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.
NASA Astrophysics Data System (ADS)
Biagetti, Matteo; Desjacques, Vincent; Kehagias, Alex; Racco, Davide; Riotto, Antonio
2016-04-01
Dark matter halos are the building blocks of the universe as they host galaxies and clusters. The knowledge of the clustering properties of halos is therefore essential for the understanding of the galaxy statistical properties. We derive an effective halo Boltzmann equation which can be used to describe the halo clustering statistics. In particular, we show how the halo Boltzmann equation encodes a statistically biased gravitational force which generates a bias in the peculiar velocities of virialized halos with respect to the underlying dark matter, as recently observed in N-body simulations.
He, Xiaoyi; Lou, Li-Shi Lou, Li-Shi
1997-12-01
In this paper, the lattice Boltzmann equation is directly derived from the Boltzmann equation. It is shown that the lattice Boltzmann equation is a special discretized form of the Boltzmann equation. Various approximations for the discretization of the Boltzmann equation in both time and phase space are discussed in detail. A general procedure to derive the lattice Boltzmann model from the continuous Boltzmann equation is demonstrated explicitly. The lattice Boltzmann models derived include the two-dimensional 6-bit, 7-bit, and 9-bit, and three-dimensional 27-bit models. {copyright} {ital 1997} {ital The American Physical Society}
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.
Boltzmann equation and hydrodynamic fluctuations.
Colangeli, Matteo; Kröger, Martin; Ottinger, Hans Christian
2009-11-01
We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics. PMID:20364972
Closure of fluid equations in collisionless magnetoplasmas
Chust, T.; Belmont, G.
2006-01-15
The possibility of using fluid equations in collisionless plasmas is revisited, and the conditions of validity of several possible closure equations are investigated. A new derivation of the well-known 'double-adiabatic' Chew-Goldberger-Low (CGL) laws is first presented. These laws are shown to demand two different kinds of conditions for ensuring (1) particle gyrotropy and (2) adiabaticity. Both kinds of conditions are investigated in detail. The usual slow and large-scales conditions (hereafter 'sls'), which are shown to be necessary for gyrotropy, are provided in a rigorous form. The role of the fundamental symmetries of the system, especially in the directions parallel and perpendicular to the magnetic field, is also emphasized for determining any 'fluid-type' behavior of a collisionless magnetoplasma. More general closure equations are afterwards proposed, relaxing first the conditions for adiabaticity and then, more speculatively, the sls conditions for gyrotropy. The dependence of these new closure equations on the shape of the velocity distribution functions is discussed, the CGL case being singular since it is shown to be fully independent of this shape.
Lattice Boltzmann equation for relativistic quantum mechanics.
Succi, Sauro
2002-03-15
Relativistic versions of the quantum lattice Boltzmann equation are discussed. It is shown that the inclusion of nonlinear interactions requires the standard collision operator to be replaced by a pair of dynamic fields coupling to the relativistic wave function in a way which can be described by a multicomponent complex lattice Boltzmann equation. PMID:16210189
Analytical methods for solving the Boltzmann equation
NASA Astrophysics Data System (ADS)
Struminskii, V. V.
The principal analytical methods for solving the Boltzmann equation are reviewed, and a very general solution is proposed. The method makes it possible to obtain a solution to the Cauchy problem for the nonlinear Boltzmann equation and thus determine the applicability regions for the various analytical methods. The method proposed here also makes it possible to demonstrate that Hilbert's theorem of macroscopic causality does not apply and Hilbert's paradox does not exist.
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.
Boltzmann kinetic equation for filtered fluid turbulence.
Girimaji, Sharath S
2007-07-20
We develop a kinetic Boltzmann equation for describing filtered fluid turbulence applicable for continuum and noncontinuum effects. The effect of unresolved turbulent motion on the resolved distribution function is elucidated and closure modeling issues of kinetic Boltzmann and Navier-Stokes descriptions are reconciled. This could pave the way for unifying turbulence modeling at kinetic and continuum levels and the development of numerical methods that are valid over a wide range of flow physics. PMID:17678288
The Boltzmann equation in the difference formulation
Szoke, Abraham; Brooks III, Eugene D.
2015-05-06
First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.
Classical non-Markovian Boltzmann equation
Alexanian, Moorad
2014-08-01
The modeling of particle transport involves anomalous diffusion, (x²(t) ) ∝ t{sup α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.
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. PMID:25493907
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.
Full Boltzmann equations for leptogenesis including scattering
Hahn-Woernle, F.; Plümacher, M.; Wong, Y.Y.Y. E-mail: pluemi@mppmu.mpg.de
2009-08-01
We study the evolution of a cosmological baryon asymmetry produced via leptogenesis by means of the full classical Boltzmann equations, without the assumption of kinetic equilibrium and including all quantum statistical factors. Beginning with the full mode equations, we derive the usual equations of motion for the right-handed neutrino number density and integrated lepton asymmetry, and show explicitly the impact of each assumption on these quantities. For the first time, we investigate also the effects of scattering of the right-handed neutrino with the top quark to leading order in the Yukawa couplings by means of the full Boltzmann equations. We find that in our full Boltzmann treatment the final lepton asymmetry can be suppressed by as much as a factor of ∼ 1.5 in the weak wash-out regime (K ∼< 1), compared to the usual integrated approach which assumes kinetic equilibrium and neglects quantum statistics. This suppression is in contrast with the enhancement seen in some previous studies that considered only decay and inverse decay of the right-handed neutrino. However, this suppression quickly decreases as we increase K. In the strong wash-out regime (K ∼> 1), the full Boltzmann treatment and the integrated approach give nearly identical final lepton asymmetries (within 10% of each other at K > 3). Finally, we show that the opposing effects of quantum statistics on decays/inverse decays and the scattering processes tend to reduce the net importance of scattering on leptogenesis in the full treatment compared to the integrated approach.
NASA Astrophysics Data System (ADS)
Held, M.; Kendl, A.
2015-10-01
A lattice Boltzmann method (LBM) approach to the Charney-Hasegawa-Mima (CHM) model for adiabatic drift wave turbulence in magnetised plasmas is implemented. The CHM-LBM model contains a barotropic equation of state for the potential, a force term including a cross-product analogous to the Coriolis force in quasigeostrophic models, and a density gradient source term. Expansion of the resulting lattice Boltzmann model equations leads to cold-ion fluid continuity and momentum equations, which resemble CHM dynamics under drift ordering. The resulting numerical solutions of standard test cases (monopole propagation, stable drift modes and decaying turbulence) are compared to results obtained by a conventional finite difference scheme that directly discretizes the CHM equation. The LB scheme resembles characteristic CHM dynamics apart from an additional shear in the density gradient direction. The occurring shear reduces with the drift ratio and is ascribed to the compressible limit of the underlying LBM.
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. PMID:21230596
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.
Asymptotic-preserving Boltzmann model equations for binary gas mixture
NASA Astrophysics Data System (ADS)
Liu, Sha; Liang, Yihua
2016-02-01
An improved system of Boltzmann model equations is developed for binary gas mixture. This system of model equations has a complete asymptotic preserving property that can strictly recover the Navier-Stokes equations in the continuum limit with the correct constitutive relations and the correct viscosity, thermal conduction, diffusion, and thermal diffusion coefficients. In this equation system, the self- and cross-collision terms in Boltzmann equations are replaced by single relaxation terms. In monocomponent case, this system of equations can be reduced to the commonly used Shakhov equation. The conservation property and the H theorem which are important for model equations are also satisfied by this system of model equations.
Asymptotic-preserving Boltzmann model equations for binary gas mixture.
Liu, Sha; Liang, Yihua
2016-02-01
An improved system of Boltzmann model equations is developed for binary gas mixture. This system of model equations has a complete asymptotic preserving property that can strictly recover the Navier-Stokes equations in the continuum limit with the correct constitutive relations and the correct viscosity, thermal conduction, diffusion, and thermal diffusion coefficients. In this equation system, the self- and cross-collision terms in Boltzmann equations are replaced by single relaxation terms. In monocomponent case, this system of equations can be reduced to the commonly used Shakhov equation. The conservation property and the H theorem which are important for model equations are also satisfied by this system of model equations. PMID:26986408
Comparison of Boltzmann equations with quantum dynamics for scalar fields
Lindner, Manfred; Mueller, Markus Michael
2006-06-15
Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after relativistic heavy ion collisions. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this paper a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar {phi}{sup 4} quantum field theory in 3+1 space-time dimensions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Apart from quantitative discrepancies, on a qualitative level the universality respected by the Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations. Furthermore, the Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. This separation of time scales is absent for the Boltzmann equation.
Asymptotic behaviour of the Boltzmann equation as a cosmological model
NASA Astrophysics Data System (ADS)
Lee, Ho
2016-08-01
As a Newtonian cosmological model the Vlasov-Poisson-Boltzmann system is considered, and a slightly modified Boltzmann equation, which describes the stability of an expanding universe, is derived. Asymptotic behaviour of solutions turns out to depend on the expansion of the universe, and in this paper we consider the soft potential case and will obtain asymptotic behaviour.
Thermal equation of state for lattice Boltzmann gases
NASA Astrophysics Data System (ADS)
Ran, Zheng
2009-06-01
The Galilean invariance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar-Gross-Krook model are proposed together with their rigorous theoretical background. From the viewpoint of group invariance, recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar-Gross-Krook equation (LBGKE) induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.
Simplified simulation of Boltzmann-Langevin equation
Ayik, S.; Randrup, J.
1994-06-01
We briefly recall the Boltzmann-Langevin model of nuclear dynamics. We then summarize recent progress in deriving approximate analytical expressions for the associated transport coefficients and describe a numerical method for simulating the stochastic evolution of the phase-space density.
Properties of the Boltzmann equation in the classical approximation
Epelbaum, Thomas; Gelis, François; Tanji, Naoto; Wu, Bin
2014-12-30
We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since onemore » has also access to the non-approximated result for comparison.« less
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.
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.
Asymptotic analysis of the Boltzmann equation for dark matter relics
NASA Astrophysics Data System (ADS)
Bender, Carl M.; Sarkar, Sarben
2012-10-01
This paper presents an asymptotic analysis of the Boltzmann equations (Riccati differential equations) that describe the physics of thermal dark-matter-relic abundances. Two different asymptotic techniques are used, boundary-layer theory, which makes use of asymptotic matching, and the delta expansion, which is a powerful technique for solving nonlinear differential equations. Two different Boltzmann equations are considered. The first is derived from general relativistic considerations and the second arises in dilatonic string cosmology. The global asymptotic analysis presented here is used to find the long-time behavior of the solutions to these equations. In the first case, the nature of the so-called freeze-out region and the post-freeze-out behavior is explored. In the second case, the effect of the dilaton on cold dark-matter abundances is calculated and it is shown that there is a large-time power-law fall off of the dark-matter abundance.
Modeling adsorption with lattice Boltzmann equation
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-01-01
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied. PMID:27256325
Modeling adsorption with lattice Boltzmann equation.
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-01-01
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied. PMID:27256325
Monitoring derivation of the quantum linear Boltzmann equation
Hornberger, Klaus; Vacchini, Bassano
2008-02-15
We show how the effective equation of motion for a distinguished quantum particle in an ideal gas environment can be obtained by means of the monitoring approach introduced by Hornberger [EPL 77, 50007 (2007)]. The resulting Lindblad master equation accounts for the quantum effects of the scattering dynamics in a nonperturbative fashion and it describes decoherence and dissipation in a unified framework. It incorporates various established equations as limiting cases and reduces to the classical linear Boltzmann equation once the state is diagonal in momentum.
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.
From Boltzmann equations to steady wall velocities
Konstandin, Thomas; Nardini, Germano; Rues, Ingo E-mail: germano.nardini@desy.de
2014-09-01
By means of a relativistic microscopic approach we calculate the expansion velocity of bubbles generated during a first-order electroweak phase transition. In particular, we use the gradient expansion of the Kadanoff-Baym equations to set up the fluid system. This turns out to be equivalent to the one found in the semi-classical approach in the non-relativistic limit. Finally, by including hydrodynamic deflagration effects and solving the Higgs equations of motion in the fluid, we determine velocity and thickness of the bubble walls. Our findings are compared with phenomenological models of wall velocities. As illustrative examples, we apply these results to three theories providing first-order phase transitions with a particle content in the thermal plasma that resembles the Standard Model.
Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics.
García de Soria, María Isabel; Maynar, Pablo; Schehr, Grégory; Barrat, Alain; Trizac, Emmanuel
2008-05-01
We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzmann operator, which characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of molecular dynamics simulations that validate the theoretical predictions. PMID:18643046
{Lp} stability for the Boltzmann equation near vacuum
NASA Astrophysics Data System (ADS)
Deng, Wei; Lu, Lu
2016-04-01
The paper is concerned with the uniform time stability in the Lebesgue space {Lp(R3 × R3)} of solutions to the Boltzmann equation near vacuum. Precisely, for the soft potential case {-2 < γ < 0}, there exists p_{γ} > 1 such that the nonnegative solution with algebraic decay rate in x, v at infinity is stable with respect to small initial data uniformly in time in {Lp} with {1 ≤ p < p_{γ}}.
Lattice Boltzmann model for generalized nonlinear wave equations
NASA Astrophysics Data System (ADS)
Lai, Huilin; Ma, Changfeng
2011-10-01
In this paper, a lattice Boltzmann model is developed to solve a class of the nonlinear wave equations. Through selecting equilibrium distribution function and an amending function properly, the governing evolution equation can be recovered correctly according to our proposed scheme, in which the Chapman-Enskog expansion is employed. We validate the algorithm on some problems where analytic solutions are available, including the second-order telegraph equation, the nonlinear Klein-Gordon equation, and the damped, driven sine-Gordon equation. It is found that the numerical results agree well with the analytic solutions, which indicates that the present algorithm is very effective and can be used to solve more general nonlinear problems.
Numerical solution of Boltzmann equation using discrete velocity grids
NASA Astrophysics Data System (ADS)
Vedula, Prakash
2015-11-01
An importance sampling based approach for numerical solution of the (single species) Boltzmann equation using discrete velocity grids is proposed. This approach involves a stochastic method for evaluation of the collision integral based on sampling of depleting/replenishing collisions and is designed to preserve important symmetries of the collision operator, including collision invariants. The underlying particle distribution function is represented as a collection of delta functions with associated weights that are non-negative. A key feature in the construction of the proposed method is that it ensures that the weights associated with the distribution function remain non-negative during collisional relaxation, thereby satisfying an important realizability condition. Performance of the proposed approach will be studied using test problems involving spatially homogeneous collisional relaxation flow and microchannel flows. Results obtained from the proposed method will be compared with those obtained from the (deterministic) collisional Lattice Boltzmann Method (cLBM) and the traditional direct simulation Monte Carlo (DSMC) method for solution of Boltzmann equation. Extension of the proposed method using discrete velocity grids for multicomponent mixtures will also be discussed.
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
NASA Astrophysics Data System (ADS)
Molnár, Etele; Niemi, Harri; Rischke, Dirk H.
2016-06-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break down. Nevertheless, one can still derive a fluid-dynamical theory, called anisotropic dissipative fluid dynamics, in terms of an expansion around a single-particle distribution function, f^0 k, which incorporates (at least parts of) the momentum anisotropy via a suitable parametrization. We construct such an expansion in terms of polynomials in energy and momentum in the direction of the anisotropy and of irreducible tensors in the two-dimensional momentum subspace orthogonal to both the fluid velocity and the direction of the anisotropy. From the Boltzmann equation we then derive the set of equations of motion for the irreducible moments of the deviation of the single-particle distribution function from f^0 k. Truncating this set via the 14-moment approximation, we obtain the equations of motion of anisotropic dissipative fluid dynamics.
Lattice Boltzmann method for the fractional advection-diffusion equation
NASA Astrophysics Data System (ADS)
Zhou, J. G.; Haygarth, P. M.; Withers, P. J. A.; Macleod, C. J. A.; Falloon, P. D.; Beven, K. J.; Ockenden, M. C.; Forber, K. J.; Hollaway, M. J.; Evans, R.; Collins, A. L.; Hiscock, K. M.; Wearing, C.; Kahana, R.; Villamizar Velez, M. L.
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β , the fractional order α , and the single relaxation time τ , the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Lattice Boltzmann method for the fractional advection-diffusion equation.
Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering. PMID:27176431
Velocity-Field Theory, Boltzmann's Transport Equation and Geometry
NASA Astrophysics Data System (ADS)
Ichinose, Shoichi
Boltzmann equation describes the time development of the velocity distribution in the continuum fluid matter. We formulate the equation using the field theory where the velocity-field plays the central role. The matter (constituent particles) fields appear as the density and the viscosity. Fluctuation is examined, and is clearly discriminated from the quantum effect. The time variable is emergently introduced through the computational process step. The collision term, for the (velocity)**4 potential (4-body interaction), is explicitly obtained and the (statistical) fluctuation is closely explained. The present field theory model does not conserve energy and is an open-system model. (One dimensional) Navier-Stokes equation or Burger's equation, appears. In the latter part, we present a way to directly define the distribution function by use of the geometry, appearing in the mechanical dynamics, and Feynman's path-integral.
Lattice Boltzmann model for a steady radiative transfer equation.
Yi, Hong-Liang; Yao, Feng-Ju; Tan, He-Ping
2016-08-01
A complete lattice Boltzmann model (LBM) is proposed for the steady radiative transfer equation (RTE). The RTE can be regarded as a pure convection equation with a source term. To derive the expressions for the equilibrium distribution function and the relaxation time, an artificial isotropic diffusion term is introduced to form a convection-diffusion equation. When the dimensionless relaxation time has a value of 0.5, the lattice Boltzmann equation (LBE) is exactly applicable to the original steady RTE. We also perform a multiscale analysis based on the Chapman-Enskog expansion to recover the macroscopic RTE from the mesoscopic LBE. The D2Q9 model is used to solve the LBE, and the numerical results obtained by the LBM are comparable to the results obtained by other methods or analytical solutions, which demonstrates that the proposed model is highly accurate and stable in simulating multidimensional radiative transfer. In addition, we find that the convergence rate of the LBM depends on the transport properties of RTE: for diffusion-dominated RTE with a large optical thickness, the LBM shows a second-order convergence rate in space, while for convection-dominated RTE with a small optical thickness, a lower convergence rate is observed. PMID:27627417
Hydrodynamic limit with geometric correction of stationary Boltzmann equation
NASA Astrophysics Data System (ADS)
Wu, Lei
2016-05-01
We consider the hydrodynamic limit of a stationary Boltzmann equation in a unit plate with in-flow boundary. The classical theory claims that the solution can be approximated by the sum of interior solution which satisfies steady incompressible Navier-Stokes-Fourier system, and boundary layer derived from Milne problem. In this paper, we construct counterexamples to disprove such formulation in L∞ both for its proof and result. Also, we show the hydrodynamic limit with a different boundary layer expansion with geometric correction.
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. PMID:20590325
Generalizing the Boltzmann equation in complex phase space.
