, the authors' previous work on the rescaling of variables is employed and the electromagnetic fields are calculated from the Maxwell equations by using the finite-difference time-domain method. To calculate temperature fields, energy equations are derived separately from the Boltzmann equations. In this way, we attempt to solve the full governing equations for plasma dynamics. With the developed model, the continuous-wave CO{sub 2} laser interaction with helium is simulated successfully.
Lattice-Boltzmann simulation of laser interaction with weakly ionized helium plasmas.
Li, Huayu; Ki, Hyungson
2010-07-01
This paper presents a lattice Boltzmann method for laser interaction with weakly ionized plasmas considering electron impact ionization and three-body recombination. To simulate with physical properties of plasmas, the authors' previous work on the rescaling of variables is employed and the electromagnetic fields are calculated from the Maxwell equations by using the finite-difference time-domain method. To calculate temperature fields, energy equations are derived separately from the Boltzmann equations. In this way, we attempt to solve the full governing equations for plasma dynamics. With the developed model, the continuous-wave CO2 laser interaction with helium is simulated successfully.
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.
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.
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.
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.
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.
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.
Evolution of magnetization in relativistic collisionless shocks
NASA Astrophysics Data System (ADS)
Keshet, Uri; Katz, Boaz; Spitkovsky, Anatoly; Waxman, Eli
Magnetic field generation and particle acceleration in relativistic collisionless shocks are believed to be essential to the emission of radiation in various astronomical sources. These processes are tightly connected but no self-consistent model simultaneously accounts for both. We have used unprecedentedly large ab-initio particle-in-cell simulations to evolve shocks to late times where particle acceleration has a strong effect on the shock structure. The energetic particles accelerated by the shock are found to drive magnetic field evolution on a time scale 104 plasma times. Progressively stronger magnetic fields are generated on larger scales in a growing region around the shock, in which magnetic fields and accelerated particles carry 1% and 10% of the downstream energy flux, respectively. These results suggest limits on acceleration and magnetization in astronomical flows. A possible self-similar steady-state configuration which naturally reproduces the flat particle spectra observed will be discussed.
The stability of a collisionless cosmological shell
NASA Technical Reports Server (NTRS)
White, Simon D. M.; Ostriker, J. P.
1990-01-01
The P3 M technique is used here to simulate the evolution of collisionless shells in an Omega = 1 universe. Starting from the spherical similarity solution, a bootstrap technique is used to follow the evolution over very large expansion factors. It is found that the overall structure follows the similarity solution for a long period during which bound clumps grow within the shell. At late times the growth of structure depends on induced velocity perturbations in material outside the shell. If such perturbations are suppressed, structure on the shell becomes self-similar. When induced motions in the background medium are included, the evolution at late times is dominated by large-scale modes as predicted by linear stability analysis. The stable final state appears to consist of one or two massive clumps on the edge of a spherical void. The possible application of these results to the origin of galaxies and large-scale structure is discussed.
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.
The merging of quasiperpendicular collisionless shocks
NASA Technical Reports Server (NTRS)
Cargill, P. J.
1990-01-01
The overtaking of one collisionless shock by another is studied by means of hybrid numerical simulations. The two shocks merge into a stronger shock and trailing nonshock discontinuities. The strong shock continues to propagate in the same direction as the two weaker shocks. The merging is shown to occur by a self-consistent process involving the interaction of ions reflected at the overtaking shock with the plasma upstream of the leading shock. The characteristic time scale for the merging is typically 1/Omega(i), where Omega(i) is the ion gyrofrequency. For exactly perpendicular shocks, the trailing discontinuity is a tangential discontinuity. It has a width of 2-3 ion Larmor radii. For oblique shocks, a contact discontinuity is present in the downstream plasma state. These results are of relevance to shock interactions in the very distant solar wind as well as in other energetic astrophysical situations such as solar flares.
A collisionless plasma thruster plume expansion model
NASA Astrophysics Data System (ADS)
Merino, Mario; Cichocki, Filippo; Ahedo, Eduardo
2015-06-01
A two-fluid model of the unmagnetized, collisionless far region expansion of the plasma plume for gridded ion thrusters and Hall effect thrusters is presented. The model is integrated into two semi-analytical solutions valid in the hypersonic case. These solutions are discussed and compared against the results from the (exact) method of characteristics; the relative errors in density and velocity increase slowly axially and radially and are of the order of 10-2-10-3 in the cases studied. The plasma density, ion flux and ambipolar electric field are investigated. A sensitivity analysis of the problem parameters and initial conditions is carried out in order to characterize the far plume divergence angle in the range of interest for space electric propulsion. A qualitative discussion of the physics of the secondary plasma plume is also provided.
Nanoflare heating model for collisionless solar corona
NASA Astrophysics Data System (ADS)
Visakh Kumar, U. L.; Varghese, Bilin Susan; Kurian, P. J.
2017-02-01
The problem of coronal heating remains one of the greatest unresolved problems in space science. Magnetic reconnection plays a significant role in heating the solar corona. When two oppositely directed magnetic fields come closer to form a current sheet, the current density of the plasma increases due to which magnetic reconnection and conversion of magnetic energy into thermal energy takes place. The present paper deals with a model for reconnection occurring in the solar corona under steady state in collisionless regime. The model predicts that reconnection time in the solar corona varies inversely with the cube of magnetic field and varies directly with the Lindquist number. Our analysis shows that reconnections are occurring within a time interval of 600 s in the solar corona, producing nanoflares in the energy range 10 21-10 23 erg /s which matches with Yohkoh X-ray observations.
Entropy production and collisionless fluid closure
NASA Astrophysics Data System (ADS)
Sarazin, Y.; Dif-Pradalier, G.; Zarzoso, D.; Garbet, X.; Ghendrih, Ph; Grandgirard, V.
2009-11-01
A novel method is proposed to construct collisionless fluid closures accounting for some kinetic properties. The idea consists in optimizing the agreement between the fluid and kinetic quasi-linear entropy production rates, so as to constrain the closure coefficients. This procedure is applied to the slab branch of the ion temperature gradient driven instability. Focusing on the kinetic regime characterized by slow waves, the closure proposed by Hammett and Perkins (Hammett and Perkins 1990 Phys. Rev. Lett. 64 3019) naturally emerges from the systematic identification of the kinetic and fluid entropy production rates. This closure is revealed to be extremely powerful well beyond the kinetic regime. Besides, it reconciles the fluid and kinetic linear stability diagrams in the two-dimensional space of the density and temperature gradient lengths. Such a method is systematic and generic. As such, it is applicable to other models and classes of instabilities.
Collisionless resistivity in a bifurcated current sheet
NASA Astrophysics Data System (ADS)
Andriyas, T.; Spencer, E.
2014-06-01
The collisionless resistivity due to charged particle chaos in spatially inhomogeneous magnetic fields is calculated for two frequently observed magnetotail current sheets, the X line, and a bifurcated current sheet (BCS) over varying strengths of cross-tail electric field. The calculation is done for two charged species, protons and O+ ions, found in the magnetotail specially during active times. Chaotic behavior of the particles is studied for the chaos parameter κ≈1defined by Buchner and Zelenyi (1989) as the square root of minimum radius of curvature to the maximum particle gyroradius. The surface of section plot and maximal Lyapunov exponents are analyzed to compare the particle behavior in the two magnetic field topologies. It is found that the particle behavior in a BCS is chaotic around the two humps located at ±z0 and that the configuration is more chaotic than the X line (maximum Lyapunov exponent for X line is around 0.25 compared to 0.34 for BCS). The collisionless resistivity is calculated using the technique developed by Numata and Yoshida (2003). The method relies on a phenomenological equivalence between the particle loss rate from the chaos region and rate of change of momentum in the chaos region under steady state conditions. Rapid decay of the particles from the chaos region is modeled with exponential fits, and it is found that a double exponent is needed for O+ ions in an X line and for both species in the BCS. The resistivity thus calculated is found to be 9-10 orders of magnitude higher than the Spitzer resistivity.