Zadehgol, Abed
2016-08-01
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014)JCTPAH0021-999110.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015)PLEEE81539-375510.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others. PMID:27627421
Floquet-Boltzmann equation for periodically driven Fermi systems
NASA Astrophysics Data System (ADS)
Genske, Maximilian; Rosch, Achim
2015-12-01
Periodically driven quantum systems can be used to realize quantum pumps, ratchets, artificial gauge fields, and novel topological states of matter. Starting from the Keldysh approach, we develop a formalism, the Floquet-Boltzmann equation, to describe the dynamics and the scattering of quasiparticles in such systems. The theory builds on a separation of time scales. Rapid, periodic oscillations occurring on a time scale T0=2 π /Ω are treated using the Floquet formalism and quasiparticles are defined as eigenstates of a noninteracting Floquet Hamiltonian. The dynamics on much longer time scales, however, is modeled by a Boltzmann equation which describes the semiclassical dynamics of the Floquet quasiparticles and their scattering processes. As the energy is conserved only modulo ℏ Ω , the interacting system heats up in the long-time limit. As a first application of this approach, we compute the heating rate for a cold-atom system, where a periodical shaking of the lattice was used to realize the Haldane model [G. Jotzu et al., Nature (London) 515, 237 (2014)], 10.1038/nature13915.
Reciprocal relations based on the non-stationary Boltzmann equation
NASA Astrophysics Data System (ADS)
Sharipov, Felix
2012-03-01
The reciprocal relations for open gaseous systems are obtained on the basis of main properties of the non-stationary Boltzmann equation and gas-surface interaction law. It is shown that the main principles to derive the kinetic coefficients satisfying the reciprocal relations remain the same as those used for time-independent gaseous systems [F. Sharipov, Onsager-Casimir reciprocal relations based on the Boltzmann equation and gas-surface interaction law single gas, Phys. Rev. 73 (2006) 026110]. First, the kinetic coefficients are obtained from the entropy production expression; then it is proved that the coefficient matrix calculated for time reversed source functions is symmetric. The proof is based on the reversibility of the gas-gas and gas-surface interactions. Three examples of applications of the present theory are given. None of these examples can be treated in the frame of the classical Onsager-Casimir reciprocal relations, which are valid only in a particular case, when the kinetic coefficients are odd or even with respect to the time reversion. The approach is generalized for gaseous mixtures.
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)
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.
Immersed boundary method for Boltzmann model kinetic equations
NASA Astrophysics Data System (ADS)
Pekardan, Cem; Chigullapalli, Sruti; Sun, Lin; Alexeenko, Alina
2012-11-01
Three different immersed boundary method formulations are presented for Boltzmann model kinetic equations such as Bhatnagar-Gross-Krook (BGK) and Ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) model equations. 1D unsteady IBM solution for a moving piston is compared with the DSMC results and 2D quasi-steady microscale gas damping solutions are verified by a conformal finite volume method solver. Transient analysis for a sinusoidally moving beam is also carried out for the different pressure conditions (1 atm, 0.1 atm and 0.01 atm) corresponding to Kn=0.05,0.5 and 5. Interrelaxation method (Method 2) is shown to provide a faster convergence as compared to the traditional interpolation scheme used in continuum IBM formulations. Unsteady damping in rarefied regime is characterized by a significant phase-lag which is not captured by quasi-steady approximations.
Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation
NASA Astrophysics Data System (ADS)
Lods, Bertrand; Pistone, Giovanni
2015-06-01
Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set $\\mathcal E$ of mutually absolutely continuous densities. We study in particular the fine properties of the Kullback-Liebler divergence in this context. We also show that this setting is well-suited for the study of the spatially homogeneous Boltzmann equation if $\\mathcal E$ is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyse the Boltzmann operator in the geometric setting from the point of its Maxwell's weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts i.e., the H-theorem. We also illustrate the robustness of our method by discussing, besides the Kullback-Leibler divergence, also the property of Hyv\\"arinen divergence. This requires to generalise our approach to Orlicz-Sobolev spaces to include derivatives.%
NASA Astrophysics Data System (ADS)
Saveliev, V. L.
2011-05-01
Pair collisions is the main interaction process in the Boltzmann gas dynamics. By making use of exactly the same physical assumptions as was used by Ludwig Boltzmann we write the kinetic equation for two-particle distribution function of molecules in the gas mixtures. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. Because the scattering operator is more simple then Boltzmann collision integral this equation opens new opportunities for mathematical description of the Boltzmann gas dynamics.
Global Solutions of the Boltzmann Equation Over {{R}^D} Near Global Maxwellians with Small Mass
NASA Astrophysics Data System (ADS)
Bardos, Claude; Gamba, Irene M.; Golse, François; Levermore, C. David
2016-07-01
We study the dynamics defined by the Boltzmann equation set in the Euclidean space {{R}^D} in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory.
Lattice Boltzmann Equation On a 2D Rectangular Grid
NASA Technical Reports Server (NTRS)
Bouzidi, MHamed; DHumieres, Dominique; Lallemand, Pierre; Luo, Li-Shi; Bushnell, Dennis M. (Technical Monitor)
2002-01-01
We construct a multi-relaxation lattice Boltzmann model on a two-dimensional rectangular grid. The model is partly inspired by a previous work of Koelman to construct a lattice BGK model on a two-dimensional rectangular grid. The linearized dispersion equation is analyzed to obtain the constraints on the isotropy of the transport coefficients and Galilean invariance for various wave propagations in the model. The linear stability of the model is also studied. The model is numerically tested for three cases: (a) a vortex moving with a constant velocity on a mesh periodic boundary conditions; (b) Poiseuille flow with an arbitrasy inclined angle with respect to the lattice orientation: and (c) a cylinder &symmetrically placed in a channel. The numerical results of these tests are compared with either analytic solutions or the results obtained by other methods. Satisfactory results are obtained for the numerical simulations.
A stochastic Galerkin method for the Boltzmann equation with uncertainty
NASA Astrophysics Data System (ADS)
Hu, Jingwei; Jin, Shi
2016-06-01
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.
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.
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.
Harleston, H. Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Distrito Federal ); Vishniac, E.T. )
1992-06-15
The Arnowitt-Deser-Misner formalism is used to write the Einstein-Boltzmann coupled system of equations. The sources of gravitational field are represented by ordinary matter described by a perfect-fluid approximation together with a particle gas described by a phase-space distribution function obeying the general-relativistic Boltzmann transport equation. Through the use of the Liouville operator in phase space, we obtain a form of the Boltzmann equation that makes it very amenable for numerical treatment. The resulting system of equations can be used for the numerical study of either massless or massive particles interacting with ordinary matter.
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.
Multigrid lattice Boltzmann method for accelerated solution of elliptic equations
NASA Astrophysics Data System (ADS)
Patil, Dhiraj V.; Premnath, Kannan N.; Banerjee, Sanjoy
2014-05-01
A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice Boltzmann method (LBM) and the multigrid (MG) technique is presented. Several benchmark elliptic equations are solved numerically with the inclusion of multiple grid-levels in two-dimensional domains at an optimal computational cost within the LB framework. The results are compared with the corresponding analytical solutions and numerical solutions obtained using the Stone's strongly implicit procedure. The classical PDEs considered in this article include the Laplace and Poisson equations with Dirichlet boundary conditions, with the latter involving both constant and variable coefficients. A detailed analysis of solution accuracy, convergence and computational efficiency of the proposed solver is given. It is observed that the use of a high-order stencil (for smoothing) improves convergence and accuracy for an equivalent number of smoothing sweeps. The effect of the type of scheduling cycle (V- or W-cycle) on the performance of the MG-LBM is analyzed. Next, a parallel algorithm for the MG-LBM solver is presented and then its parallel performance on a multi-core cluster is analyzed. Lastly, a practical example is provided wherein the proposed elliptic PDE solver is used to compute the electro-static potential encountered in an electro-chemical cell, which demonstrates the effectiveness of this new solver in complex coupled systems. Several orders of magnitude gains in convergence and parallel scaling for the canonical problems, and a factor of 5 reduction for the multiphysics problem are achieved using the MG-LBM.
Multilevel Methods for the Poisson-Boltzmann Equation
NASA Astrophysics Data System (ADS)
Holst, Michael Jay
We consider the numerical solution of the Poisson -Boltzmann equation (PBE), a three-dimensional second order nonlinear elliptic partial differential equation arising in biophysics. This problem has several interesting features impacting numerical algorithms, including discontinuous coefficients representing material interfaces, rapid nonlinearities, and three spatial dimensions. Similar equations occur in various applications, including nuclear physics, semiconductor physics, population genetics, astrophysics, and combustion. In this thesis, we study the PBE, discretizations, and develop multilevel-based methods for approximating the solutions of these types of equations. We first outline the physical model and derive the PBE, which describes the electrostatic potential of a large complex biomolecule lying in a solvent. We next study the theoretical properties of the linearized and nonlinear PBE using standard function space methods; since this equation has not been previously studied theoretically, we provide existence and uniqueness proofs in both the linearized and nonlinear cases. We also analyze box-method discretizations of the PBE, establishing several properties of the discrete equations which are produced. In particular, we show that the discrete nonlinear problem is well-posed. We study and develop linear multilevel methods for interface problems, based on algebraic enforcement of Galerkin or variational conditions, and on coefficient averaging procedures. Using a stencil calculus, we show that in certain simplified cases the two approaches are equivalent, with different averaging procedures corresponding to different prolongation operators. We also develop methods for nonlinear problems based on a nonlinear multilevel method, and on linear multilevel methods combined with a globally convergent damped-inexact-Newton method. We derive a necessary and sufficient descent condition for the inexact-Newton direction, enabling the development of extremely
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.
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.
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.
NASA Astrophysics Data System (ADS)
Sospedra-Alfonso, Reinel; Shizgal, Bernie D.
2012-11-01
We use a finite difference discretization method to solve the space homogeneous, isotropic nonlinear Boltzmann equation. We study the time evolution of the distribution function in relation to the solution of the linearized Boltzmann equation for three different initial conditions. The relaxation process is described in terms of the Laguerre moments and the spectral properties of the linearized collision operator. The motivation is the need to include self-collisions in the study of suprathermal oxygen atoms in the terrestrial exosphere.
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.
Recovering Navier–Stokes Equations from Asymptotic Limits of the Boltzmann Gas Mixture Equation
NASA Astrophysics Data System (ADS)
Carlo, Bianca; Christian, Dogbe
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].
The lattice Boltzmann model for the second-order Benjamin-Ono equations
NASA Astrophysics Data System (ADS)
Lai, Huilin; Ma, Changfeng
2010-04-01
In this paper, in order to extend the lattice Boltzmann method to deal with more complicated nonlinear equations, we propose a 1D lattice Boltzmann scheme with an amending function for the second-order (1 + 1)-dimensional Benjamin-Ono equation. With the Taylor expansion and the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The equilibrium distribution function and the amending function are obtained. Numerical simulations are carried out for the 'good' Boussinesq equation and the 'bad' one to validate the proposed model. It is found that the numerical results agree well with the analytical solutions. The present model can be used to solve more kinds of nonlinear partial differential equations.
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.
An asymptotic preserving Monte Carlo method for the multispecies Boltzmann equation
NASA Astrophysics Data System (ADS)
Zhang, Bin; Liu, Hong; Jin, Shi
2016-01-01
An asymptotic preserving (AP) scheme is efficient in solving multiscale kinetic equations with a wide range of the Knudsen number. In this paper, we generalize the asymptotic preserving Monte Carlo method (AP-DSMC) developed in [25] to the multispecies Boltzmann equation. This method is based on the successive penalty method [26] originated from the BGK-penalization-based AP scheme developed in [7]. For the multispecies Boltzmann equation, the penalizing Maxwellian should use the unified Maxwellian as suggested in [12]. We give the details of AP-DSMC for multispecies Boltzmann equation, show its AP property, and verify through several numerical examples that the scheme can allow time step much larger than the mean free time, thus making it much more efficient for flows with possibly small Knudsen numbers than the classical DSMC.
NASA Astrophysics Data System (ADS)
Chen, Lin-Jie; Ma, Chang-Feng
2010-01-01
This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations (NPDEs) in the form ut + αuux + βunux + γuxx + δuxxx + ζuxxxx = 0. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.
NASA Astrophysics Data System (ADS)
Suriyawichitseranee, A.; Grigoriev, Yu. N.; Meleshko, S. V.
2015-03-01
The paper is devoted to group analysis of the spatially homogeneous and isotropic Boltzmann equation with a source term. In fact, the Fourier transform of the Boltzmann equation with respect to the molecular velocity variable is considered. Using a particular class of solutions, the determining equation for the admitted Lie group is reduced to a partial differential equation for the source function. The latter equation is analyzed by an algebraic method. A complete group classification of the Fourier transform of the Boltzmann equation is given. All invariant solutions of this equation are also presented in the paper.
NASA Astrophysics Data System (ADS)
Grigoriev, Yurii N.; Meleshko, Sergey V.; Suriyawichitseranee, Amornrat
2015-06-01
Group analysis of the spatially homogeneous and molecular energy dependent Boltzmann equations with source term is carried out. The Fourier transform of the Boltzmann equation with respect to the molecular velocity variable is considered. The correspondent determining equation of the admitted Lie group is reduced to a partial differential equation for the admitted source. The latter equation is analyzed by an algebraic method. A complete group classification of the Fourier transform of the Boltzmann equation with respect to a source function is given. The representation of invariant solutions and corresponding reduced equations for all obtained source functions are also presented.
A note on a Discrete Boltzmann Equation with multiple collisions
NASA Astrophysics Data System (ADS)
Oliveira, Filipe; Soares, Ana Jacinta
2008-05-01
We compute a non-trivial explicit solution for the one-dimensional plane 6-velocity discrete Boltzmann model with multiple collisions introduced in [E. Longo, R. Monaco, On the discrete kinetic theory with multiple collisions: Plane six-velocity and unsteady Couette flow, in: Muntz, et al. (Eds.), The Proceedings of Rarefied Gas Dynamics, in: AIAA Publ., vol. 118, 1989, pp. 118-130] which asymptotically connects two particular equilibrium states. We prove that such a solution exists provided that a suitable condition on the differential elastic cross sections holds.
Entropy inequality and hydrodynamic limits for the Boltzmann equation.
Saint-Raymond, Laure
2013-12-28
Boltzmann brought a fundamental contribution to the understanding of the notion of entropy, by giving a microscopic formulation of the second principle of thermodynamics. His ingenious idea, motivated by the works of his contemporaries on the atomic nature of matter, consists of describing gases as huge systems of identical and indistinguishable elementary particles. The state of a gas can therefore be described in a statistical way. The evolution, which introduces couplings, loses part of the information, which is expressed by the decay of the so-called mathematical entropy (the opposite of physical entropy!). PMID:24249776
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.
Numerical scheme for a spatially inhomogeneous matrix-valued quantum Boltzmann equation
NASA Astrophysics Data System (ADS)
Lu, Jianfeng; Mendl, Christian B.
2015-06-01
We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are 2 × 2 matrix-valued to accommodate the spin degree of freedom, and the scalar quantum Boltzmann equation is recovered as a special case when all matrices are proportional to the identity. We use Fourier discretization and fast Fourier transform to efficiently evaluate the collision kernel with spectral accuracy, and numerically investigate periodic, Dirichlet and Maxwell boundary conditions. Model simulations quantify the convergence to local and global thermal equilibrium.
NASA Astrophysics Data System (ADS)
Watanabe, Hirofumi; Okiyama, Yoshio; Nakano, Tatsuya; Tanaka, Shigenori
2010-11-01
We developed FMO-PB method, which incorporates solvation effects into the Fragment Molecular Orbital calculation with the Poisson-Boltzmann equation. This method retains good accuracy in energy calculations with reduced computational time. We calculated the solvation free energies for polyalanines, Alpha-1 peptide, tryptophan cage, and complex of estrogen receptor and 17 β-estradiol to show the applicability of this method for practical systems. From the calculated results, it has been confirmed that the FMO-PB method is useful for large biomolecules in solution. We also discussed the electric charges which are used in solving the Poisson-Boltzmann equation.
High-accuracy deterministic solution of the Boltzmann equation for the shock wave structure
NASA Astrophysics Data System (ADS)
Malkov, E. A.; Bondar, Ye. A.; Kokhanchik, A. A.; Poleshkin, S. O.; Ivanov, M. S.
2015-07-01
A new deterministic method of solving the Boltzmann equation has been proposed. The method has been employed in numerical studies of the plane shock wave structure in a hard sphere gas. Results for Mach numbers and have been compared with predictions of the direct simulation Monte Carlo (DSMC) method, which has been used to obtain the reference solution. Particular attention in estimating the solution accuracy has been paid to a fine structural effect: the presence of a total temperature peak exceeding the temperature value further downstream. The results of solving the Boltzmann equation for the shock wave structure are in excellent agreement with the DSMC predictions.
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.
Equations of motion of test particles for solving the spin-dependent Boltzmann-Vlasov equation
NASA Astrophysics Data System (ADS)
Xia, Yin; Xu, Jun; Li, Bao-An; Shen, Wen-Qing
2016-08-01
A consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-dependent Boltzmann-Vlasov equation is presented. The resulting EOMs in phase space are similar to the canonical equations in Hamiltonian dynamics, and the EOM of spin is the same as that in the Heisenburg picture of quantum mechanics. Considering further the quantum nature of spin and choosing the direction of total angular momentum in heavy-ion reactions as a reference of measuring nucleon spin, the EOMs of spin-up and spin-down nucleons are given separately. The key elements affecting the spin dynamics in heavy-ion collisions are identified. The resulting EOMs provide a solid foundation for using the test-particle approach in studying spin dynamics in heavy-ion collisions at intermediate energies. Future comparisons of model simulations with experimental data will help to constrain the poorly known in-medium nucleon spin-orbit coupling relevant for understanding properties of rare isotopes and their astrophysical impacts.
Regularized lattice Boltzmann model for a class of convection-diffusion equations.
Wang, Lei; Shi, Baochang; Chai, Zhenhua
2015-10-01
In this paper, a regularized lattice Boltzmann model for a class of nonlinear convection-diffusion equations with variable coefficients is proposed. The main idea of the present model is to introduce a set of precollision distribution functions that are defined only in terms of macroscopic moments. The Chapman-Enskog analysis shows that the nonlinear convection-diffusion equations can be recovered correctly. Numerical tests, including Fokker-Planck equations, Buckley-Leverett equation with discontinuous initial function, nonlinear convection-diffusion equation with anisotropic diffusion, are carried out to validate the present model, and the results show that the present model is more accurate than some available lattice Boltzmann models. It is also demonstrated that the present model is more stable than the traditional single-relaxation-time model for the nonlinear convection-diffusion equations. PMID:26565368
Numerical investigations of low-density nozzle flow by solving the Boltzmann equation
NASA Astrophysics Data System (ADS)
Deng, Zheng-Tao; Liaw, Goang-Shin; Chou, Lynn Chen
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.
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.
NASA Astrophysics Data System (ADS)
Shizgal, Bernie D.
2011-05-01
The study of the solution of the linearized Boltzmann equation has a very long history arising from the classic work by Chapman and Cowling. For small departures from a Maxwellian, the nonlinear Boltzmann equation can be linearized and the transport coefficients calculated with the Chapman-Enskog approach. This procedure leads to a set of linear integral equations which are generally solved with the expansion of the departure from Maxwellian in Sonine polynomials. The method has been used successfully for many decades to compare experimental transport data in atomic gases with theory generally carried out for realistic atom-atom differential cross sections. There are alternate pseudospectral methods which involve the discretization of the distribution function on a discrete grid. This paper considers a pseudospectral method of solution of the linearized hard sphere Boltzmann equation for the viscosity in a simple gas. The relaxation of a small departure from a Maxwellian is also considered for the linear test particle problem with unit mass ratio which is compared with the relaxation for the linearized one component Boltzmann equation.
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)
BV-Regularity of the Boltzmann Equation in Non-Convex Domains
NASA Astrophysics Data System (ADS)
Guo, Y.; Kim, C.; Tonon, D.; Trescases, A.
2016-06-01
We consider the Boltzmann equation in a general non-convex domain with the diffuse boundary condition. We establish optimal BV estimates for such solutions. Our method consists of a new W 1,1-trace estimate for the diffuse boundary condition and a delicate construction of an {\\varepsilon}-tubular neighborhood of the singular set.
The Green's function for the three-dimensional linear Boltzmann equation via Fourier transform
NASA Astrophysics Data System (ADS)
Machida, Manabu
2016-04-01
The linear Boltzmann equation with constant coefficients in the three-dimensional infinite space is revisited. It is known that the Green's function can be calculated via the Fourier transform in the case of isotropic scattering. In this paper, we show that the three-dimensional Green's function can be computed with the Fourier transform even in the case of arbitrary anisotropic scattering.
The Boltzmann equation for gluons at early times after a heavy ion collision
NASA Astrophysics Data System (ADS)
Mueller, A. H.
2000-03-01
A Boltzmann equation is given for the early stages of evolution of the gluon system produced in a head-on heavy ion collision. The collision term is taken from gluon-gluon scattering in the one-gluon approximation.
and
A Revisiting of the -Stability Theory of the Boltzmann Equation Near Global Maxwellians
NASA Astrophysics Data System (ADS)
Ha, Seung-Yeal; Xiao, Qinghua
2015-07-01
We study the -stability theory of the Boltzmann equation near a global Maxwellian. When an initial datum is a perturbation of a global Maxwellian, we show that the -distance between two classical solutions can be controlled by the initial data in a Lipschitz manner, which illustrates the Lipschitz continuity of the solution operator for the Boltzmann equation in -topology. Our local-in-time -stability results cover cutoff very soft potentials as well as non-cutoff hard and soft potentials. These cases were not treated in the previous work (Ha et al. in Arch Ration Mech Anal 197:657-688, 2010). Thus, our results together with the results in Ha et al. (2010) complete the -stability theory for the Boltzmann equation near a global Maxwellian. For this -stability estimate, we use the coercivity estimate of the linearized collision operator, the smallness of perturbation in a mixed Lebesgue norm, and Strichartz-type estimates of perturbation. We also show that for all classical solutions available in the literature, the Lipschitz constant can be chosen as independent of time to obtain the uniform -stability of the Boltzmann equation.
The charge conserving Poisson-Boltzmann equations: Existence, uniqueness, and maximum principle
Lee, Chiun-Chang
2014-05-15
The present article is concerned with the charge conserving Poisson-Boltzmann (CCPB) equation in high-dimensional bounded smooth domains. The CCPB equation is a Poisson-Boltzmann type of equation with nonlocal coefficients. First, under the Robin boundary condition, we get the existence of weak solutions to this equation. The main approach is variational, based on minimization of a logarithm-type energy functional. To deal with the regularity of weak solutions, we establish a maximum modulus estimate for the standard Poisson-Boltzmann (PB) equation to show that weak solutions of the CCPB equation are essentially bounded. Then the classical solutions follow from the elliptic regularity theorem. Second, a maximum principle for the CCPB equation is established. In particular, we show that in the case of global electroneutrality, the solution achieves both its maximum and minimum values at the boundary. However, in the case of global non-electroneutrality, the solution may attain its maximum value at an interior point. In addition, under certain conditions on the boundary, we show that the global non-electroneutrality implies pointwise non-electroneutrality.
NASA Astrophysics Data System (ADS)
Zhang, Jianying; Yan, Guangwu
2016-04-01
A lattice Boltzmann model for solving the (2+1) dimensional cubic-quintic complex Ginzburg-Landau equation (CQCGLE) is proposed. Different from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a two-dimensional space, and the evolution of the model is about a spatial axis rather than time. The algorithm provides advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg-Landau equations. Numerical results reproduce the phenomena of the fusion of necklace-ring pattern and the effect of non-linearity on the soliton in the CQCGLE.
NASA Astrophysics Data System (ADS)
Zhang, Jianying; Yan, Guangwu
2015-12-01
A spatiotemporal lattice Boltzmann model for solving the three-dimensional cubic-quintic complex Ginzburg-Landau equation (CQCGLE) is proposed. Different from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a three-dimensional spatiotemporal space, and the evolution of the model is about a spatial axis rather than time. The algorithm possesses advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg-Landau equations. Examples show that the model reproduces the phenomena in the CQCGLE accurately.
Radiative or neutron transport modeling using a lattice Boltzmann equation framework
NASA Astrophysics Data System (ADS)
Bindra, H.; Patil, D. V.
2012-07-01
In this paper, the lattice Boltzmann equation (LBE)-based framework is used to obtain the solution for the linear radiative or neutron transport equation. The LBE framework is devised for the integrodifferential forms of these equations which arise due to the inclusion of the scattering terms. The interparticle collisions are neglected, hence omitting the nonlinear collision term. Furthermore, typical representative examples for one-dimensional or two-dimensional geometries and inclusion or exclusion of the scattering term (isotropic and anisotropic) in the Boltzmann transport equation are illustrated to prove the validity of the method. It has been shown that the solution from the LBE methodology is equivalent to the well-known Pn and Sn methods. This suggests that the LBE can potentially provide a more convenient and easy approach to solve the physical problems of neutron and radiation transport.
On the validity of the Boltzmann equation for short range potentials
NASA Astrophysics Data System (ADS)
Pulvirenti, M.; Saffirio, C.; Simonella, S.
2014-02-01
We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low-density (Boltzmann-Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King [9] (that appeared after the well-known result of Lanford [10] for hard spheres) and of a recent paper by Gallagher et al. [5]. Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence.
Convergence of Solutions to the Boltzmann Equation in the Incompressible Euler Limit
NASA Astrophysics Data System (ADS)
SAINT-RAYMOND, LAURE
We consider here the problem of deriving rigorously, for well-prepared initial data and without any additional assumption, dissipative or smooth solutions of the incompressible Euler equations from renormalized solutions of the Boltzmann equation. This completes the partial results obtained by Golse [B. Perthame and L. Desvillettes eds., Series in Applied Mathematics 4 (2000), Gauthier-Villars, Paris] and Lions & Masmoudi [Arch. Rational Mech. Anal. 158 (2001), 195-211].
Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method
NASA Technical Reports Server (NTRS)
Chen, Hudong; Chen, Shiyi; Matthaeus, William H.
1992-01-01
A lattice Boltzmann model is presented which gives the complete Navier-Stokes equation and may provide an efficient parallel numerical method for solving various fluid problems. The model uses the single-time relaxation approximation and a particular Maxwell-type distribution. The model eliminates exactly (1) the non-Galilean invariance caused by a density-dependent coefficient in the convection term and (2) a velocity-dependent equation of state.
The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium
NASA Astrophysics Data System (ADS)
Briant, Marc; Daus, Esther S.
2016-07-01
We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in {L^1_vL^∞_x(m)} , where {m˜ (1+ |v|^k)} is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an {L^2-L^E&infty}; theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the {L^1_vL^∞_x} framework is dealt with for any {k > k_0} , recovering the optimal physical threshold of finite energy {k_0=2} in the particular case of a multi-species hard spheres mixture with the same masses.
Fully-Lagrangian and Lattice-Boltzmann Methods for Solving Systems of Conservation Equations
NASA Astrophysics Data System (ADS)
Ancona, M. G.
1994-11-01
A class of "fully-Lagrangian" methods for solving systems of conservation equations is defined. The key step in formulating these methods is the definition of a new set of field variables for which Lagrangian discretization is trivial. Recently popular lattice-Boltzmann simulation schemes for solving such systems are shown to be a useful sub-class of these fully-Lagrangian methods in which (a) the conservation laws are satisfied at each grid point, (b) the Lagrangian variables are expanded perturbatively, and (c) discretization error is used to represent physics. Such schemes are typically derived using methods of kinetic theory. Our numerical analysis approach shows that the conventional physical derivation, while certainly valid and fruitful, is not essential, that it often confuses physics and numerics and that it can be unnecessarily constraining. For example, we show that lattice-Boltzmann-like methods can be non-perturbative and can be made higher-order, implicit and/or with non-uniform grids. Furthermore, our approach provides new perspective on the relationship between lattice-Boltzmann methods and finite-difference techniques. Among other things, we show that the lattice-Boltzmann schemes are only conditionally consistent and in some cases are identical to the well-known Dufort-Frankel method. Through this connection, the lattice-Boltzmann method provides a rational basis for understanding Dufort-Frankel and gives a pathway for its generalization. At the same time, that Dufort Frankel is no longer much used suggests that the lattice-Boltzmann approach might also share this fate.
Numerical Treatment of the Boltzmann Equation for Self-Propelled Particle Systems
NASA Astrophysics Data System (ADS)
Thüroff, Florian; Weber, Christoph A.; Frey, Erwin
2014-10-01
Kinetic theories constitute one of the most promising tools to decipher the characteristic spatiotemporal dynamics in systems of actively propelled particles. In this context, the Boltzmann equation plays a pivotal role, since it provides a natural translation between a particle-level description of the system's dynamics and the corresponding hydrodynamic fields. Yet, the intricate mathematical structure of the Boltzmann equation substantially limits the progress toward a full understanding of this equation by solely analytical means. Here, we propose a general framework to numerically solve the Boltzmann equation for self-propelled particle systems in two spatial dimensions and with arbitrary boundary conditions. We discuss potential applications of this numerical framework to active matter systems and use the algorithm to give a detailed analysis to a model system of self-propelled particles with polar interactions. In accordance with previous studies, we find that spatially homogeneous isotropic and broken-symmetry states populate two distinct regions in parameter space, which are separated by a narrow region of spatially inhomogeneous, density-segregated moving patterns. We find clear evidence that these three regions in parameter space are connected by first-order phase transitions and that the transition between the spatially homogeneous isotropic and polar ordered phases bears striking similarities to liquid-gas phase transitions in equilibrium systems. Within the density-segregated parameter regime, we find a novel stable limit-cycle solution of the Boltzmann equation, which consists of parallel lanes of polar clusters moving in opposite directions, so as to render the overall symmetry of the system's ordered state nematic, despite purely polar interactions on the level of single particles.
NASA Astrophysics Data System (ADS)
Feng, Yue
Plasma is currently a hot topic and it has many significant applications due to its composition of both positively and negatively charged particles. The energy distribution function is important in plasma science since it characterizes the ability of the plasma to affect chemical reactions, affect physical outcomes, and drive various applications. The Boltzmann Transport Equation is an important kinetic equation that provides an accurate basis for characterizing the distribution function---both in energy and space. This dissertation research proposes a multi-term approximation to solve the Boltzmann Transport Equation by treating the relaxation process using an expansion of the electron distribution function in Legendre polynomials. The elastic and 29 inelastic cross sections for electron collisions with nitrogen molecules (N2) and singly ionized nitrogen molecules ( N+2 ) have been used in this application of the Boltzmann Transport Equation. Different numerical methods have been considered to compare the results. The numerical methods discussed in this thesis are the implicit time-independent method, the time-dependent Euler method, the time-dependent Runge-Kutta method, and finally the implicit time-dependent relaxation method by generating the 4-way grid with a matrix solver. The results show that the implicit time-dependent relaxation method is the most accurate and stable method for obtaining reliable results. The results were observed to match with the published experimental data rather well.
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.
NASA Astrophysics Data System (ADS)
Briant, Marc; Einav, Amit
2016-04-01
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.
NASA Astrophysics Data System (ADS)
Marcozzi, M.; Nota, A.
2016-03-01
We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with a magnetic transport term. Moreover, we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. We compare these results with those ones obtained for a system of hard disks in Bobylev et al. (Phys Rev Lett 75:2, 1995), which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.
Yuan Jiankui; Jette, David; Chen Weimin
2008-09-15
A photon transport algorithm for fully three-dimensional radiotherapy treatment planning has been developed based on the discrete ordinates (S{sub N}) solution of the Boltzmann equation. The algorithm is characterized by orthogonal adaptive meshes, which place additional points where large gradients occur and a procedure to evaluate the collided flux using the representation of spherical harmonic expansion instead of the summation of the volume-weighted contribution from discrete angles. The Boltzmann equation was solved in the form of S{sub N} spatial, energy, and angular discretization with mitigation of ray effects by the first-collision source method. Unlike existing S{sub N} codes, which were designed for general purpose for multiparticle transport in areas such as nuclear engineering, our code is optimized for medical radiation transport. To validate the algorithm, several examples were employed to calculate the photon flux distribution. Numerical results show good agreement with the Monte Carlo calculations using EGSnrc.
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.
Coulomb collisions in the Boltzmann equation for electrons in low-temperature gas discharge plasmas
NASA Astrophysics Data System (ADS)
Hagelaar, G. J. M.
2016-02-01
This paper investigates the effects of electron-electron and electron-ion Coulomb collisions on the electron distribution function and transport coefficients obtained from the Boltzmann equation for simple dc gas discharge conditions. Expressions are provided for the full Coulomb collision terms acting on both the isotropic and anisotropic parts of the electron distribution function, which are then incorporated in the freeware Boltzmann equation solver BOLSIG+. Different Coulomb collision effects are demonstrated and discussed on the basis of BOLSIG+ results for argon gas. It is shown that the anisotropic part of the electron-electron collision term, neglected in previous work, can in certain cases have a large effect on the electron mobility and is essential when describing the transition towards the Coulomb-collision dominated regime characterized by Spitzer transport coefficients. Finally, a brief overview is presented of the discharge conditions for which different Coulomb collision effects occur in different gases.
Moment Closure Approximations of the Boltzmann Equation Based on \\varphi -Divergences
NASA Astrophysics Data System (ADS)
Abdelmalik, M. R. A.; van Brummelen, E. H.
2016-07-01
This paper is concerned with approximations of the Boltzmann equation based on the method of moments. We propose a generalization of the setting of the moment-closure problem from relative entropy to \\varphi -divergences and a corresponding closure procedure based on minimization of \\varphi -divergences. The proposed description encapsulates as special cases Grad's classical closure based on expansion in Hermite polynomials and Levermore's entropy-based closure. We establish that the generalization to divergence-based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment-closure formulations such as inadmissibility of the approximate phase-space distribution, potential loss of hyperbolicity and singularity of flux functions at local equilibrium. The divergence-based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the Boltzmann equation.
Blow-up rate estimates for the solutions of the bosonic Boltzmann-Nordheim equation
NASA Astrophysics Data System (ADS)
Bandyopadhyay, J.; Velázquez, J. J. L.
2015-06-01
In this paper, we study the behavior of a class of mild solutions of the homogeneous and isotropic bosonic Boltzmann-Nordheim equation near the blow-up. We obtain some estimates on the blow-up rate of the solutions and prove that, as long as a solution is bounded above by the critical singularity /1 x (the equilibrium solutions behave like this power law near the origin), it remains bounded in the uniform norm. In Sec. III of the paper, we prove a local existence result for a class of measure-valued mild solutions, which is of independent interest and which allows us to solve the Boltzmann-Nordheim equation for some classes of unbounded densities.
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
Dilaton and off-shell (non-critical string) effects in Boltzmann equation for species abundances
NASA Astrophysics Data System (ADS)
Lahanas, Ab; Mavromatos, Ne; Nanopoulos, Dv
In this work we derive the modifications to the Boltzmann equation governing the cosmic evolution of relic abundances induced by dilaton dissipative-source and non-critical-string terms in dilaton-driven non-equilibrium string Cosmologies. We also discuss briefly the most important phenomenological consequences, including modifications of the constraints on the available parameter space of cosmologically appealing particle physics models, imposed by recent precision data of astrophysical measurements.
Recent applications of the Boltzmann master equation to heavy ion precompound decay phenomena
Blann, M.; Remington, B.A.
1988-06-01
The Boltzmann master equation (BME) is described and used as a tool to interpret preequilibrium neutron emission from heavy ion collisions gated on evaporation residue or fission fragments. The same approach is used to interpret neutron spectra gated on deep inelastic and quasi-elastic heavy ion collisions. Less successful applications of BME to proton inclusive data with 40 MeV/u incident /sup 12/C ions are presented, and improvements required in the exciton injection term are discussed.
NASA Astrophysics Data System (ADS)
Bouchard, Hugo; Bielajew, Alex
2015-07-01
To establish a theoretical framework for generalizing Monte Carlo transport algorithms by adding external electromagnetic fields to the Boltzmann radiation transport equation in a rigorous and consistent fashion. Using first principles, the Boltzmann radiation transport equation is modified by adding a term describing the variation of the particle distribution due to the Lorentz force. The implications of this new equation are evaluated by investigating the validity of Fano’s theorem. Additionally, Lewis’ approach to multiple scattering theory in infinite homogeneous media is redefined to account for the presence of external electromagnetic fields. The equation is modified and yields a description consistent with the deterministic laws of motion as well as probabilistic methods of solution. The time-independent Boltzmann radiation transport equation is generalized to account for the electromagnetic forces in an additional operator similar to the interaction term. Fano’s and Lewis’ approaches are stated in this new equation. Fano’s theorem is found not to apply in the presence of electromagnetic fields. Lewis’ theory for electron multiple scattering and moments, accounting for the coupling between the Lorentz force and multiple elastic scattering, is found. However, further investigation is required to develop useful algorithms for Monte Carlo and deterministic transport methods. To test the accuracy of Monte Carlo transport algorithms in the presence of electromagnetic fields, the Fano cavity test, as currently defined, cannot be applied. Therefore, new tests must be designed for this specific application. A multiple scattering theory that accurately couples the Lorentz force with elastic scattering could improve Monte Carlo efficiency. The present study proposes a new theoretical framework to develop such algorithms.
Bouchard, Hugo; Bielajew, Alex
2015-07-01
To establish a theoretical framework for generalizing Monte Carlo transport algorithms by adding external electromagnetic fields to the Boltzmann radiation transport equation in a rigorous and consistent fashion. Using first principles, the Boltzmann radiation transport equation is modified by adding a term describing the variation of the particle distribution due to the Lorentz force. The implications of this new equation are evaluated by investigating the validity of Fano's theorem. Additionally, Lewis' approach to multiple scattering theory in infinite homogeneous media is redefined to account for the presence of external electromagnetic fields. The equation is modified and yields a description consistent with the deterministic laws of motion as well as probabilistic methods of solution. The time-independent Boltzmann radiation transport equation is generalized to account for the electromagnetic forces in an additional operator similar to the interaction term. Fano's and Lewis' approaches are stated in this new equation. Fano's theorem is found not to apply in the presence of electromagnetic fields. Lewis' theory for electron multiple scattering and moments, accounting for the coupling between the Lorentz force and multiple elastic scattering, is found. However, further investigation is required to develop useful algorithms for Monte Carlo and deterministic transport methods. To test the accuracy of Monte Carlo transport algorithms in the presence of electromagnetic fields, the Fano cavity test, as currently defined, cannot be applied. Therefore, new tests must be designed for this specific application. A multiple scattering theory that accurately couples the Lorentz force with elastic scattering could improve Monte Carlo efficiency. The present study proposes a new theoretical framework to develop such algorithms. PMID:26061045
Parallel FE Approximation of the Even/Odd Parity Form of the Linear Boltzmann Equation
Drumm, Clifton R.; Lorenz, Jens
1999-07-21
A novel solution method has been developed to solve the linear Boltzmann equation on an unstructured triangular mesh. Instead of tackling the first-order form of the equation, this approach is based on the even/odd-parity form in conjunction with the conventional mdtigroup discrete-ordinates approximation. The finite element method is used to treat the spatial dependence. The solution method is unique in that the space-direction dependence is solved simultaneously, eliminating the need for the conventional inner iterations, and the method is well suited for massively parallel computers.