Collisionless Magnetic Reconnection in Space Plasmas
NASA Astrophysics Data System (ADS)
Treumann, Rudolf A.; Baumjohann, Wolfgang
2013-12-01
Magnetic reconnection, the merging of oppositely directed magnetic fields that leads to field reconfiguration, plasma heating, jetting and acceleration, is one of the most celebrated processes in collisionless plasmas. It requires the violation of the frozen-in condition which ties gyrating charged particles to the magnetic field inhibiting diffusion. Ongoing reconnection has been identified in near-Earth space as being responsible for the excitation of substorms, magnetic storms, generation of field aligned currents and their consequences, the wealth of auroral phenomena. Its theoretical understanding is now on the verge of being completed. Reconnection takes place in thin current sheets. Analytical concepts proceeded gradually down to the microscopic scale, the scale of the electron skin depth or inertial length, recognizing that current layers that thin do preferentially undergo spontaneous reconnection. Thick current layers start reconnecting when being forced by plasma inflow to thin. For almost half a century the physical mechanism of reconnection has remained a mystery. Spacecraft in situ observations in combination with sophisticated numerical simulations in two and three dimensions recently clarified the mist, finding that reconnection produces a specific structure of the current layer inside the electron inertial (also called electron diffusion) region around the reconnection site, the X line. Onset of reconnection is attributed to pseudo-viscous contributions of the electron pressure tensor aided by electron inertia and drag, creating a complicated structured electron current sheet, electric fields, and an electron exhaust extended along the current layer. We review the general background theory and recent developments in numerical simulation on collisionless reconnection. It is impossible to cover the entire field of reconnection in a short space-limited review. The presentation necessarily remains cursory, determined by our taste, preferences, and kn
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.
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)
Rax, J.M.
1992-04-01
The dynamics of electrons in two-dimensional, linearly or circularly polarized, ultra-high intensity (above 10{sup 18}W/cm{sup 2}) laser waves, is investigated. The Compton harmonic resonances are identified as the source of various stochastic instabilities. Both Arnold diffusion and resonance overlap are considered. The quasilinear kinetic equation, describing the evolution of the electron distribution function, is derived, and the associated collisionless damping coefficient is calculated. The implications of these new processes are considered and discussed.
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.
Exploring Numerical (Naked) Singularity Formation with Collisionless Matter
NASA Astrophysics Data System (ADS)
Okounkova, Maria; Hemberger, Daniel; Scheel, Mark
2017-01-01
A proposed channel for the formation of naked singularities (singularities without event horizons) is the collapse of collisionless matter. In 1991, Shapiro and Teukolsky numerically investigated the collapse of prolate spheroids of collisionless matter in axisymmetry, and found that for certain initial configurations, a singularity formed on the domain without the appearance of an apparent horizon. While this may be a candidate for naked singularity formation, the role of the axisymmetry of the configuration and the termination of the simulation at singularity formation leave the question of generically forming an event horizon open. We have implemented (fully backreacting, fully 3-dimensional) collisionless matter evolution in SpEC, the Spectral Einstein Code, and present our results for the collapse of various configurations of collisionless matter. We expand on previous results by excising singularities, giving more time for the appearance of an apparent horizon, and by considering a variety of initial configurations.
Bianchi I solutions of the Einstein-Boltzmann system with a positive cosmological constant
NASA Astrophysics Data System (ADS)
Lee, Ho; Nungesser, Ernesto
2017-09-01
In this paper, we study the future global existence and late-time behaviour of the Einstein-Boltzmann system with Bianchi I symmetry and a positive cosmological constant Λ >0 . For the Boltzmann equation, we consider the scattering kernel of Israel particles which are the relativistic counterpart of Maxwellian particles. Under a smallness assumption on initial data in a suitable norm, we show that solutions exist globally in time and isotropize at late times.
Kinetic model for the collisionless sheath of a collisional plasma
Tang, Xian-Zhu Guo, Zehua
2016-08-15
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.
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.
Pointwise Description for the Linearized Fokker-Planck-Boltzmann Model
NASA Astrophysics Data System (ADS)
Wu, Kung-Chien
2015-09-01
In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker-Planck-Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543-1608, 2004) for Boltzmann equation, but the Fokker-Planck term in this paper creates some technical difficulties.
Preconditioned lattice-Boltzmann method for steady flows
NASA Astrophysics Data System (ADS)
Guo, Zhaoli; Zhao, T. S.; Shi, Yong
2004-12-01
In this paper we propose a preconditioned lattice Boltzmann (LB) method for steady incompressible flows. For steady flows, the macroscopic equations derived from this LB model are equivalent to those from the standard LB model, but with an improved eigenvalue system. The proposed model can be viewed as an explicit solver for preconditioned compressible Navier-Stokes equations. Linear stability analysis is performed and the results show that the stability of the model is the same as that of the standard LB model for low Mach numbers. The proposed model retains the structure of the standard LB model and, hence, possesses all the advantages. Numerical tests show that the convergence rate can be enhanced as much as an order of magnitude compared to the standard lattice Boltzmann method. The accuracy of the solutions is improved as well.