NASA Astrophysics Data System (ADS)
Ma, John Z. G.; St.-Maurice, J.-P.
2015-06-01
By applying a backward mapping technique, we solve the Boltzmann equation to investigate the effects of ion-neutral collisions on the ion velocity distribution and related transport properties in cylindrically symmetric, uniformly charged auroral ionosphere. Such a charge geometry introduces a radial electric field which increases linearly with distance from the axis of symmetry. In order to obtain complete analytical solutions for gaining physical insights into more complicated problems, we have substituted a relaxation collision model for the Boltzmann collision integral in the Boltzmann equation. Our calculations show that collisions drive the velocity distribution to a "horseshoe" shape after a few collision times. This feature extends to all radial positions as long as the electric field keeps increasing linearly versus radius. If the electric field is introduced suddenly, there is a transition from the collision-free pulsating Maxwellian distributions obtained in previous work (Ma and St.-Maurice, J. Geophys. Res., 113:A05312, 2008) to the "horseshoe" shapes on a time scale of within the few collision times. We also show how the transport properties evolve in a similar fashion, from oscillating to a non-oscillating features over the same time interval.
Conservative phase-field lattice Boltzmann model for interface tracking equation.
Geier, Martin; Fakhari, Abbas; Lee, Taehun
2015-06-01
Based on the phase-field theory, we propose a conservative lattice Boltzmann method to track the interface between two different fluids. The presented model recovers the conservative phase-field equation and conserves mass locally and globally. Two entirely different approaches are used to calculate the gradient of the phase field, which is needed in computation of the normal to the interface. One approach uses finite-difference stencils similar to many existing lattice Boltzmann models for tracking the two-phase interface, while the other one invokes central moments to calculate the gradient of the phase field without any finite differences involved. The former approach suffers from the nonlocality of the collision operator while the latter is entirely local making it highly suitable for massive parallel implementation. Several benchmark problems are carried out to assess the accuracy and stability of the proposed model. PMID:26172824
Theory of the Lattice Boltzmann Equation: Symmetry properties of Discrete Velocity Sets
NASA Technical Reports Server (NTRS)
Rubinstein, Robert; Luo, Li-Shi
2007-01-01
In the lattice Boltzmann equation, continuous particle velocity space is replaced by a finite dimensional discrete set. The number of linearly independent velocity moments in a lattice Boltzmann model cannot exceed the number of discrete velocities. Thus, finite dimensionality introduces linear dependencies among the moments that do not exist in the exact continuous theory. Given a discrete velocity set, it is important to know to exactly what order moments are free of these dependencies. Elementary group theory is applied to the solution of this problem. It is found that by decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group, it becomes relatively straightforward to assess the behavior of moments in the theory. The construction of some standard two- and three-dimensional models is reviewed from this viewpoint, and procedures for constructing some new higher dimensional models are suggested.
NASA Astrophysics Data System (ADS)
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.
NASA Astrophysics Data System (ADS)
Takane, Yositake; Hayashi, Masahiko; Ebisawa, Hiromichi
2016-08-01
The time-dependent Ginzburg-Landau equation and the Boltzmann transport equation for charge-density-wave (CDW) conductors are derived from a microscopic one-dimensional model by applying the Keldysh Green's function approach under a quasiclassical approximation. The effects of an external electric field and impurity pinning of the CDW are fully taken into account without relying on a phenomenological argument. These equations simultaneously describe the spatiotemporal dynamics of both the CDW and quasiparticles; thus, they serve as a starting point to develop a general framework to analyze various nonequilibrium phenomena, such as current conversion between the CDW condensate and quasiparticles, in realistic CDW conductors. It is shown that, in typical situations, the equations correctly describe the nonlinear behavior of electric conductivity in a simpler manner.
A conservative multi-group approach to the Boltzmann equations for reactive gas mixtures
NASA Astrophysics Data System (ADS)
Bisi, M.; Rossani, A.; Spiga, G.
2015-11-01
Starting from a simple kinetic model for a quaternary mixture of gases undergoing a bimolecular chemical reaction, multi-group integro-differential equations are derived for the particle distribution functions of all species. The procedure takes advantage of a suitable probabilistic formulation, based on the underlying collision frequencies and transition probabilities, of the relevant reactive kinetic equations of Boltzmann type. Owing to an appropriate choice of a sufficiently large number of weight functions, it is shown that the proposed multi-group equations are able to fulfil exactly, at any order of approximation, the correct conservation laws that must be inherited from the original kinetic equations, where speed was a continuous variable. Future developments are also discussed.
Derivation of quantum mechanics from the Boltzmann equation for the Planch aether
Winterberg, F.
1995-10-01
The Planck aether hypothesis assumes that space is densely filled with an equal number of locally interacting positive and negative Planck masses obeying an exactly nonrelativistic law of motion. The Planck masses can be described by a quantum mechanical two-component nonrelativistic operator field equation having the form of a two-component nonlinear Schroedinger equation, with a spectrum of quasiparticles obeying Lorentz invariance as a dynamic symmetry for energies small compared to the Planck energy. We show that quantum mechanics itself can be derived from the Newtonian mechanics of the Planck aether as an approximate solution of Boltzmann`s equation for the locally interacting positive and negative Planck masses, and that the validity of the nonrelativistic Schroedinger equation depends on Lorentz invariance as a dynamic symmetry. We also show how the many-body Schroedinger wave function can be factorized into a product of quasiparticles of the Planck aether with separable quantum potentials. Finally, we present a possible explanation of wave function collapse as a kind of enhanced gravitational collapse in the presence of the negative Planck masses.
NASA Astrophysics Data System (ADS)
Asinari, P.
2011-03-01
Boltzmann equation is one the most powerful paradigms for explaining transport phenomena in fluids. Since early fifties, it received a lot of attention due to aerodynamic requirements for high altitude vehicles, vacuum technology requirements and nowadays, micro-electro-mechanical systems (MEMs). Because of the intrinsic mathematical complexity of the problem, Boltzmann himself started his work by considering first the case when the distribution function does not depend on space (homogeneous case), but only on time and the magnitude of the molecular velocity (isotropic collisional integral). The interest with regards to the homogeneous isotropic Boltzmann equation goes beyond simple dilute gases. In the so-called econophysics, a Boltzmann type model is sometimes introduced for studying the distribution of wealth in a simple market. Another recent application of the homogeneous isotropic Boltzmann equation is given by opinion formation modeling in quantitative sociology, also called socio-dynamics or sociophysics. The present work [1] aims to improve the deterministic method for solving homogenous isotropic Boltzmann equation proposed by Aristov [2] by two ideas: (a) the homogeneous isotropic problem is reformulated first in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium).
NASA Astrophysics Data System (ADS)
Tessarotto, Massimo; Cremaschini, Claudio
2014-07-01
In this investigation, exact particular realizations are sought for the microscopic statistical description which is associated with the classical dynamical system (CDS) formed by N identical smooth hard spheres subject to elastic collisions ( S N -CDS). The problem is posed in the framework of the ab initio statistical description of S N -CDS recently developed. It is shown that the Liouville equation associated with SN-CDS admits an exact particular solution for the N-body probability density function (PDF). This is factorized in terms of the i-th particle 1-body PDF (for all i = 1, N) via suitable weighting factors, which are denoted here as particle occupation coefficients. The latter are found to depend functionally only on the 1-body PDFs which are associated with each of the remaining particles belonging to S N -CDS. Furthermore, the 1-body PDF is proved to obey a well-defined statistical equation, referred to here as Master kinetic equation. This is an exact kinetic equation which takes into account the occurrence of configuration-space correlations due to the finite size of the extended particles, while depending functionally on the same 1-body PDF only. The asymptotic approximation of the Master equation, which holds in validity of the Boltzmann-Grad limit, is shown to recover in a suitable asymptotic sense the customary Boltzmann equation. Finally, a critical analysis is presented of the original and modified versions of the Enskog kinetic equation, as well as of some of the non-linear kinetic approaches formulated in the past for dense granular gases. Their conditions of validity and main differences with respect to the present theory are pointed out.
NASA Astrophysics Data System (ADS)
Yano, Ryosuke; Matsumoto, Jun; Suzuki, Kojiro
2011-06-01
Thermally relativistic flow with dissipation was analyzed by solving the rarefied supersonic flow of thermally relativistic matter around a triangle prism by Yano and Suzuki [Phys. Rev. DPRVDAQ1550-7998 83, 023517 (2011)10.1103/PhysRevD.83.023517], where the Anderson-Witting (AW) model was used as a solver. In this paper, we solve the same problem, which was analyzed by Yano and Suzuki, using the relativistic Boltzmann equation (RBE). To solve the RBE, the conventional direct simulation Monte Carlo method for the nonrelativistic Boltzmann equation is extended to a new direct simulation Monte Carlo method for the RBE. Additionally, we solve the modified Marle (MM) model proposed by Yano-Suzuki-Kuroda for comparisons. The solution of the thermally relativistic shock layer around the triangle prism obtained using the relativistic Boltzmann equation is considered by focusing on profiles of macroscopic quantities, such as the density, velocity, temperature, heat flux and dynamic pressure along the stagnation streamline (SSL). Differences among profiles of the number density, velocity and temperature along the SSL obtained using the RBE, the AW and MM. models are described in the framework of the relativistic Navier-Stokes-Fourier law. Finally, distribution functions on the SSL obtained using the RBE are compared with those obtained using the AW and MM models. The distribution function inside the shock wave obtained using the RBE does not indicate a bimodal form, which is obtained using the AW and MM models, but a smooth deceleration of thermally relativistic matter inside a shock wave.
NASA Astrophysics Data System (ADS)
Ayissi, Raoul Domingo; Noutchegueme, Norbert
2015-01-01
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
Ayissi, Raoul Domingo Noutchegueme, Norbert
2015-01-15
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
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.
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)
Sharipov, Felix
2011-12-01
The Boltzmann equation subject to a general boundary condition is expanded in a power series with respect to a thermodynamic force disturbing a gaseous system. Recurrence relations between the terms of the expansion are obtained using the main properties of the collision integral and of the gas-surface interaction kernel. The reciprocal relation for nonlinear irreversible phenomena, i.e., a relation between the terms of different orders, is obtained. The relations can be used to estimate the range of applicability of linearized solutions and to predict nonlinear phenomena in gaseous systems.
On the linear Boltzmann equation with rough granular collisions and spin
NASA Astrophysics Data System (ADS)
Pettersson, Rolf
2012-11-01
This paper considers the time-and space-dependent linear Boltzmann equation with general boundary conditions in the case of inelastic rough granular collisions. First, in the angular cut-off case or hard sphere case, mild L1-solutions are constructed as limits of the iterate functions and boundedness of higher velocity moments are discussed in the case of hard inverse power collisions or hard sphere collisions. Furthermore, convergence of solutions to a stationary state, when time goes to infinity, is discussed, using a generalized H-theorem.
Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation.
Khurana, Saheba; Thachuk, Mark
2016-03-14
A numerical method using B-splines is used to solve the linear Boltzmann equation describing the energy relaxation of massive tracer particles moving through a dilute bath gas. The smooth and rough hard sphere and Maxwell molecule models are used with a variety of mass ratios and initial energies to test the capability of the numerical method. Massive tracers are initialized with energies typically found in energy loss experiments in mass spectrometry using biomolecules. The method is also used to examine the applicability of known expressions for the kinetic energy decay from the Fokker-Planck equation for the Rayleigh gas, where we find that results are generally good provided that the initial energy is properly bounded. Otherwise, the energy decay is not constant and a more complex behaviour occurs. The validity of analytical expressions for drag coefficients for spherical particles under specular and diffuse scattering is also tested. We find such expressions are generally good for hard spheres but cannot account, as expected, for the softer repulsive walls of the Maxwell (and real) molecules. Overall, the numerical method performed well even when tracers more than 400 times as massive as the bath were initialized with energies very far from equilibrium. This is a range of applicability beyond many of the standard methods for solving the Boltzmann equation. PMID:26979675
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.
NASA Astrophysics Data System (ADS)
Huang, Rongzong; Wu, Huiying
2015-03-01
A lattice Boltzmann (LB) model for the convection-diffusion equation (CDE) with divergence-free velocity field is proposed, and the Chapman-Enskog analysis shows that the CDE can be recovered correctly. In the present model, the convection term is treated as a source term in the lattice Boltzmann equation (LBE) rather than being directly recovered by LBE; thus the CDE is intrinsically solved as a pure diffusion equation with a corresponding source term. To avoid the adoption of a nonlocal finite-difference scheme for computing the convection term, a local scheme is developed based on the Chapman-Enskog analysis. Most importantly, by properly specifying the discrete source term in the moment space, the local scheme can reach the same order (ɛ2) at which the CDE is recovered by a LB model. Numerical tests, including a one-dimensional periodic problem, diffusion of a Gaussian hill, diffusion of a rectangular pulse, and natural convection in a square cavity, are carried out to verify the present model. Numerical results are satisfactorily consistent with analytical solutions or previous numerical results, and show higher accuracy due to the correct recovery of CDE.
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.
A new lattice Boltzmann model for solving the coupled viscous Burgers’ equation
NASA Astrophysics Data System (ADS)
Lai, Huilin; Ma, Changfeng
2014-02-01
In this paper, a new lattice Boltzmann model for the coupled nonlinear system of viscous Burgers’ equation is proposed by using the double evolutionary equations. Through selecting equilibrium distribution functions and amending functions properly, the governing evolution system can be recovered correctly according to our proposed scheme, in which the Chapman-Enskog expansion is employed. The effects of space and time resolutions on the accuracy and stability of the model are numerically investigated in detail. The numerical solutions for various initial and boundary conditions are calculated and validated against analytic solutions or other numerical solutions reported in previous studies. It is found that the numerical results agree well with the analytic solutions, which indicates the potential of the present algorithm for solving the coupled nonlinear system of viscous Burgers’ equation.
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.
Nanoscale roughness effect on Maxwell-like boundary conditions for the Boltzmann equation
NASA Astrophysics Data System (ADS)
Brull, S.; Charrier, P.; Mieussens, L.
2016-08-01
It is well known that the roughness of the wall has an effect on microscale gas flows. This effect can be shown for large Knudsen numbers by using a numerical solution of the Boltzmann equation. However, when the wall is rough at a nanometric scale, it is necessary to use a very small mesh size which is much too expansive. An alternative approach is to incorporate the roughness effect in the scattering kernel of the boundary condition, such as the Maxwell-like kernel introduced by the authors in a previous paper. Here, we explain how this boundary condition can be implemented in a discrete velocity approximation of the Boltzmann equation. Moreover, the influence of the roughness is shown by computing the structure scattering pattern of mono-energetic beams of the incident gas molecules. The effect of the angle of incidence of these molecules, of their mass, and of the morphology of the wall is investigated and discussed in a simplified two-dimensional configuration. The effect of the azimuthal angle of the incident beams is shown for a three-dimensional configuration. Finally, the case of non-elastic scattering is considered. All these results suggest that our approach is a promising way to incorporate enough physics of gas-surface interaction, at a reasonable computing cost, to improve kinetic simulations of micro- and nano-flows.
Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation
NASA Astrophysics Data System (ADS)
Harris, Robert C.; Boschitsch, Alexander H.; Fenley, Marcia O.
2014-02-01
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.
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. PMID:24559370
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.
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.
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.
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
NASA Astrophysics Data System (ADS)
Tsumura, Kyosuke; Kikuchi, Yuta; Kunihiro, Teiji
2015-10-01
We derive the second-order hydrodynamic equation and the microscopic formulas of the relaxation times as well as the transport coefficients systematically from the relativistic Boltzmann equation. Our derivation is based on a novel development of the renormalization-group method, a powerful reduction theory of dynamical systems, which has been applied successfully to derive the nonrelativistic second-order hydrodynamic equation. Our theory nicely gives a compact expression of the deviation of the distribution function in terms of the linearized collision operator, which is different from those used as an ansatz in the conventional fourteen-moment method. It is confirmed that the resultant microscopic expressions of the transport coefficients coincide with those derived in the Chapman-Enskog expansion method. Furthermore, we show that the microscopic expressions of the relaxation times have natural and physically plausible forms. We prove that the propagating velocities of the fluctuations of the hydrodynamical variables do not exceed the light velocity, and hence our second-order equation ensures the desired causality. It is also confirmed that the equilibrium state is stable for any perturbation described by our equation.
Sun, Hongguang; Meerschaert, Mark M; Zhang, Yong; Zhu, Jianting; Chen, Wen
2013-02-01
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
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
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.
Obliger, Amaël; Duvail, Magali; Jardat, Marie; Coelho, Daniel; Békri, Samir; Rotenberg, Benjamin
2013-07-01
We report the calculation of all the transfer coefficients which couple the solvent and ionic fluxes through a charged pore under the effect of pressure, electrostatic potential, and concentration gradients. We use a combination of analytical calculations at the Poisson-Nernst-Planck and Navier-Stokes levels of description and mesoscopic lattice simulations based on kinetic theory. In the absence of added salt, i.e., when the only ions present in the fluid are the counterions compensating the charge of the surface, exact analytical expressions for the fluxes in cylindrical pores allow us to validate a new lattice-Boltzmann electrokinetics (LBE) scheme which accounts for the osmotic contribution to the transport of all species. The influence of simulation parameters on the numerical accuracy is thoroughly investigated. In the presence of an added salt, we assess the range of validity of approximate expressions of the fluxes computed from the linearized Poisson-Boltzmann equation by a systematic comparison with LBE simulations. PMID:23944561
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. 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
Yong, Wen-An; Zhao, Weifeng; Luo, Li-Shi
2016-03-01
We propose using the Maxwell iteration to derive the hydrodynamic equations from the lattice Boltzmann equation (LBE) with an external forcing term. The proposed methodology differs from existing approaches in several aspects. First, it need not explicitly introduce multiple-timescales or the Knudsen number, both of which are required in the Chapman-Enskog analysis. Second, it need not use the Hilbert expansion of the hydrodynamic variables, which is necessary in the asymptotic analysis of the LBE. The proposed methodology assumes the acoustic scaling (or the convective scaling) δ_{t}∼δ_{x}, thus δ_{t} is the only expansion parameter in the analysis of the LBE system, and it leads to the Navier-Stokes equations in compressible form. The forcing density derived in this work can reproduce existing forcing schemes by adjusting appropriate parameters. The proposed methodology also analyzes the numerical accuracy of the LBE. In particular, it shows the Mach number Ma should scale as O(δ_{t}^{1/3}) to maintain the truncation errors due to Ma and δ_{t} in balance when δ_{t}→0, so that the LBE can converge to the expected hydrodynamic equations effectively and efficiently. PMID:27078487
NASA Astrophysics Data System (ADS)
Yong, Wen-An; Zhao, Weifeng; Luo, Li-Shi
2016-03-01
We propose using the Maxwell iteration to derive the hydrodynamic equations from the lattice Boltzmann equation (LBE) with an external forcing term. The proposed methodology differs from existing approaches in several aspects. First, it need not explicitly introduce multiple-timescales or the Knudsen number, both of which are required in the Chapman-Enskog analysis. Second, it need not use the Hilbert expansion of the hydrodynamic variables, which is necessary in the asymptotic analysis of the LBE. The proposed methodology assumes the acoustic scaling (or the convective scaling) δt˜δx , thus δt is the only expansion parameter in the analysis of the LBE system, and it leads to the Navier-Stokes equations in compressible form. The forcing density derived in this work can reproduce existing forcing schemes by adjusting appropriate parameters. The proposed methodology also analyzes the numerical accuracy of the LBE. In particular, it shows the Mach number Ma should scale as O (δt1 /3) to maintain the truncation errors due to Ma and δt in balance when δt→0 , so that the LBE can converge to the expected hydrodynamic equations effectively and efficiently.