High performance computing with a conservative spectral Boltzmann solver
NASA Astrophysics Data System (ADS)
Haack, Jeffrey R.; Gamba, Irene M.
2012-11-01
We present new results building on the conservative deterministic spectral method for the space inhomogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. We extend this method to second order accuracy in space and time, and explore how to leverage the structure of the collisional formulation for high performance computing environments. The locality in space of the collisional term provides a straightforward memory decomposition, and we perform some initial scaling tests on high performance computing resources. We also use the improved computational power of this method to investigate a boundary-layer generated shock problem that cannot be described by classical hydrodynamics.
Dissipative Boltzmann-Robertson-Walker cosmologies
Hiscock, W.A.; Salmonson, J. )
1991-05-15
The equations governing a flat Robertson-Walker cosmological model containing a dissipative Boltzmann gas are integrated numerically. The bulk viscous stress is modeled using the Eckart and Israel-Stewart theories of dissipative relativistic fluids; the resulting cosmologies are compared and contrasted. The Eckart models are shown to always differ in a significant quantitative way from the Israel-Stewart models. It thus appears inappropriate to use the pathological (nonhyperbolic) Eckart theory for cosmological applications. For large bulk viscosities, both cosmological models approach asymptotic nonequilibrium states; in the Eckart model the total pressure is negative, while in the Israel-Stewart model the total pressure is asymptotically zero. The Eckart model also expands more rapidly than the Israel-Stewart models. These results suggest that bulk-viscous'' inflation may be an artifact of using a pathological fluid theory such as the Eckart theory.
Lattice Boltzmann model with nearly constant density.
Fang, Hai-ping; Wan, Rong-zheng; Lin, Zhi-fang
2002-09-01
An improved lattice Boltzmann model is developed to simulate fluid flow with nearly constant fluid density. The ingredient is to incorporate an extra relaxation for fluid density, which is realized by introducing a feedback equation in the equilibrium distribution functions. The pressure is dominated by the moving particles at a node, while the fluid density is kept nearly constant and explicit mass conservation is retained as well. Numerical simulation based on the present model for the (steady) plane Poiseuille flow and the (unsteady) two-dimensional Womersley flow shows a great improvement in simulation results over the previous models. In particular, the density fluctuation has been reduced effectively while achieving a relatively large pressure gradient.
NASA Astrophysics Data System (ADS)
Mahmoodi-Darian, Masoomeh; Ettehadi-Abari, Mehdi; Sedaghat, Mahsa
2016-03-01
Laser absorption in the interaction between ultra-intense femtosecond laser and solid density plasma is studied theoretically here in the intensity range I{λ^2} ˜eq 10^{14}{-}10^{16}{{W}}{{{cm}}^{-2}} \\upmu{{{m}}2} . The collisionless effect is found to be significant when the incident laser intensity is less than 10^{16}{{W}}{{{cm}}^{-2}}\\upmu{{{m}}2} . In the current work, the propagation of a high-frequency electromagnetic wave, for underdense collisionless plasma in the presence of an external magnetic field is investigated. When a constant magnetic field parallel to the laser pulse propagation direction is applied, the electrons rotate along the magnetic field lines and generate the electromagnetic part in the wake with a nonzero group velocity. Here, by considering the ponderomotive force in attendance of the external magnetic field and assuming the isothermal collisionless plasma, the nonlinear permittivity of the plasma medium is obtained and the equation of electromagnetic wave propagation in plasma is solved. Here, by considering the effect of the ponderomotive force in isothermal collisionless magnetized plasma, it is shown that by increasing the laser pulse intensity, the electrons density profile leads to steepening and the electron bunches of plasma become narrower. Moreover, it is found that the wavelength of electric and magnetic field oscillations increases by increasing the external magnetic field and the density distribution of electrons also grows in comparison to the unmagnetized collisionless plasma.