NASA Astrophysics Data System (ADS)
Khoromskij, Boris N.
2007-09-01
We develop efficient data-sparse representations to a class of high order tensors via a block many-fold Kronecker product decomposition. Such a decomposition is based on low separation-rank approximations of the corresponding multivariate generating function. We combine the Sinc interpolation and a quadrature-based approximation with hierarchically organised block tensor-product formats. Different matrix and tensor operations in the generalised Kronecker tensor-product format including the Hadamard-type product can be implemented with the low cost. An application to the collision integral from the deterministic Boltzmann equation leads to an asymptotical cost O(n^4log^beta n) - O(n^5log^beta n) in the one-dimensional problem size n (depending on the model kernel function), which noticeably improves the complexity O(n^6log^beta n) of the full matrix representation.
Boundary conditions of the lattice Boltzmann method for convection-diffusion equations
NASA Astrophysics Data System (ADS)
Huang, Juntao; Yong, Wen-An
2015-11-01
In this paper, we employ an asymptotic analysis technique and construct two boundary schemes accompanying the lattice Boltzmann method for convection-diffusion equations with general Robin boundary conditions. One scheme is for straight boundaries, with the boundary points locating at any distance from the lattice nodes, and has second-order accuracy. The other is for curved boundaries, has only first-order accuracy and is much simpler than the existing schemes. Unlike those in the literature, our schemes involve only the current lattice node. Such a "single-node" boundary schemes are highly desirable for problems with complex geometries. The two schemes are validated numerically with a number of examples. The numerical results show the utility of the constructed schemes and very well support our theoretical predications.
NASA Astrophysics Data System (ADS)
Hu, Jingwei; Wang, Li
2015-01-01
We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for electron mass and energy as mean free path goes to zero. As opposed to the classical drift-diffusion limit where the stiff collisions are all in one scale, new difficulties arise in the two-scale stiff collision terms because the simple BGK penalization [15] fails to drive the solution to the correct limit. We propose to set up a spatially dependent threshold on the penalization of the stiffer collision operator such that the evolution of the solution resembles a Hilbert expansion at the continuous level. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of our scheme.
Lattice Boltzmann method for the age concentration equation in shallow water
NASA Astrophysics Data System (ADS)
Liu, Haifei; Ding, Yu; Wang, Hongda; Zhang, Jie
2015-10-01
Water age is a critical parameter in reflecting the extent of water exchange. It represents the time that water parcels or contaminants are transported from source to current positions. In this study, an equilibrium distribution function for water age concentration is proposed within the Eulerian framework based on the existing theory of water age, and it can recover the age concentration equation. In addition, the lattice Boltzmann method for water age in the Lagrangian procedures is developed. This method also enables the Lagrangian age to be fundamentally simulated under computationally expensive conditions. In numerical tests, cubic and circular reservoirs with narrow inflow-outflow boundaries are used to verify the applicability of the model. Finally, the proposed approaches are applied to the Baiyangdian Lake, the biggest freshwater lake in northern China. The result is compared with that acquired by the Environmental Fluid Dynamic Code (EFDC) as well.
Radtke, Gregg A; Hadjiconstantinou, Nicolas G
2009-05-01
We present an efficient variance-reduced particle simulation technique for solving the linearized Boltzmann transport equation in the relaxation-time approximation used for phonon, electron, and radiative transport, as well as for kinetic gas flows. The variance reduction is achieved by simulating only the deviation from equilibrium. We show that in the limit of small deviation from equilibrium of interest here, the proposed formulation achieves low relative statistical uncertainty that is also independent of the magnitude of the deviation from equilibrium, in stark contrast to standard particle simulation methods. Our results demonstrate that a space-dependent equilibrium distribution improves the variance reduction achieved, especially in the collision-dominated regime where local equilibrium conditions prevail. We also show that by exploiting the physics of relaxation to equilibrium inherent in the relaxation-time approximation, a very simple collision algorithm with a clear physical interpretation can be formulated. PMID:19518597
NASA Astrophysics Data System (ADS)
Rhyner, Reto; Luisier, Mathieu
2013-12-01
We propose to check and validate the approximations made in dissipative quantum transport (QT) simulations solved in the Non-equilibrium Green's Function formalism by comparing them with the exact solution of the linearized Boltzmann Transport Equation (LB) in the stationary regime. For that purpose, we calculate the phonon-limited electron and hole mobility in bulk Si and ultra-scaled Si nanowires for different crystal orientations ⟨100⟩, ⟨110⟩, and ⟨111⟩. In both QT and LB simulations, we use the same sp3d5s* tight-binding model to describe the electron/hole properties and the same valence-force-field approach to account for the phonon properties. It is found that the QT simplifications work well for electrons, but are less accurate for holes, where a renormalization of the phonon scattering strength is proved useful to improve the results.
A numerical solution of the linear Boltzmann equation using cubic B-splines
NASA Astrophysics Data System (ADS)
Khurana, Saheba; Thachuk, Mark
2012-03-01
A numerical method using cubic B-splines is presented for solving the linear Boltzmann equation. The collision kernel for the system is chosen as the Wigner-Wilkins kernel. A total of three different representations for the distribution function are presented. Eigenvalues and eigenfunctions of the collision matrix are obtained for various mass ratios and compared with known values. Distribution functions, along with first and second moments, are evaluated for different mass and temperature ratios. Overall it is shown that the method is accurate and well behaved. In particular, moments can be predicted with very few points if the representation is chosen well. This method produces sparse matrices, can be easily generalized to higher dimensions, and can be cast into efficient parallel algorithms.
NASA Astrophysics Data System (ADS)
Alekseenko, A.; Josyula, E.
2012-11-01
We propose an approach for high order discretization of the Boltzmann equation in the velocity space using discontinuous Galerkin methods. Our approach employs a reformulation of the collision integral in the form of a bilinear operator with a time-independent kernel. In the fully non-linear case the complexity of the method is O(n8) operations per spatial cell where n is the number of degrees of freedom in one velocity direction. The new method is suitable for parallelization to a large number of processors. Techniques of automatic perturbation decomposition and linearisation are developed to achieve additional performance improvement. The number of operations per spatial cell in the linearised regime is O(n6). The method is applied to the solution of the spatially homogeneous relaxation problem. Mass momentum and energy is conserved to a good precision in the computed solutions.
Evaluation of the Lattice-Boltzmann Equation Solver PowerFLOW for Aerodynamic Applications
NASA Technical Reports Server (NTRS)
Lockard, David P.; Luo, Li-Shi; Singer, Bart A.; Bushnell, Dennis M. (Technical Monitor)
2000-01-01
A careful comparison of the performance of a commercially available Lattice-Boltzmann Equation solver (Power-FLOW) was made with a conventional, block-structured computational fluid-dynamics code (CFL3D) for the flow over a two-dimensional NACA-0012 airfoil. The results suggest that the version of PowerFLOW used in the investigation produced solutions with large errors in the computed flow field; these errors are attributed to inadequate resolution of the boundary layer for reasons related to grid resolution and primitive turbulence modeling. The requirement of square grid cells in the PowerFLOW calculations limited the number of points that could be used to span the boundary layer on the wing and still keep the computation size small enough to fit on the available computers. Although not discussed in detail, disappointing results were also obtained with PowerFLOW for a cavity flow and for the flow around a generic helicopter configuration.
NASA Astrophysics Data System (ADS)
Wu, Lei; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.
2015-08-01
The Boltzmann equation with an arbitrary intermolecular potential is solved by the fast spectral method. As examples, noble gases described by the Lennard-Jones potential are considered. The accuracy of the method is assessed by comparing both transport coefficients with variational solutions and mass/heat flow rates in Poiseuille/thermal transpiration flows with results from the discrete velocity method. The fast spectral method is then applied to Fourier and Couette flows between two parallel plates, and the influence of the intermolecular potential on various flow properties is investigated. It is found that for gas flows with the same rarefaction parameter, differences in the heat flux in Fourier flow and the shear stress in Couette flow are small. However, differences in other quantities such as density, temperature, and velocity can be very large.
NASA Astrophysics Data System (ADS)
Fournier, Nicolas; Mouhot, Clément
2009-08-01
We prove an inequality on the Kantorovich-Rubinstein distance-which can be seen as a particular case of a Wasserstein metric-between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, but with a moderate angular singularity. Our method is in the spirit of [7]. We deduce some well-posedness and stability results in the physically relevant cases of hard and moderately soft potentials. In the case of hard potentials, we relax the regularity assumption of [6], but we need stronger assumptions on the tail of the distribution (namely some exponential decay). We thus obtain the first uniqueness result for measure initial data. In the case of moderately soft potentials, we prove existence and uniqueness assuming only that the initial datum has finite energy and entropy (for very moderately soft potentials), plus sometimes an additionnal moment condition. We thus improve significantly on all previous results, where weighted Sobolev spaces were involved.
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(+). PMID:25941952
A modified multiple-relaxation-time lattice Boltzmann model for convection-diffusion equation
NASA Astrophysics Data System (ADS)
Huang, Rongzong; Wu, Huiying
2014-10-01
A modified lattice Boltzmann model with multiple relaxation times (MRT) for the convection-diffusion equation (CDE) is proposed. By modifying the relaxation matrix, as well as choosing the corresponding equilibrium distribution function properly, the present model can recover the CDE with anisotropic diffusion coefficient with no deviation term even when the velocity vector varies generally with space or time through the Chapman-Enskog analysis. This model is firstly validated by simulating the diffusion of a Gaussian hill, which demonstrates it can handle the anisotropic diffusion problem correctly. Then it is adopted to calculate the longitudinal dispersion coefficient of the Taylo-Aris dispersion. Numerical results show that the present model can further reduce the numerical error under the condition of non-zero velocity vector, especially when the dimensionless relaxation time is relatively large.
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.
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.
NASA Astrophysics Data System (ADS)
Bakaleinikov, L. A.; Flegontova, E. Yu.; Ender, A. Ya.; Ender, I. A.
2016-04-01
A recurrence procedure for a sequential construction of kernels G_{{l_1},{l_2}}^l ( c, c 1, c 2) appearing upon the expansion of a nonlinear collision integral of the Boltzmann equation in spherical harmonics is developed. The starting kernel for this procedure is kernel G 0,0 0 ( c, c 1, c 2) of the collision integral for the distribution function isotropic with respect to the velocities. Using the recurrence procedure, a set of kernels G_{{l_1},{l_2}}^{ + l} ( c, c 1, c 2) for a gas consisting of hard spheres and Maxwellian molecules is constructed. It is shown that the resultant kernels exhibit similarity and symmetry properties and satisfy the relations following from the conservation laws.
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)
Faghaninia, Alireza; Ager, Joel W.; Lo, Cynthia S.
2015-06-01
Accurate models of carrier transport are essential for describing the electronic properties of semiconductor materials. To the best of our knowledge, the current models following the framework of the Boltzmann transport equation (BTE) either rely heavily on experimental data (i.e., semiempirical), or utilize simplifying assumptions, such as the constant relaxation time approximation (BTE-cRTA). While these models offer valuable physical insights and accurate calculations of transport properties in some cases, they often lack sufficient accuracy—particularly in capturing the correct trends with temperature and carrier concentration. We present here a transport model for calculating low-field electrical drift mobility and Seebeck coefficient of n -type semiconductors, by explicitly considering relevant physical phenomena (i.e., elastic and inelastic scattering mechanisms). We first rewrite expressions for the rates of elastic scattering mechanisms, in terms of ab initio properties, such as the band structure, density of states, and polar optical phonon frequency. We then solve the linear BTE to obtain the perturbation to the electron distribution—resulting from the dominant scattering mechanisms—and use this to calculate the overall mobility and Seebeck coefficient. Therefore, we have developed an ab initio model for calculating mobility and Seebeck coefficient using the Boltzmann transport (aMoBT) equation. Using aMoBT, we accurately calculate electrical transport properties of the compound n -type semiconductors, GaAs and InN, over various ranges of temperature and carrier concentration. aMoBT is fully predictive and provides high accuracy when compared to experimental measurements on both GaAs and InN, and vastly outperforms both semiempirical models and the BTE-cRTA. Therefore, we assert that this approach represents a first step towards a fully ab initio carrier transport model that is valid in all compound semiconductors.
Evolution of Central Moments for a General-Relativistic Boltzmann Equation
NASA Astrophysics Data System (ADS)
Banach, Zbigniew; Larecki, Wieslaw
Beginning from the relativistic Boltzmann equation in a curved space-time, and assuming that there exists a fiducial congruence of timelike world lines with four-velocity vector field u, it is the aim of this paper to present a systematic derivation of a hierarchy of closed systems of moment equations. These systems are found by using the closure by entropy maximization. Our concepts are primarily applied to the formalism of central moments because if an alternative and more familiar theory of covariant moments is taken into account, then the method of maximum entropy is ill-defined in a neighborhood of equilibrium states. The central moments are not covariant in the following sense: two observers looking at the same relativistic gas will, in general, extract two different sets of central moments, not related to each other by a tensorial linear transformation. After a brief review of the formalism of trace-free symmetric spacelike tensors, the differential equations for irreducible central moments are obtained and compared with those of Ellis et al. [Ann. Phys. (NY) 150 (1983) 455]. We derive some auxiliary algebraic identities which involve the set of central moments and the corresponding set of Lagrange multipliers; these identities enable us to show that there is an additional balance law interpreted as the equation of balance of entropy. The above results are valid for an arbitrary choice of the Lorentzian metric g and the four-velocity vector field u. Later, the definition of u as in the well-known theory of Arnowitt, Deser, and Misner is proposed in order to construct a hierarchy of symmetric hyperbolic systems of field equations. Also, the Eckart and Landau-Lifshitz definitions of u are discussed. Specifically, it is demonstrated that they lead, in general, to the systems of nonconservative equations.
Fillion-Gourdeau, F; Herrmann, H J; Mendoza, M; Palpacelli, S; Succi, S
2013-10-18
We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and nonrelativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wave function) propagates in space-time as a classical statistical process (Boltzmann distribution). PMID:24182245
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.
Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow
NASA Astrophysics Data System (ADS)
Zheng, Lin; Zheng, Song; Zhai, Qinglan
2016-02-01
In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface force (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter via Cahn-Hilliard equation which is solved in the frame work of LBE. The scalar convection-diffusion equation for temperature field is resolved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two/three dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results.
Fast Maximum Entropy Moment Closure Approach to Solving the Boltzmann Equation
NASA Astrophysics Data System (ADS)
Summy, Dustin; Pullin, Dale
2015-11-01
We describe a method for a moment-based solution of the Boltzmann Equation (BE). This is applicable to an arbitrary set of velocity moments whose transport is governed by partial-differential equations (PDEs) derived from the BE. The equations are unclosed, containing both higher-order moments and molecular-collision terms. These are evaluated using a maximum-entropy reconstruction of the velocity distribution function f (c , x , t) , from the known moments, within a finite-box domain of single-particle velocity (c) space. Use of a finite-domain alleviates known problems (Junk and Unterreiter, Continuum Mech. Thermodyn., 2002) concerning existence and uniqueness of the reconstruction. Unclosed moments are evaluated with quadrature while collision terms are calculated using any desired method. This allows integration of the moment PDEs in time. The high computational cost of the general method is greatly reduced by careful choice of the velocity moments, allowing the necessary integrals to be reduced from three- to one-dimensional in the case of strictly 1D flows. A method to extend this enhancement to fully 3D flows is discussed. Comparison with relaxation and shock-wave problems using the DSMC method will be presented. Partially supported by NSF grant DMS-1418903.
Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation.
Ren, Feng; Song, Baowei; Sukop, Michael C; Hu, Haibao
2016-08-01
The primary and key task of binary fluid flow modeling is to track the interface with good accuracy, which is usually challenging due to the sharp-interface limit and numerical dispersion. This article concentrates on further development of the conservative Allen-Cahn equation (ACE) [Geier et al., Phys. Rev. E 91, 063309 (2015)10.1103/PhysRevE.91.063309] under the framework of the lattice Boltzmann method (LBM), with incorporation of the incompressible hydrodynamic equations [Liang et al., Phys. Rev. E 89, 053320 (2014)10.1103/PhysRevE.89.053320]. Utilizing a modified equilibrium distribution function and an additional source term, this model is capable of correctly recovering the conservative ACE through the Chapman-Enskog analysis. We also simulate four phase-tracking benchmark cases, including one three-dimensional case; all show good accuracy as well as low numerical dispersion. By coupling the incompressible hydrodynamic equations, we also simulate layered Poiseuille flow and the Rayleigh-Taylor instability, illustrating satisfying performance in dealing with complex flow problems, e.g., high viscosity ratio, high density ratio, and high Reynolds number situations. The present work provides a reliable and efficient solution for binary flow modeling. PMID:27627416
Improved Multiple-Coarsening Methods for Sn Discretizations of the Boltzmann Equation
Lee, Barry
2010-06-01
In a recent series of articles, the author presented a multiple-coarsening multigrid method for solving $S_n$ discretizations of the Boltzmann transport equation. This algorithm is applied to an integral equation for the scalar flux or moments. Although this algorithm is very efficient over parameter regimes that describe realistic neutron/photon transport applications, improved methods that can reduce the computational cost are presented in this paper. These improved methods are derived through a careful examination of the frequencies, particularly the near-nullspace, of the integral equation. In the earlier articles, the near-nullspace components were shown to be smooth in angle in the sense that the angular fluxes generated by these components are smooth in angle. In this paper, we present a spatial description of these near-nullspace components. Using the angular description of the earlier papers together with the spatial description reveals the intrinsic space-angle dependence of the integral equation's frequencies. This space-angle dependence is used to determine the appropriate space-angle grids to represent and efficiently attenuate the near-nullspace error components on. It will be shown that these components can have multiple spatial scales. By using only the appropriate space-angle grids that can represent these spatial scales in the original multiple-coarsening algorithm, an improved algorithm is obtained. Moreover, particularly for anisotropic scattering, recognizing the strong angle dependence of the angular fluxes generated by the high frequencies of the integral equation, another improved multiple-coarsening scheme is derived. Restricting this scheme to the appropriate space-angle grids produces a very efficient method.
Improved Multiple-Coarsening Methods for Sn Discretizations of the Boltzmann Equation
Lee, B
2008-12-01
In a recent series of articles, the author presented a multiple-coarsening multigrid method for solving S{sub n} discretizations of the Boltzmann transport equation. This algorithm is applied to an integral equation for the scalar flux or moments. Although this algorithm is very efficient over parameter regimes that describe realistic neutron/photon transport applications, improved methods that can reduce the computational cost are presented in this paper. These improved methods are derived through a careful examination of the frequencies, particularly the near-nullspace, of the integral equation. In the earlier articles, the near-nullspace components were shown to be smooth in angle in the sense that the angular fluxes generated by these components are smooth in angle. In this paper, we present a spatial description of these near-nullspace components. Using the angular description of the earlier papers together with the spatial description reveals the intrinsic space-angle dependence of the integral equation's frequencies. This space-angle dependence is used to determine the appropriate space-angle grids to represent and efficiently attenuate the near-nullspace error components on. It will be shown that these components can have multiple spatial scales. By using only the appropriate space-angle grids that can represent these spatial scales in the original multiple-coarsening algorithm, an improved algorithm is obtained. Moreover, particularly for anisotropic scattering, recognizing the strong angle dependence of the angular fluxes generated by the high frequencies of the integral equation, another improved multiple-coarsening scheme is derived. Restricting this scheme to the appropriate space-angle grids produces a very efficient method.
NASA Astrophysics Data System (ADS)
Zhang, Jianying; Yan, Guangwu; Wang, Moran
2016-02-01
A lattice Boltzmann model for solving the three-dimensional cubic-quintic complex Ginzburg-Landau equation (CQCGLE) is proposed. Differently from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a three-dimensional space, and the evolution of the model is about a spatial axis rather than time. The algorithm provides advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg-Landau equations. Examples show that the model accurately reproduces the vortex tori pattern in the CQCGLE.
Exact solution of the unidimensional Poisson-Boltzmann equation for a 1:2 (2:1) electrolyte.
Andrietti, F; Peres, A; Pezzotta, R
1976-01-01
The unidimensional Poisson-Boltzmann equation for a 1:2 (2:1) electrolyte has been solved analytically. The results have been compared with those obtained from the linearized equation. It is shown that in physiological conditions the difference may be greater than 10%. The value of the derivative of the potential in x=0, (dpsi/dx)x=0, has been used by many authors in the evaluation of the superficial charges of biological membranes. The value of (dpsi/dx)x-0 have also been compared with the ones derived from the linearized equation. The difference may be greater than 25%. Our results suggest that the linearization of the Poisson-Boltzmann equation for a 1:2(2:1) electrolyte may be greatly misleading. PMID:963209
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
Modeling of RF glow discharges by direct numerical procedure of the Boltzmann equation
Nakano, N.; Makabe, T.
1995-12-31
In the previous papers, the authors performed a direct numerical procedure (DNP) of the Boltzmann equation with an algorithm of their own without any expansion of the velocity distribution in spatial homogeneity, in order to study and radio frequency (rf) electron transport. They further developed ;and applied the DNP to the modeling of an rf glow discharge with space and time variation of the electron velocity distribution. It has a great advantage in that the nonequilibrium characteristics of electrons in transport can be exactly expressed without statistical error. Only collision cross sections of the electron with the molecular are required for calculation without swarm parameters as for electrons, when the governing equation system is numerically simulated. It is of practical advantage to a gas mixture which results from the dissociation of an initial gas after discharge initiation. Capacitively couples parallel plate discharge with 20mm spacing is considered in the present modeling in order to simplify the model and to compare the plasma structure with the previous result using relaxation continuum model. Non reactive Ar with simple cross sections is employed as the source gas.
Boltzmann-equation simulations of radio-frequency-driven, low-temperature plasmas
Drallos, P.J.; Riley, M.E.
1995-01-01
We present a method for the numerical solution of the Boltzmann equation (BE) describing plasma electrons. We apply the method to a capacitively-coupled, radio-frequency-driven He discharge in parallel-plate (quasi-1D) geometry which contains time scales for physical processes spanning six orders of magnitude. Our BE solution procedure uses the method of characteristics for the Vlasov operator with interpolation in phase space at early time, allowing storage of the distribution function on a fixed phase-space grid. By alternating this BE method with a fluid description of the electrons, or with a novel time-cycle-average equation method, we compute the periodic steady state of a He plasma by time evolution from startup conditions. We find that the results compare favorably with measured current-voltage, plasma density, and ``cited state densities in the ``GEC`` Reference Cell. Our atomic He model includes five levels (some are summed composites), 15 electronic transitions, radiation trapping, and metastable-metastable collisions.
NASA Astrophysics Data System (ADS)
Mandt, Stephan
2014-11-01
Variational solutions of the Boltzmann equation usually rely on the concept of linear response. We extend the variational approach for tight-binding models at high entropies to a regime far beyond linear response. We analyze both weakly interacting fermions and incoherent bosons on a lattice. We consider a case where the particles are driven by a constant force, leading to the well-known Bloch oscillations, and we consider interactions that are weak enough not to overdamp these oscillations. This regime is computationally demanding and relevant for ultracold atoms in optical lattices. We derive a simple theory in terms of coupled dynamic equations for the particle density, energy density, current, and heat current, allowing for analytic solutions. As an application, we identify damping coefficients for Bloch oscillations in the Hubbard model at weak interactions and compute them for a one-dimensional toy model. We also approximately solve the long-time dynamics of a weakly interacting, strongly Bloch-oscillating cloud of fermionic particles in a tilted lattice, leading to a subdiffusive scaling exponent.
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. PMID:26871187
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.
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. PMID:12059718
A simple Boltzmann transport equation for ballistic to diffusive transient heat transport
NASA Astrophysics Data System (ADS)
Maassen, Jesse; Lundstrom, Mark
2015-04-01
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.
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.
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.
Catt, B; Snyder, M
2014-06-15
Purpose: To investigate the use of the linear Boltzmann transport equation as a dose calculation tool which can account for interface effects, while still having faster computation times than Monte Carlo methods. In particular, we introduce a forward scattering approximation, in hopes of improving calculation time without a significant hindrance to accuracy. Methods: Two coupled Boltzmann transport equations were constructed, one representing the fluence of photons within the medium, and the other, the fluence of electrons. We neglect the scattering term within the electron transport equation, resulting in an extreme forward scattering approximation to reduce computational complexity. These equations were then solved using a numerical technique for solving partial differential equations, known as a finite difference scheme, where the fluence at each discrete point in space is calculated based on the fluence at the previous point in the particle's path. Using this scheme, it is possible to develop a solution to the Boltzmann transport equations by beginning with boundary conditions and iterating across the entire medium. The fluence of electrons can then be used to find the dose at any point within the medium. Results: Comparisons with Monte Carlo simulations indicate that even simplistic techniques for solving the linear Boltzmann transport equation yield expected interface effects, which many popular dose calculation algorithms are not capable of predicting. Implementation of a forward scattering approximation does not appear to drastically reduce the accuracy of this algorithm. Conclusion: Optimized implementations of this algorithm have been shown to be very accurate when compared with Monte Carlo simulations, even in build up regions where many models fail. Use of a forward scattering approximation could potentially give a reasonably accurate dose distribution in a shorter amount of time for situations where a completely accurate dose distribution is not
A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations
NASA Astrophysics Data System (ADS)
Liu, Chang; Xu, Kun; Sun, Quanhua; Cai, Qingdong
2016-06-01
Fluid dynamic equations are valid in their respective modeling scales, such as the particle mean free path scale of the Boltzmann equation and the hydrodynamic scale of the Navier-Stokes (NS) equations. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. Even though the Boltzmann equation is claimed to be valid in all scales, many Boltzmann solvers, including direct simulation Monte Carlo method, require the cell resolution to the order of particle mean free path scale. Therefore, they are still single scale methods. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. Even though the UGKS is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time, it still has space to develop more accurate flow solver in the region, where the time step is comparable with the local particle mean free time. In such a scale, there is dynamic difference from the full Boltzmann collision term and the model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region
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
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.
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.
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
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.
Boltzmann equation analysis of electron-molecule collision cross sections in water vapor and ammonia
Yousfi, M.; Benabdessadok, M.D.
1996-12-01
Sets of electron-molecule collision cross sections for H{sub 2}O and NH{sub 3} have been determined from a classical technique of electron swarm parameter unfolding. This deconvolution method is based on a simplex algorithm using a powerful multiterm Boltzmann equation analysis established in the framework of the classical hydrodynamic approximation. It is well adapted for the simulation of the different classes of swarm experiments (i.e., time resolved, time of flight, and steady state experiments). The sets of collision cross sections that exist in the literature are reviewed and analyzed. Fitted sets of cross sections are determined for H{sub 2}O and NH{sub 3} which exhibit features characteristic of polar molecules such as high rotational excitation collision cross sections. The hydrodynamic swarm parameters (i.e., drift velocity, longitudinal and transverse diffusion coefficients, ionization and attachment coefficients) calculated from the fitted sets are in excellent agreement with the measured ones. These sets are finally used to calculate the transport and reaction coefficients needed for discharge modeling in two cases of typical gas mixtures for which experimental swarm data are very sparse or nonexistent (i.e., flue gas mixtures and gas mixtures for rf plasma surface treatment). {copyright} {ital 1996 American Institute of Physics.}
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.
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.
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.
Shrinkage of bubbles and drops in the lattice Boltzmann equation method for nonideal gases.
Zheng, Lin; Lee, Taehun; Guo, Zhaoli; Rumschitzki, David
2014-03-01
One characteristic of multiphase lattice Boltzmann equation (LBE) methods is that the interfacial region has a finite (i.e., noninfinitesimal) thickness known as a diffuse interface. In simulations of, e.g., bubble or drop dynamics, for problems involving nonideal gases, one frequently observes that the diffuse interface method produces a spontaneous, nonphysical shrinkage of the bubble or drop radius. In this paper, we analyze in detail a single-fluid two-phase model and use a LBE model for nonideal gases in order to explain this fundamental problem. For simplicity, we only investigate the static bubble or droplet problem. We find that the method indeed produces a density shift, bubble or droplet shrinkage, as well as a critical radius below which the bubble or droplet eventually vanishes. Assuming that the ratio between the interface thickness D and the initial bubble or droplet radius r0 is small, we analytically show the existence of this density shift, bubble or droplet radius shrinkage, and critical bubble or droplet survival radius. Numerical results confirm our analysis. We also consider droplets on a solid surface with different curvatures, contact angles, and initial droplet volumes. Numerical results show that the curvature, contact angle, and the initial droplet volume have an effect on this spontaneous shrinkage process, consistent with the survival criterion. PMID:24730962
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.
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
Influence of Grid Spacing in Poisson-Boltzmann Equation Binding Energy Estimation.
Harris, Robert C; Boschitsch, Alexander H; Fenley, Marcia O
2013-08-13
Grid-based solvers of the Poisson-Boltzmann, PB, equation are routinely used to estimate electrostatic binding, ΔΔGel, and solvation, ΔGel, free energies. The accuracies of such estimates are subject to grid discretization errors from the finite difference approximation to the PB equation. Here, we show that the grid discretization errors in ΔΔGel are more significant than those in ΔGel, and can be divided into two parts: (i) errors associated with the relative positioning of the grid and (ii) systematic errors associated with grid spacing. The systematic error in particular is significant for methods, such as the molecular mechanics PB surface area, MM-PBSA, approach that predict electrostatic binding free energies by averaging over an ensemble of molecular conformations. Although averaging over multiple conformations can control for the error associated with grid placement, it will not eliminate the systematic error, which can only be controlled by reducing grid spacing. The present study indicates that the widely-used grid spacing of 0.5 Å produces unacceptable errors in ΔΔGel, even though its predictions of ΔGel are adequate for the cases considered here. Although both grid discretization errors generally increase with grid spacing, the relative sizes of these errors differ according to the solute-solvent dielectric boundary definition. The grid discretization errors are generally smaller on the Gaussian surface used in the present study than on either the solvent-excluded or van der Waals surfaces, which both contain more surface discontinuities (e.g., sharp edges and cusps). Additionally, all three molecular surfaces converge to very different estimates of ΔΔGel. PMID:23997692
Yousfi, M.; Himoudi, A.; Gaouar, A. )
1992-12-15
Electron distribution functions in nonthermal cold plasmas generated by classical electrical discharges have been calculated from a powerful Boltzmann equation solution and an original Monte Carlo simulation. In these two methods both classical (i.e., elastic, inelastic, and superelastic) electron-atom (or molecule) collisions and electron-electron interactions are taken into account. The approximations considered to include long-range (electron-electron) and short-range (electron-atom) interactions in the same Monte Carlo algorithm are first validated by comparing with Boltzmann equation results. Then, the influence of electron-electron interactions on electron distribution functions, swarm parameters, and reaction rates under nonthermal cold plasma conditions are analyzed and discussed as a function of reduced electric field [ital E]/[ital N] and ionization degree [ital n][sub [ital e
Hashimoto, K; Kanki, K; Tanaka, S; Petrosky, T
2016-02-01
Irreversible processes of weakly coupled one-dimensional quantum perfect Lorentz gas are studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouville-von Neumann operator. Without any phenomenological operations, such as a coarse-graining of space-time, or a truncation of the higher order correlation, we obtained irreversible processes in a purely dynamical basis in all space and time scale including the microscopic atomic interaction range that is much smaller than the mean-free length. Based on this solution, a limitation of the usual phenomenological Boltzmann equation, as well as an extension of the Boltzmann equation to entire space-time scale, is discussed. PMID:26986313
NASA Astrophysics Data System (ADS)
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.
NASA Astrophysics Data System (ADS)
Dodulad, O. I.; Kloss, Yu. Yu.; Potapov, A. P.; Tcheremissine, F. G.; Shuvalov, P. V.
2016-06-01
Flows of a simple rarefied gas and gas mixtures are computed on the basis of the Boltzmann kinetic equation, which is solved by applying various versions of the conservative projection method, namely, a two-point method for a simple gas and gas mixtures with a small difference between the molecular masses and a multipoint method in the case of a large mass difference. Examples of steady and unsteady flows are computed in a wide range of Mach and Knudsen numbers.
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.
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
NASA Astrophysics Data System (ADS)
Wang, Mingliang; Wong, Chung F.; Liu, Jianhong; Zhang, Peixin
2007-07-01
We have successfully coupled the Kohn-Sham with Poisson-Boltzmann equations to predict the solvation free energy, where the Kohn-Sham equations were solved by implementing the flexible pseudo atomic orbitals as in S IESTA package. It was found that the calculated solvation free energy is in good agreement with experimental results for small neutral molecules, and its standard error is 1.33 kcal/mol, the correlation coefficient is 0.97. Due to its high efficiency and accuracy, the proposed model can be a promising tool for computing solvation free energies in computer aided drug design in future.
A lattice-Boltzmann scheme of the Navier-Stokes equations on a 3D cuboid lattice
NASA Astrophysics Data System (ADS)
Min, Haoda; Peng, Cheng; Wang, Lian-Ping
2015-11-01
The standard lattice-Boltzmann method (LBM) for fluid flow simulation is based on a square (in 2D) or cubic (in 3D) lattice grids. Recently, two new lattice Boltzmann schemes have been developed on a 2D rectangular grid using the MRT (multiple-relaxation-time) collision model, by adding a free parameter in the definition of moments or by extending the equilibrium moments. Here we developed a lattice Boltzmann model on 3D cuboid lattice, namely, a lattice grid with different grid lengths in different spatial directions. We designed our MRT-LBM model by matching the moment equations from the Chapman-Enskog expansion with the Navier-Stokes equations. The model guarantees correct hydrodynamics. A second-order term is added to the equilibrium moments in order to restore the isotropy of viscosity on a cuboid lattice. The form and the coefficients of the extended equilibrium moments are determined through an inverse design process. An additional benefit of the model is that the viscosity can be adjusted independent of the stress-moment relaxation parameter, thus improving the numerical stability of the model. The resulting cuboid MRT-LBM model is then validated through benchmark simulations using laminar channel flow, turbulent channel flow, and the 3D Taylor-Green vortex flow.
NASA Astrophysics Data System (ADS)
Ausloos, M.
2000-09-01
Recent observations have indicated that the traditional equilibrium market hypothesis (EMH; also known as Efficient Market Hypothesis) is unrealistic. It is shown here that it is the analog of a Boltzmann equation in physics, thus having some bad properties of mean-field approximations like a Gaussian distribution of price fluctuations. A kinetic theory for prices can be simply derived, considering in a first approach that market actors have all identical relaxation times, and solved within a Chapman-Enskog like formalism. In closing the set of equations, (i) an equation of state with a pressure and (ii) the equilibrium (isothermal) equation for the price (taken as the order parameter) of a stock as a function of the volume of money available are obtained.
Conjugate heat and mass transfer in the lattice Boltzmann equation method
Li, LK; Chen, C; Mei, RW; Klausner, JF
2014-04-22
An interface treatment for conjugate heat and mass transfer in the lattice Boltzmann equation method is proposed based on our previously proposed second-order accurate Dirichlet and Neumann boundary schemes. The continuity of temperature (concentration) and its flux at the interface for heat (mass) transfer is intrinsically satisfied without iterative computations, and the interfacial temperature (concentration) and their fluxes are conveniently obtained from the microscopic distribution functions without finite-difference calculations. The present treatment takes into account the local geometry of the interface so that it can be directly applied to curved interface problems such as conjugate heat and mass transfer in porous media. For straight interfaces or curved interfaces with no tangential gradient, the coupling between the interfacial fluxes along the discrete lattice velocity directions is eliminated and thus the proposed interface schemes can be greatly simplified. Several numerical tests are conducted to verify the applicability and accuracy of the proposed conjugate interface treatment, including (i) steady convection-diffusion in a channel containing two different fluids, (ii) unsteady convection-diffusion in the channel, (iii) steady heat conduction inside a circular domain with two different solid materials, and (iv) unsteady mass transfer from a spherical droplet in an extensional creeping flow. The accuracy and order of convergence of the simulated interior temperature (concentration) field, the interfacial temperature (concentration), and heat (mass) flux are examined in detail and compared with those obtained from the "half-lattice division" treatment in the literature. The present analysis and numerical results show that the half-lattice division scheme is second-order accurate only when the interface is fixed at the center of the lattice links, while the present treatment preserves second-order accuracy for arbitrary link fractions. For curved
NASA Astrophysics Data System (ADS)
Rukolaine, Sergey A.
2016-05-01
In classical kinetic models a particle free path distribution is exponential, but this is more likely to be an exception than a rule. In this paper we derive a generalized linear Boltzmann equation (GLBE) for a general free path distribution in the framework of Alt's model. In the case that the free path distribution has at least first and second finite moments we construct an asymptotic solution to the initial value problem for the GLBE for small mean free paths. In the special case of the one-speed transport problem the asymptotic solution results in a diffusion approximation to the GLBE.
NASA Astrophysics Data System (ADS)
Li, Zhi-Hui; Peng, Ao-Ping; Zhang, Han-Xin; Yang, Jaw-Yen
2015-04-01
This article reviews rarefied gas flow computations based on nonlinear model Boltzmann equations using deterministic high-order gas-kinetic unified algorithms (GKUA) in phase space. The nonlinear Boltzmann model equations considered include the BGK model, the Shakhov model, the Ellipsoidal Statistical model and the Morse model. Several high-order gas-kinetic unified algorithms, which combine the discrete velocity ordinate method in velocity space and the compact high-order finite-difference schemes in physical space, are developed. The parallel strategies implemented with the accompanying algorithms are of equal importance. Accurate computations of rarefied gas flow problems using various kinetic models over wide ranges of Mach numbers 1.2-20 and Knudsen numbers 0.0001-5 are reported. The effects of different high resolution schemes on the flow resolution under the same discrete velocity ordinate method are studied. A conservative discrete velocity ordinate method to ensure the kinetic compatibility condition is also implemented. The present algorithms are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the steady normal shock wave structures for different Mach numbers, the two-dimensional flows past a circular cylinder and a NACA 0012 airfoil to verify the present methodology and to simulate gas transport phenomena covering various flow regimes. Illustrations of large scale parallel computations of three-dimensional hypersonic rarefied flows over the reusable sphere-cone satellite and the re-entry spacecraft using almost the largest computer systems available in China are also reported. The present computed results are compared with the theoretical prediction from gas dynamics, related DSMC results, slip N-S solutions and experimental data, and good agreement can be found. The numerical experience indicates that although the direct model Boltzmann equation solver in phase space can be computationally expensive
NASA Astrophysics Data System (ADS)
de Urquijo, Jaime; Basurto, E.; Juarez, A. M.; Ness, Kevin; Robson, Robert; Brunger, Michael; White, Ron
2014-10-01
The drift velocity of electrons in mixtures of gaseous water with helium and argon are measured over the range of reduced electric fields from 0--300 Td using a pulsed-Townsend technique. Small admixtures of water to both helium and argon are found to produce negative differential conductivity (NDC), despite NDC being absent from the pure gases. Comparison of the measured drift velocities with those calculated from a multi-term solution of Boltzmann's equation provides a further discriminative assessment on the accuracy and completeness of electron water vapour cross-sections. Funding acknowledgements: ARC, Mexican govt (PAPIIT IN 111014).