NASA Astrophysics Data System (ADS)
Brosens, Fons; Magnus, Wim
2009-03-01
In principle, transport of charged carriers in nanometer sized solid-state devices can be fully characterized once the non- equilibrium distribution function describing the carrier ensemble is known. In this light, we have revisited the Boltzmann and the Wigner distribution functions and the framework in which they emerge from the classical respectively quantum mechanical Liouville equation. We have assessed the method of the characteristic curves as a potential workhorse to solve the time dependent Boltzmann equation for carriers propagating through spatially non-uniform systems, such as nanodevices. In order to validate the proposed solution strategy, we numerically solve the Boltzmann equation for a one- dimensional conductor mimicking the basic features of a biased low-dimensional transistor operating in the on-state. Finally, we propose a computational scheme capable of extending the benefits of the above mentioned solution strategy when it comes to solve the Wigner-Liouville equation.
Assessment of Linear Finite-Difference Poisson-Boltzmann Solvers
Wang, Jun; Luo, Ray
2009-01-01
CPU time and memory usage are two vital issues that any numerical solvers for the Poisson-Boltzmann equation have to face in biomolecular applications. In this study we systematically analyzed the CPU time and memory usage of five commonly used finite-difference solvers with a large and diversified set of biomolecular structures. Our comparative analysis shows that modified incomplete Cholesky conjugate gradient and geometric multigrid are the most efficient in the diversified test set. For the two efficient solvers, our test shows that their CPU times increase approximately linearly with the numbers of grids. Their CPU times also increase almost linearly with the negative logarithm of the convergence criterion at very similar rate. Our comparison further shows that geometric multigrid performs better in the large set of tested biomolecules. However, modified incomplete Cholesky conjugate gradient is superior to geometric multigrid in molecular dynamics simulations of tested molecules. We also investigated other significant components in numerical solutions of the Poisson-Boltzmann equation. It turns out that the time-limiting step is the free boundary condition setup for the linear systems for the selected proteins if the electrostatic focusing is not used. Thus, development of future numerical solvers for the Poisson-Boltzmann equation should balance all aspects of the numerical procedures in realistic biomolecular applications. PMID:20063271
Extended lattice Boltzmann scheme for droplet combustion.
Ashna, Mostafa; Rahimian, Mohammad Hassan; Fakhari, Abbas
2017-05-01
The available lattice Boltzmann (LB) models for combustion or phase change are focused on either single-phase flow combustion or two-phase flow with evaporation assuming a constant density for both liquid and gas phases. To pave the way towards simulation of spray combustion, we propose a two-phase LB method for modeling combustion of liquid fuel droplets. We develop an LB scheme to model phase change and combustion by taking into account the density variation in the gas phase and accounting for the chemical reaction based on the Cahn-Hilliard free-energy approach. Evaporation of liquid fuel is modeled by adding a source term, which is due to the divergence of the velocity field being nontrivial, in the continuity equation. The low-Mach-number approximation in the governing Navier-Stokes and energy equations is used to incorporate source terms due to heat release from chemical reactions, density variation, and nonluminous radiative heat loss. Additionally, the conservation equation for chemical species is formulated by including a source term due to chemical reaction. To validate the model, we consider the combustion of n-heptane and n-butanol droplets in stagnant air using overall single-step reactions. The diameter history and flame standoff ratio obtained from the proposed LB method are found to be in good agreement with available numerical and experimental data. The present LB scheme is believed to be a promising approach for modeling spray combustion.
Extended lattice Boltzmann scheme for droplet combustion
NASA Astrophysics Data System (ADS)
Ashna, Mostafa; Rahimian, Mohammad Hassan; Fakhari, Abbas
2017-05-01
The available lattice Boltzmann (LB) models for combustion or phase change are focused on either single-phase flow combustion or two-phase flow with evaporation assuming a constant density for both liquid and gas phases. To pave the way towards simulation of spray combustion, we propose a two-phase LB method for modeling combustion of liquid fuel droplets. We develop an LB scheme to model phase change and combustion by taking into account the density variation in the gas phase and accounting for the chemical reaction based on the Cahn-Hilliard free-energy approach. Evaporation of liquid fuel is modeled by adding a source term, which is due to the divergence of the velocity field being nontrivial, in the continuity equation. The low-Mach-number approximation in the governing Navier-Stokes and energy equations is used to incorporate source terms due to heat release from chemical reactions, density variation, and nonluminous radiative heat loss. Additionally, the conservation equation for chemical species is formulated by including a source term due to chemical reaction. To validate the model, we consider the combustion of n-heptane and n -butanol droplets in stagnant air using overall single-step reactions. The diameter history and flame standoff ratio obtained from the proposed LB method are found to be in good agreement with available numerical and experimental data. The present LB scheme is believed to be a promising approach for modeling spray combustion.