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.
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
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.
Noronha, Jorge; Denicol, Gabriel S.
2015-12-30
In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime AdS2 Ⓧ S2. We further derive explicit analytic expressions for the momentum dependence of the single-particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density does not matchmore » the equilibrium form. The nonequilibrium contribution to the entropy density is shown to be due to higher-order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Furthermore, in this system the slowly moving hydrodynamic degrees of freedom can exhibit true perfect fluidity while being totally decoupled from the fast moving, nonhydrodynamical microscopic degrees of freedom that lead to entropy production.« less
NASA Astrophysics Data System (ADS)
Ordonez-Miranda, Jose; Yang, Ronggui; Volz, Sebastian; Alvarado-Gil, J. J.
2015-08-01
Based on the phonon Boltzmann transport equation under the relaxation time approximation, analytical expressions for the temperature profiles of both the steady state and modulated heat conduction inside a thin film deposited on a substrate are derived and analyzed. It is shown that these components of the temperature depend strongly on the ratio between the film thickness and the average phonon mean free path (MFP), and they exhibit the diffusive behavior as predicted by the Fourier's law of heat conduction when this ratio is much larger than unity. In contrast, in the ballistic regime when this ratio is comparable to or smaller than unity, the steady-state temperature tends to be independent of position, while the amplitude and the phase of the modulated temperature appear to be lower than those determined by the Fourier's law. Furthermore, we derive an invariant of heat conduction and a simple formula for the cross-plane thermal conductivity of dielectric thin films, which could be a useful guide for understanding and optimizing the thermal performance of the layered systems. This work represents the Boltzmann transport equation-based extension of the Rosencwaig and Gersho work [J. Appl. Phys. 47, 64 (1976)], which is based on the Fourier's law and has widely been used as the theoretical framework for the development of photoacoustic and photothermal techniques. This work might shed some light on developing a theoretical basis for the determination of the phonon MFP and relaxation time using ultrafast laser-based transient heating techniques.
NASA Astrophysics Data System (ADS)
Sun, Hao; Wu, Yi; Rong, Mingzhe; Guo, Anxiang; Han, Guiquan; Lu, Yanhui
2016-03-01
In this paper, the dielectric properties of CO2, CO2/air, CO2/O2, CO2/N2, CO2/CF4, CO2/CH4, CO2/He, CO2/H2, CO2/NH3 and CO2/CO were investigated based on the Boltzmann equation analysis, in which the reduced critical electric field strength (E/N)cr of the gases was derived from the calculated electron energy distribution function (EEDF) by solving the Boltzmann transport equation. In this work, it should be noted that the fundamental data were carefully selected by the published experimental results and calculations to ensure the validity of the calculation. The results indicate that if He, H2, N2 and CH4, in which there are high ionization coefficients or a lack of attachment reactions, are added into CO2, the dielectric properties will decrease. On the other hand, air, O2, NH3 and CF4 (ranked in terms of (E/N)cr value in increasing order) have the potential to improve the dielectric property of CO2 at room temperature. supported in part by the National Key Basic Research Program of China (973 Program) (No. 2015CB251002), the Science and Technology Project Funds of the Grid State Corporation of China (No. SGSNK00KJJS1501564), National Natural Science Foundation of China (Nos. 51221005, 51577145), the Fundamental Research Funds for the Central Universities of China, and the Program for New Century Excellent Talents in University, China
NASA Astrophysics Data System (ADS)
Nassios, Jason; Sader, John
2012-11-01
Kinetic theory provides a rigorous foundation for calculating the dynamics of gas flow at arbitrary degrees of rarefaction. Solutions to the Boltzmann equation require numerical methods in many cases of practical interest. However, the near-continuum regime has been analyzed analytically using asymptotic techniques. These asymptotic analyses often assume steady flow, for which analytical slip models have been derived. Recent developments in nanofabrication have stimulated research into the study of oscillatory flows, drawing into question the applicability of the steady flow assumption. In this talk, I will discuss some key findings of a formal asymptotic analysis of the unsteady linearized Boltzmann-BGK equation, which generalizes existing theory to the unsteady case. The near-continuum limit is considered where the mean free path and oscillation frequency are small. A brief exploration of the implications of this theory for the oscillatory thermal creep problem will be presented, where temperature gradients along adjacent walls generate a flow. The authors acknowledge financial support from an Australian Postgraduate Award and the Australian Research Council Grants Scheme.
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)
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)
Blais, Bruno; Tucny, Jean-Michel; Vidal, David; Bertrand, François
2015-08-01
The volume-averaged Navier-Stokes (VANS) equations are at the basis of numerous models used to investigate flows in porous media or systems containing multiple phases, one of which is made of solid particles. Although they are traditionally solved using the finite volume, finite difference or finite element method, the lattice Boltzmann method is an interesting alternative solver for these equations since it is explicit and highly parallelizable. In this work, we first show that the most common implementation of the VANS equations in the LBM, based on a redefined collision operator, is not valid in the case of spatially varying void fractions. This is illustrated through five test cases designed using the so-called method of manufactured solutions. We then present an LBM scheme for these equations based on a novel collision operator. Using the Chapman-Enskog expansion and the same five test cases, we show that this scheme is second-order accurate, explicit and stable for large void fraction gradients.
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. PMID:21613667
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.
NASA Astrophysics Data System (ADS)
Donko, Zoltan; Dyatko, Nikolay
2016-06-01
The Negative Differential Conductivity and Transient Negative Mobility effects in xenon gas are analyzed by a first-principles particle simulation technique and via an approximate solution of the Boltzmann transport equation (BE). The particle simulation method is devoid of the approximations that are traditionally adopted in the BE solutions in which: (i) the distribution function is searched for in a two-term form; (ii) the Coulomb part of the collision integral for the anisotropic part of the distribution function is neglected; (iii) Coulomb collisions are treated as binary events; and (iv) the range of the electron-electron interaction is limited to a cutoff distance. The results obtained from the two methods are, for both effects, in good qualitative agreement, small differences are attributed to the approximations listed above.
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. PMID:25122406
NASA Astrophysics Data System (ADS)
White, R. D.; Robson, R. E.; Nicoletopoulos, P.; Dujko, S.
2012-05-01
The Franck-Hertz experiment with neon gas is modelled as an idealised steady-state Townsend experiment and analysed theoretically using (a) multi-term solution of Boltzmann equation and (b) Monte-Carlo simulation. Theoretical electron periodic electron structures, together with the `window' of reduced fields in which they occur, are compared with experiment, and it is explained why it is necessary to account for all competing scattering processes in order to explain the observed experimental `wavelength'. The study highlights the fundamental flaws in trying to explain the observations in terms of a single, assumed dominant electronic excitation process, as is the case in text books and the myriad of misleading web sites.
Li, Yunqi; Zhao, Qin; Huang, Qingrong
2014-01-30
A combination of turbidimetric titration, a sigmoidal Boltzmann equation approach and Monte Carlo simulation has been used to study the complex coacervation in serum albumin and pectin mixtures. The effects of the mass ratio of protein to polysaccharide on the critical pH values, the probability of complex coacervation and the electrostatic interaction from charge patches in serum albumin were investigated. Turbidimetric titration results showed an optimum pH for complex coacervation (pHm), which corresponded to the maximum turbidity in the protein/polysaccharide mixture. The pHm monotonically decreased as the ratio decreased, and could be fitted using the sigmoidal Boltzmann equation. It suggests that pHm could be a good ordering parameter to characterize the phase behavior associated with protein/polysaccharide complex coacervation. Qualitative understanding of pHm by taking into account the minimization of electrostatic interaction, as well as quantitative matching of pHm according to the concept of charge neutralization were both achieved. Our results suggest that the serum albumin/pectin complexes were ultimately neutralized by the partial charges originated from the titratable residues in protein and polysaccharide chains at pHm. The Monte Carlo simulation provided consistent phase boundaries for complex coacervation in the same system, and the intermolecular association strength was determined to be several kBT below the given ionic strength. The strongest binding site in the protein is convergent to the largest positive charge patch if pure electrostatic interaction was considered. Further inclusion of contribution from excluded volume resulted in the binding site distribution over five different positive charge patches at different protein/polysaccharide ratios and pH values. PMID:24299810
NASA Astrophysics Data System (ADS)
Vassiliev, Oleg N.; Wareing, Todd A.; McGhee, John; Failla, Gregory; Salehpour, Mohammad R.; Mourtada, Firas
2010-02-01
A new grid-based Boltzmann equation solver, Acuros™, was developed specifically for performing accurate and rapid radiotherapy dose calculations. In this study we benchmarked its performance against Monte Carlo for 6 and 18 MV photon beams in heterogeneous media. Acuros solves the coupled Boltzmann transport equations for neutral and charged particles on a locally adaptive Cartesian grid. The Acuros solver is an optimized rewrite of the general purpose Attila© software, and for comparable accuracy levels, it is roughly an order of magnitude faster than Attila. Comparisons were made between Monte Carlo (EGSnrc) and Acuros for 6 and 18 MV photon beams impinging on a slab phantom comprising tissue, bone and lung materials. To provide an accurate reference solution, Monte Carlo simulations were run to a tight statistical uncertainty (σ ≈ 0.1%) and fine resolution (1-2 mm). Acuros results were output on a 2 mm cubic voxel grid encompassing the entire phantom. Comparisons were also made for a breast treatment plan on an anthropomorphic phantom. For the slab phantom in regions where the dose exceeded 10% of the maximum dose, agreement between Acuros and Monte Carlo was within 2% of the local dose or 1 mm distance to agreement. For the breast case, agreement was within 2% of local dose or 2 mm distance to agreement in 99.9% of voxels where the dose exceeded 10% of the prescription dose. Elsewhere, in low dose regions, agreement for all cases was within 1% of the maximum dose. Since all Acuros calculations required less than 5 min on a dual-core two-processor workstation, it is efficient enough for routine clinical use. Additionally, since Acuros calculation times are only weakly dependent on the number of beams, Acuros may ideally be suited to arc therapies, where current clinical algorithms may incur long calculation times.
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
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→∞.
Ringe, Stefan; Oberhofer, Harald; Hille, Christoph; Matera, Sebastian; Reuter, Karsten
2016-08-01
The size-modified Poisson-Boltzmann (MPB) equation is an efficient implicit solvation model which also captures electrolytic solvent effects. It combines an account of the dielectric solvent response with a mean-field description of solvated finite-sized ions. We present a general solution scheme for the MPB equation based on a fast function-space-oriented Newton method and a Green's function preconditioned iterative linear solver. In contrast to popular multigrid solvers, this approach allows us to fully exploit specialized integration grids and optimized integration schemes. We describe a corresponding numerically efficient implementation for the full-potential density-functional theory (DFT) code FHI-aims. We show that together with an additional Stern layer correction the DFT+MPB approach can describe the mean activity coefficient of a KCl aqueous solution over a wide range of concentrations. The high sensitivity of the calculated activity coefficient on the employed ionic parameters thereby suggests to use extensively tabulated experimental activity coefficients of salt solutions for a systematic parametrization protocol. PMID:27323006
Goffin, Mark A.; Baker, Christopher M.J.; Buchan, Andrew G.; Pain, Christopher C.; Eaton, Matthew D.; Smith, Paul N.
2013-06-01
This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, k{sub eff}, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for k{sub eff} with directional dependence. General error estimators are derived for any given functional of the flux and applied to k{sub eff} to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The k{sub eff} goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.
Shestakov, A I; Milovich, J L; Noy, A
2002-03-01
The nonlinear Poisson-Boltzmann (PB) equation is solved using Newton-Krylov iterations coupled with pseudo-transient continuation. The PB potential is used to compute the electrostatic energy and evaluate the force on a user-specified contour. The PB solver is embedded in a existing, 3D, massively parallel, unstructured-grid, finite element code. Either Dirichlet or mixed boundary conditions are allowed. The latter specifies surface charges, approximates far-field conditions, or linearizes conditions "regulating" the surface charge. Stability and robustness are proved using results for backward Euler differencing of diffusion equations. Potentials and energies of charged spheres and plates are computed and results compared to analysis. An approximation to the potential of the nonlinear, spherical charge is derived by combining two analytic formulae. The potential and force due to a conical probe interacting with a flat plate are computed for two types of boundary conditions: constant potential and constant charge. The second case is compared with direct force measurements by chemical force microscopy. The problem is highly nonlinear-surface potentials of the linear and nonlinear PB equations differ by over an order of magnitude. Comparison of the simulated and experimentally measured forces shows that approximately half of the surface carboxylic acid groups, of density 1/(0.2 nm2), ionize in the electrolyte implying surface charges of 0.4 C/m2, surface potentials of 0.27 V, and a force of 0.6 nN when the probe and plate are 8.7 nm apart. PMID:16290441
NASA Astrophysics Data System (ADS)
Illg, Christian; Haag, Michael; Teeny, Nicolas; Wirth, Jens; Fähnle, Manfred
2016-03-01
Scatterings of electrons at quasiparticles or photons are very important for many topics in solid-state physics, e.g., spintronics, magnonics or photonics, and therefore a correct numerical treatment of these scatterings is very important. For a quantum-mechanical description of these scatterings, Fermi's golden rule is used to calculate the transition rate from an initial state to a final state in a first-order time-dependent perturbation theory. One can calculate the total transition rate from all initial states to all final states with Boltzmann rate equations involving Brillouin zone integrations. The numerical treatment of these integrations on a finite grid is often done via a replacement of the Dirac delta distribution by a Gaussian. The Dirac delta distribution appears in Fermi's golden rule where it describes the energy conservation among the interacting particles. Since the Dirac delta distribution is a not a function it is not clear from a mathematical point of view that this procedure is justified. We show with physical and mathematical arguments that this numerical procedure is in general correct, and we comment on critical points.
NASA Astrophysics Data System (ADS)
Paussa, A.; Esseni, D.
2013-03-01
This paper revisits the problem of the linearized Boltzmann transport equation (BTE), or, equivalently, of the momentum relaxation time, momentum relaxation time (MRT), for the calculation of low field mobility, which in previous works has been almost universally solved in approximated forms. We propose an energy driven discretization method that allows an exact determination of the relaxation time by solving a linear, algebraic problem, where multiple scattering mechanisms are naturally accounted for by adding the corresponding scattering rates before the calculation of the MRT, and without resorting to the semi-empirical Matthiessen's rule for the relaxation times. The application of our rigorous solution of the linearized BTE to a graphene bilayer reveals that, for a non monotonic energy relation, the relaxation time can legitimately take negative values with no unphysical implications. We finally compare the mobility calculations provided by an exact solution of the MRT problem with the results obtained with some of the approximations most frequently employed in the literature and so discuss their accuracy.
NASA Astrophysics Data System (ADS)
Cekmen, Z. C.; Dincer, M. S.
2009-07-01
The effective ionization coefficients and transport parameters such as electron mean energy drift velocity and transverse diffusion coefficient in binary and ultradilute SF6-Ar gas mixtures have been calculated for density reduced electric field strength E/N values from 10 to 400 Td. These calculations have been performed by using the two-term spherical harmonic expansion to obtain the numerical solution of the Boltzmann transport equation based on the finite element method under steady-state Townsend condition. In order to confirm the model and code developed in this study, the Reid ramp model has been used as a benchmark test and then effective ionization coefficients and transport parameters have been evaluated for SF6 contents of 1%, 10%, 25%, 50%, 70% and 100% in the binary mixture. Finally SF6 contents in the ultradilute mixtures of 0.1%, 0.3%, 0.5% and 0.7% are taken into account with the evaluated effective ionizations and transport parameters of electron mean energy, drift velocity and transverse diffusion coefficients.
NASA Astrophysics Data System (ADS)
Hua, Chengyun; Minnich, Austin J.
2015-05-01
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.
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.
Collisional Behaviors of Astrophysical Collisionless Plasmas
NASA Astrophysics Data System (ADS)
Bret, A.
2015-12-01
In collisional fluids, a number of key processes rely on the frequency of binary collisions. Collisions seem necessary to generate a shock wave when two fluids collide fast enough, to fulfill the Rankine-Hugoniot (RH) relations, to establish an equation of state or a Maxwellian distribution. Yet, these seemingly collisional features are routinely either observed or assumed, in relation with collisionless astrophysical plasmas. This article will review our current answers to the following questions: How do colliding collisionless plasmas end-up generating a shock as if they were fluids? To which extent are the RH relations fulfilled in this case? Do collisionless shocks propagate like fluid ones? Can we use an equation of state to describe collisionless plasmas, like MHD codes for astrophysics do? Why are Maxwellian distributions ubiquitous in particle-in-cell simulations of collisionless shocks? Time and length scales defining the border between the collisional and the collisionless behavior will be given when relevant. In general, when the time and length scales involved in the collisionless processes responsible for the fluid-like behavior may be neglected, the system may be treated like a fluid.
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.
NASA Astrophysics Data System (ADS)
Kópházi, József; Lathouwers, Danny
2015-09-01
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.
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.
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
Mikell, Justin K.; Klopp, Ann H.; Gonzalez, Graciela M. N.; Kisling, Kelly D.; Price, Michael J.; Berner, Paula A.; Eifel, Patricia J.; Mourtada, and Firas
2014-01-01
Purpose To investigate the dosimetric impact of the heterogeneity dose calculation Acuros, 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 192Ir intracavitary brachytherapy for cervical cancer with computed tomography (CT)/magnetic resonance (MR)-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 1g/cc muscle, and with and without overriding contrast materials to muscle or 2.25 g/cc 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. TG-43 and the GBBS were compared for the following clinical dosimetric parameters: Manchester points A and B, ICRU report #38 rectal and bladder points, three and nine o'clock, and D2cc to the bladder, rectum, and sigmoid. Results Points A, B, D2cc 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. The D2cc rectum (n=3), D2cc sigmoid (n=1), and ICRU rectum (n=6) had differences > 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 mapped to bone and radiopaque packing was not overridden. Conclusions The GBBS has minimal impact on clinical parameters for this cohort of GYN patients with unshielded applicators. The incorrect mapping of rectal and balloon contrast does not have a significant impact on clinical parameters. Rectal parameters may be
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
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.
Perfect fluidity of a dissipative system: Analytical solution for the Boltzmann equation in AdS2⊗S2
NASA Astrophysics Data System (ADS)
Noronha, Jorge; Denicol, Gabriel S.
2015-12-01
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 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. Thus, in this system the slowly moving hydrodynamic degrees of freedom can exhibit true perfect fluidity while being totally decoupled from the fast moving, nonhydrodynamical microscopic degrees of freedom that lead to entropy production.