Gravity in a lattice Boltzmann model
Buick; Greated
2000-05-01
In this paper we consider the introduction of a body force, in the incompressible limit, into the lattice Boltzmann model. A number of methods are considered and their suitability to our objectives determined. When there is no density variation across the fluid, gravity can be introduced in the form of an altered pressure gradient. This method correctly satisfies the Navier-Stokes equation; however, if there is a non-negligible density variation present (produced by the body force or otherwise) this method becomes less accurate as the density variation increases and the constant density approximation becomes less valid. Three other methods are also considered for application when there is a non-negligible density variation. The equations of motion satisfied by these models are found up to second order in the Knudsen number and it is seen that only one of these methods satisfies the true Navier-Stokes equation. Numerical simulations are performed to compare the different models and to assess the range of application of each.
NASA Astrophysics Data System (ADS)
Rebhan, Anton
1992-06-01
The dynamical equations for rotational (vector) perturbations of a Friedmann-Robertson-Walker universe containing a perfect fluid of massive matter and radiation together with relativistic collisionless matter are established. These equations have solutions which remain regular as the initial singularity is approached, in contrast to the purely perfect-fluid case, where small rotational perturbations cannot coexist with a Friedmann-type singularity due to the Helmholtz-Kelvin circulation theorem. With collisionless matter present (e.g., gravitons after the Planck era), this obstruction is circumvented, and solutions which exhibit a growing mode of vorticity on superhorizon scales are obtained. The anisotropies in the cosmic microwave background caused by these small vector perturbations are analyzed, and limits on admissible primordial vorticity are derived. In the radiation era, large-scale vorticity gives rise to large-scale primordial magnetic fields, which are shown potentially to have the right magnitude to act as seed fields for galactic dynamo action and thereby to explain the presently observed galactic magnetic fields.
Accretion of a relativistic, collisionless kinetic gas into a Schwarzschild black hole
NASA Astrophysics Data System (ADS)
Rioseco, Paola; Sarbach, Olivier
2017-05-01
We provide a systematic study for the accretion of a collisionless, relativistic kinetic gas into a nonrotating black hole. To this end, we first solve the relativistic Liouville equation on a Schwarzschild background spacetime. The most general solution for the distribution function is given in terms of appropriate symplectic coordinates on the cotangent bundle, and the associated observables, including the particle current density and stress energy-momentum tensor, are determined. Next, we explore the case where the flow is steady-state and spherically symmetric. Assuming that in the asymptotic region the gas is described by an equilibrium distribution function, we determine the relevant parameters of the accretion flow as a function of the particle density and the temperature of the gas at infinity. In particular, we find that in the low temperature limit the tangential pressure at the horizon is about an order of magnitude larger than the radial one, showing explicitly that a collisionless gas, despite exerting kinetic pressure, behaves very differently than an isotropic perfect fluid, and providing a partial explanation for the known fact that the accretion rate is much lower than in the hydrodynamic case of Bondi-Michel accretion. Finally, we establish the asymptotic stability of the steady-state spherical flows by proving pointwise convergence results which show that a large class of (possibly nonstationary and nonspherical) initial conditions for the distribution function lead to solutions of the Liouville equation which relax in time to a steady-state, spherically symmetric configuration.
Collisionless microinstabilities in stellarators. II. Numerical simulations
NASA Astrophysics Data System (ADS)
Proll, J. H. E.; Xanthopoulos, P.; Helander, P.
2013-12-01
Microinstabilities exhibit a rich variety of behavior in stellarators due to the many degrees of freedom in the magnetic geometry. It has recently been found that certain stellarators (quasi-isodynamic ones with maximum-J geometry) are partly resilient to trapped-particle instabilities, because fast-bouncing particles tend to extract energy from these modes near marginal stability. In reality, stellarators are never perfectly quasi-isodynamic, and the question thus arises whether they still benefit from enhanced stability. Here, the stability properties of Wendelstein 7-X and a more quasi-isodynamic configuration, QIPC, are investigated numerically and compared with the National Compact Stellarator Experiment and the DIII-D tokamak. In gyrokinetic simulations, performed with the gyrokinetic code GENE in the electrostatic and collisionless approximation, ion-temperature-gradient modes, trapped-electron modes, and mixed-type instabilities are studied. Wendelstein 7-X and QIPC exhibit significantly reduced growth rates for all simulations that include kinetic electrons, and the latter are indeed found to be stabilizing in the energy budget. These results suggest that imperfectly optimized stellarators can retain most of the stabilizing properties predicted for perfect maximum-J configurations.
On Relaxation Processes in Collisionless Mergers
NASA Astrophysics Data System (ADS)
Valluri, Monica; Vass, Ileana M.; Kazantzidis, Stelios; Kravtsov, Andrey V.; Bohn, Courtlandt L.
2007-04-01
We analyze N-body simulations of halo mergers to investigate the mechanisms responsible for driving mixing in phase space and the evolution to dynamical equilibrium. We focus on mixing in energy and angular momentum and show that mixing occurs in a steplike fashion following pericenter passages of the halos. This makes mixing during a merger unlike other well-known mixing processes such as phase mixing and chaotic mixing, whose rates scale with local dynamical time. We conclude that the mixing process that drives the system to equilibrium is primarily a response to energy and angular momentum redistribution that occurs due to impulsive tidal shocking and dynamical friction rather than a result of chaotic mixing in a changing potential. We also analyze the merger remnants to determine the degree of mixing at various radii by monitoring changes in radius, energy, and angular momentum of particles. We confirm previous findings that show that the majority of particles retain strong memory of their original kinetic energies and angular momenta, but do experience changes in their potential energies owing to the tidal shocks they experience during pericenter passages. Finally, we show that a significant fraction of mass (~40%) in the merger remnant lies outside its formal virial radius, and that this matter is ejected roughly uniformly from all radii outside the inner regions. This highlights the fact that mass, in its standard virial definition, is not additive in mergers. We discuss the implications of these results for our understanding of relaxation in collisionless dynamical systems.
Nonlinear gyrofluid simulations of collisionless reconnection
Grasso, D.; Tassi, E.; Waelbroeck, F. L.
2010-08-15
The Hamiltonian gyrofluid model recently derived by Waelbroeck et al. [Phys. Plasmas 16, 032109 (2009)] is used to investigate nonlinear collisionless reconnection with a strong guide field by means of numerical simulations. Finite ion Larmor radius gives rise to a cascade of the electrostatic potential to scales below both the ion gyroradius and the electron skin depth. This cascade is similar to that observed previously for the density and current in models with cold ions. In addition to density cavities, the cascades create electron beams at scales below the ion gyroradius. The presence of finite ion temperature is seen to modify, inside the magnetic island, the distribution of the velocity fields that advect two Lagrangian invariants of the system. As a consequence, the fine structure in the electron density is confined to a layer surrounding the separatrix. Finite ion Larmor radius effects produce also a different partition between the electron thermal, potential, and kinetic energy, with respect to the cold-ion case. Other aspects of the dynamics such as the reconnection rate and the stability against Kelvin-Helmholtz modes are similar to simulations with finite electron compressibility but cold ions.
Relativistic soliton-like collisionless ionization wave
NASA Astrophysics Data System (ADS)
Arefiev, Alexey; McCormick, Matthew; Quevedo, Hernan; Bengtson, Roger; Ditmire, Todd
2014-10-01
It has been observed in recent experiments with laser-irradiated gas jets that a plasma filament produced by the laser and containing energetic electrons can launch a relativistic ionization wave into ambient gas. Here we present a self-consistent theory that explains how a collisionless ionization wave can propagate in a self-sustaining regime. A population of hot electrons necessarily generates a sheath electric field at the plasma boundary. This field penetrates the ambient gas, ionizing the gas atoms and thus causing the plasma boundary to expand. We show that the motion of the newly generated electrons can form a potential well adjacent to the plasma boundary. The outwards motion of the well causes a bunch of energetic electrons to become trapped, while allowing the newly generated electrons to escape into the plasma without retaining much energy. The resulting soliton-like ionizing field structure propagates outwards with a bunch of hot electrons that maintain a strong sheath field despite significant plasma expansion. We also present 1D and 2D particle-in-cell simulations that illustrate the described mechanism. The simulations were performed using HPC resources provided by the Texas Advanced Computing Center. This work was supported by NNSA Contract No. DE-FC52-08NA28512 and U.S. DOE Contract No. DE-FG02-04ER54742.