NASA Astrophysics Data System (ADS)
Cui, Shuqi; Hong, Ning; Shi, Baochang; Chai, Zhenhua
2016-04-01
In this paper, we will focus on the multiple-relaxation-time (MRT) lattice Boltzmann model for two-dimensional convection-diffusion equations (CDEs), and analyze the discrete effect on the halfway bounce-back (HBB) boundary condition (or sometimes called bounce-back boundary condition) of the MRT model where three different discrete velocity models are considered. We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in the x or y direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a second-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the single-relaxation-time model or the Bhatnagar-Gross-Krook model unless the relaxation parameter related to the diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that the numerical results are consistent with our theoretical analysis. Finally, we would also like to point out the present analysis can be extended to other boundary conditions of lattice Boltzmann models for CDEs.
Cui, Shuqi; Hong, Ning; Shi, Baochang; Chai, Zhenhua
2016-04-01
In this paper, we will focus on the multiple-relaxation-time (MRT) lattice Boltzmann model for two-dimensional convection-diffusion equations (CDEs), and analyze the discrete effect on the halfway bounce-back (HBB) boundary condition (or sometimes called bounce-back boundary condition) of the MRT model where three different discrete velocity models are considered. We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in the x or y direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a second-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the single-relaxation-time model or the Bhatnagar-Gross-Krook model unless the relaxation parameter related to the diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that the numerical results are consistent with our theoretical analysis. Finally, we would also like to point out the present analysis can be extended to other boundary conditions of lattice Boltzmann models for CDEs. PMID:27176432
Numerical Studies of Collisionless Current Layers
NASA Technical Reports Server (NTRS)
Quest, Kevin B.
1993-01-01
The purpose of this proposal was to investigate collisionless current layers using a variety of analytic and numerical tools. The first year of the contract was dedicated to analytical studies, to the porting and adaption of codes being used in this study, and to the numerical simulation of collisionless current layers. The second year entailed the development of multi-dimensional hybrid algorithms as well as the re-examination of the problem of integro-differential equations that occur in the linear stage of plasma instabilities.
NASA Astrophysics Data System (ADS)
Horsten, N.; Dekeyser, W.; Samaey, G.; Baelmans, M.
2016-01-01
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.
NASA Astrophysics Data System (ADS)
Maginot, Peter G.; Morel, Jim E.; Ragusa, Jean C.
2012-08-01
We present a new nonlinear spatial finite-element method for the linearized Boltzmann transport equation with Sn angular discretization in 1-D and 2-D Cartesian geometries. This method has two central characteristics. First, it is equivalent to the linear-discontinuous (LD) Galerkin method whenever that method yields a strictly non-negative solution. Second, it always satisfies both the zeroth and first spatial moment equations. Because it yields the LD solution when that solution is non-negative, one might interpret our method as a classical fix-up to the LD scheme. However, fix-up schemes for the LD equations derived in the past have given up solution of the first moment equations when the LD solution is negative in order to satisfy positivity in a simple manner. We present computational results comparing our method in 1-D to the strictly non-negative linear exponential-discontinuous method and to the LD method. We present computational results in 2-D comparing our method to a recently developed LD fix-up scheme and to the LD scheme. It is demonstrated that our method is a valuable alternative to existing methods.
Lee, T.; Leaf, G. K.; Mathematics and Computer Science; The City Univ. of New York
2009-04-01
We propose an Eulerian description of the bounce-back boundary condition based on the high-order implicit time-marching schemes to improve the accuracy of lattice Boltzmann simulation in the vicinity of curved boundary. The Eulerian description requires only one grid spacing between fluid nodes when second-order accuracy in time and space is desired, although high-order accurate boundary conditions can be constructed on more grid-point support. The Eulerian description also provides an analytical framework for several different interpolation-based boundary conditions. For instance, the semi-Lagrangian, linear interpolation boundary condition is found to be a first-order upwind discretization that changes the time-marching schemes from implicit to explicit as the distance between the fluid boundary node and the solid boundary increases.
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.
Dielectric and permeability effects in collisionless plasmas. [in collisionless plasmas
NASA Technical Reports Server (NTRS)
Cole, K. D.
1984-01-01
Using the unabridged Maxwell equations (including vectors D, E and H) new effects in collisionless plasmas are uncovered. In a steady state, it is found that spatially varying energy density of the electric field (E perpendicular) orthogonal to B produces electric current leading, under certain conditions, to the relationship P perpendicular + B(2)/8 pi-epsilon E perpendicular(2)/8 pi = constant, where epsilon is the dielectric constant of the plasma for fields orthogonal to B. In steady state quasi-two-dimensional flows in plasmas, a general relationship between the components of electric field parallel and perpendicular to B is found. These effects are significant in geophysical and astrophysical plasmas. The general conditions for a steady state in collisionless plasma are deduced. With time variations in a plasma, slow compared to ion-gyroperiod, there is a general current, (j-asterisk), which includes the well-known polarization current, given by J-asterisk = d/dt (E x M) + (P x B) x B B(-2) where M and P are the magnetization and polarization vectors respectively.
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.
Stellar Dynamics around a Massive Black Hole I: Secular Collisionless Theory
NASA Astrophysics Data System (ADS)
Sridhar, S.; Touma, Jihad R.
2016-03-01
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 semi-major 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.
NASA Astrophysics Data System (ADS)
Sohier, Thibault; Calandra, Matteo; Park, Cheol-Hwan; Bonini, Nicola; Marzari, Nicola; Mauri, Francesco
2014-09-01
We use first-principles calculations, at the density-functional-theory (DFT) and GW levels, to study both the electron-phonon interaction for acoustic phonons and the "synthetic" vector potential induced by a strain deformation (responsible for an effective magnetic field in case of a nonuniform strain). In particular, the interactions between electrons and acoustic phonon modes, the so-called gauge-field and deformation potential, are calculated at the DFT level in the framework of linear response. The zero-momentum limit of acoustic phonons is interpreted as a strain of the crystal unit cell, allowing the calculation of the acoustic gauge-field parameter (synthetic vector potential) within the GW approximation as well. We find that using an accurate model for the polarizations of the acoustic phonon modes is crucial to obtain correct numerical results. Similarly, in the presence of a strain deformation, the relaxation of atomic internal coordinates cannot be neglected. The role of electronic screening on the electron-phonon matrix elements is carefully investigated. We then solve the Boltzmann equation semianalytically in graphene, including both acoustic and optical phonon scattering. We show that, in the Bloch-Grüneisen and equipartition regimes, the electronic transport is mainly ruled by the unscreened acoustic gauge field, while the contribution due to the deformation potential is negligible and strongly screened. We show that the contribution of acoustic phonons to resistivity is doping and substrate independent, in agreement with experimental observations. The first-principles calculations, even at the GW level, underestimate this contribution to resistivity by ≈30%. At high temperature (T >270 K), the calculated resistivity underestimates the experimental one more severely, the underestimation being larger at lower doping. We show that, besides remote phonon scattering, a possible explanation for this disagreement is the electron-electron interaction
Lattice Boltzmann morphodynamic model
NASA Astrophysics Data System (ADS)
Zhou, Jian Guo
2014-08-01
Morphological change due to sediment transport is a common natural phenomenon in real flows. It involves complex processes of erosion and deposition such as those along beaches and in river beds, imposing a strong strain on human beings. Studying and understanding morphodynamic evolution are essential to protect living environment. Although there are conventional numerical methods like finite difference method and finite volume method for forecast of morphological change by solving flow and morphodynamic equations, the methods are too complex/inefficient to be applied to a real large scale problem. To overcome this, a lattice Boltzmann method is developed to simulate morphological evolution under flows. It provides an alternative way of studying morphodynamics at the full advantages of the lattice Boltzmann methodology. The model is verified by applications to the evolution of one and two dimensional sand dunes under shallow water flows.
NASA Astrophysics Data System (ADS)
Deng, Yun-Kun; Xiao, Deng-Ming
2013-03-01
The electron swarm parameters including the density-normalized effective ionization coefficients (α-η)/N and the electron drift velocities Ve are calculated for a gas mixture of CF3I with N2 and CO2 by solving the Boltzmann equation in the condition of a steady-state Townsend (SST) experiment. The overall density-reduced electric field strength is from 100 Td to 1000 Td (1 Td = 10-17 V·cm2), while the CF3I content k in the gas mixture can be varied over the range from 0% to 100%. From the variation of (α-η)/N with the CF3I mixture ratio k, the limiting field strength (E/N)lim for each CF3I concentration is derived. It is found that for the mixtures with 70% CF3I, the values of (E/N)lim are essentially the same as that for pure SF6. Additionally, the global warming potential (GWP) and the liquefaction temperature of the gas mixtures are also taken into account to evaluate the possibility of application in the gas insulation of power equipment.
NASA Astrophysics Data System (ADS)
Deng, Yunkun; Xiao, Dengming
2014-09-01
The present study is devoted to the calculation of electron swarm parameters, including the reduced effective ionization coefficient, electron mean energy, and electron drift velocity, for the gas mixtures of CF3I with Ar, Xe, He, N2, and CO2. These data are computed by employing the Boltzmann equation method with two-term approximation in the condition of steady-state Townsend (SST) discharge. For the purpose of evaluating the insulation strength of CF3I gas mixtures, values of the limiting field strength (E/N)lim for which the ionization exactly balances the electron attachment are determined from the variation curves of (α - η)/N. The results indicate that mixtures of CF3I-N2 present the greatest insulation strength among all the combinations for CF3I content varied from 20 to 90%. Furthermore, the gas mixture with 70% CF3I can achieve a very similar dielectric strength to that of SF6. The concerned liquefaction issues are also taken into account to fully assess the possibility of applying CF3I gas mixtures in power equipment as an insulation medium.
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
Talon, Laurent; Bauer, Daniela
2013-12-01
Simulating flow of a Bingham fluid in porous media still remains a challenging task as the yield stress may significantly alter the numerical stability and precision. We present a Lattice-Boltzmann TRT scheme that allows the resolution of this type of flow in stochastically reconstructed porous media. LB methods have an intrinsic error associated to the boundary conditions. Depending on the schemes this error might be directly linked to the effective viscosity. As for non-Newtonian fluids viscosity varies in space the error becomes inhomogeneous and very important. In contrast to that, the TRT scheme does not present this deficiency and is therefore adequate to be used for simulations of non-Newtonian fluid flow. We simulated Bingham fluid flow in porous media and determined a generalized Darcy equation depending on the yield stress, the effective viscosity, the pressure drop and a characteristic length of the porous medium. By evaluating the flow in the porous structure, we distinguished three different scaling regimes. Regime I corresponds to the situation where fluid is flowing in only one channel. Here, the relation between flow rate and pressure drop is given by the non-Newtonian Poiseuille law. During Regime II an increase in pressure triggers the opening of new paths and the relation between flow rate and the difference in pressure to the critical yield pressure becomes quadratic: [Formula: see text]. Finally, Regime III corresponds to the situation where all the fluid is flowing. In this case, [Formula: see text]. PMID:24326905
NASA Astrophysics Data System (ADS)
Li, Xingwen; Zhao, Hu; Jia, Shenli; Murphy, Anthony B.
2014-10-01
The dielectric strength of c-C4F8, and mixtures of c-C4F8 with CF4, CO2, N2, O2 and air, is studied through solution of the Boltzmann equation. The reduced ionization coefficient α/N and reduced attachment coefficient η/N are calculated, allowing the reduced effective ionization coefficient (α-η)/N and the critical reduced electric field strength (E/N)cr (the reduced electric field for which (α-η)/N = 0), to be determined. A high value of (E/N)cr for an electronegative gas, such as those considered here, indicates good insulating properties. It is found that c-C4F8-N2 and c-C4F8-air have very similar (E/N)cr values, higher than those of the other three mixtures, and superior even to that of pure SF6 for c-C4F8 concentrations above 80%. Comparison of the results obtained for c-C4F8 and c-C4F8-N2 with experimental values from the literature supports the validity of the approach taken here and the parameters used.
NASA Astrophysics Data System (ADS)
Deng, Y. K.; Xiao, D. M.
2012-02-01
The present paper concerns itself with the insulation characteristics of c-C4F8/N2 gas mixtures and studies the possibility of applying in the gas insulation of power equipments. We aim to use the theoretical framework of the Boltzmann equation to calculate the density-normalized effective ionization coefficients (α-ƞ)/N and transport parameters of c-C4F8/N2 gas mixtures for E/N values from 180 to 550 Td (1 Td = 10-17 V cm2) in the condition of steady-state Townsend (SST) experiment. From the variation curve of (α-ƞ)/N with the c-C4F8 mixture ratio k, the limiting field strength (E/N)lim of the gas mixtures at different gas content is determined. In order to confirm the validity of the results obtained, comparisons with Monte Carlo simulation and experimental data have been performed. It is found that the insulation properties of c-C4F8 and N2 gas mixtures are much better than those of SF6 and N2 mixtures for applying in the high voltage apparatus as an insulation medium, especially if we take the global warming potential into account.
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
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.
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.
NASA Astrophysics Data System (ADS)
Bailey, Wm. F.; Smithtro, C. G.
1999-10-01
A two-dimensional model of the electron kinetics within a glow discharge positive column has been developed, based on the formalism of Uhrlandt and Winkler(Uhrlandt, D. and Winkler, R. "Radially Inhomogeneous Electron Kinetics in the DC Column Plasma," J. Phys. D: App. Phys., 29:155-120 (1996).). The model establishes a steady state solution, such that the net ionization rate is exactly balanced by the wall loss. After summarizing the analytic development, we present the numerical techniques used to solve the resulting elliptic partial differential equation, discussing an efficient method to treat sparse banded matrices. The model is first validated against published results in rare gases, examined in the limits of the local and nonlocal kinetic approximations and also compared to a previous Monte Carlo treatment. Current flow within the solution area of a neon column is examined. The model is then extended to consider the influence of the addition of a molecular gas, nitrogen, to the rare gas, neon. Current flow and ranges of applicability of the local and nonlocal approximations in the mixture are contrasted with the pure neon results.
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.
Collisionless tearing instability in magnetotail plasmas
NASA Technical Reports Server (NTRS)
Wang, Xiaogang; Bhattacharjee, A.; Lui, A. T. Y.
1990-01-01
The problem of the linear stability of collisionless tearing modes in the earth's magnetotail is revisited. It is found that the collisionless tearing mode is linearly unstable with wavelengths of the order of 10 R(E). It is shown that an important feature neglected in earlier theories is a nonzero equilibrium B(y)-field. The physics of the instability is elucidated in the context of a simple slab model and a sheared parabolic model which is representative of the magnetotail in which all three components of the magnetic field are nonzero. The dispersion equation for the instability is obtained by a boundary-layer analysis. The implications of the theory for recent observations on current disruption and diversion during substorms is discussed.
Expansion techniques for collisionless stellar dynamical simulations
NASA Astrophysics Data System (ADS)
Meiron, Yohai
2016-02-01
We present ETICS, a collisionless N-body code based on two kinds of series expansions of the Poisson equation, implemented for graphics processing units (GPUs). The code is publicly available and can be used as a standalone program or as a library (an AMUSE plugin is included). One of the two expansion methods available is the self-consistent field (SCF) method, which is a Fourier-like expansion of the density field in some basis set; the other is the multipole expansion (MEX) method, which is a Taylor-like expansion of the Green's function. MEX, which has been advocated in the past, has not gained as much popularity as SCF. Both are particle-field methods and optimized for collisionless galactic dynamics, but while SCF is a ``pure'' expansion, MEX is an expansion in just the angular part; thus, MEX is capable of capturing radial structure easily, while SCF needs a large number of radial terms.
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.
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
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.
Collisionless reconnection in Jupiter's magnetotail
NASA Astrophysics Data System (ADS)
Zimbardo, G.
1991-04-01
Collisionless reconnection in Jupiter's magnetotail is quantitatively studied for the first time. It is proposed that the same tearing mechanism which works in the earth magnetotail also works in Jupiter's. It is shown that collisionless reconnection may occur around 60 R(J) downtail.
Nonlinear Collisionless Magnetic Reconnection
Grasso, D.; Tassi, E.; Borgogno, D.; Pegoraro, F.
2008-10-15
We review some recent results that have been obtained in the investigation of collisionless reconnection in two and three dimensional magnetic configurations with a strong guide field in regimes of interest for laboratory plasmas. First, we adopt a two-field plasma model where two distinct regimes, laminar and turbulent, can be identified. Then, we show that these regimes may combine when we consider a more general four-field model, where perturbation of the magnetic and velocity fields are allowed also along the ignorable coordinate.
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. PMID:26565365
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.''
Accurate deterministic solutions for the classic Boltzmann shock profile
NASA Astrophysics Data System (ADS)
Yue, Yubei
The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.
Generalized Boltzmann formalism for oscillating neutrinos
Strack, P.; Burrows, A.
2005-05-01
In the standard approaches to neutrino transport in the simulation of core-collapse supernovas, one will often start from the classical Boltzmann equation for the neutrino's spatial, temporal, and spectral evolution. For each neutrino species, and its antiparticle, the classical density in phase space, or the associated specific intensity, will be calculated as a function of time. The neutrino radiation is coupled to matter by source and sink terms on the 'right-hand side' of the transport equation and together with the equations of hydrodynamics this set of coupled partial differential equations for classical densities describes, in principle, the evolution of core collapse and explosion. However, with the possibility of neutrino oscillations between species, a purely quantum-physical effect, how to generalize this set of Boltzmann equations for classical quantities to reflect oscillation physics has not been clear. To date, the formalisms developed have retained the character of quantum operator physics involving complex quantities and have not been suitable for easy incorporation into standard supernova codes. In this paper, we derive generalized Boltzmann equations for quasiclassical, real-valued phase-space densities that retain all the standard oscillation phenomenology, including the matter-enhanced resonant flavor conversion (Mikheev-Smirnov-Wolfenstein effect), neutrino self-interactions, and the interplay between decohering matter coupling and flavor oscillations. With this formalism, any code(s) that can now handle the solution of the classical Boltzmann or transport equation can easily be generalized to include neutrino oscillations in a quantum-physically consistent fashion.
Lattice gas and lattice Boltzmann computational physics
Chen, S.
1993-05-01
Recent developments of the lattice gas automata method and its extension to the lattice Boltzmann method have provided new computational schemes for solving a variety of partial differential equations and modeling different physics systems. The lattice gas method, regarded as the simplest microscopic and kinetic approach which generates meaningful macroscopic dynamics, is fully parallel and can be easily programmed on parallel machines. In this talk, the author will review basic principles of the lattice gas and lattice Boltzmann method, its mathematical foundation and its numerical implementation. A detailed comparison of the lattice Boltzmann method with the lattice gas technique and other traditional numerical schemes, including the finite-difference scheme and the pseudo-spectral method, for solving the Navier-Stokes hydrodynamic fluid flows, will be discussed. Recent achievements of the lattice gas and the the lattice Boltzmann method and their applications in surface phenomena, spinodal decomposition and pattern formation in chemical reaction-diffusion systems will be 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. PMID:23558425
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.
Multiple-Relaxation-Time Lattice Boltzmann Models in 3D
NASA Technical Reports Server (NTRS)
dHumieres, Dominique; Ginzburg, Irina; Krafczyk, Manfred; Lallemand, Pierre; Luo, Li-Shi; Bushnell, Dennis M. (Technical Monitor)
2002-01-01
This article provides a concise exposition of the multiple-relaxation-time lattice Boltzmann equation, with examples of fifteen-velocity and nineteen-velocity models in three dimensions. Simulation of a diagonally lid-driven cavity flow in three dimensions at Re=500 and 2000 is performed. The results clearly demonstrate the superior numerical stability of the multiple-relaxation-time lattice Boltzmann equation over the popular lattice Bhatnagar-Gross-Krook equation.
Boltzmann's "H"-Theorem and the Assumption of Molecular Chaos
ERIC Educational Resources Information Center
Boozer, A. D.
2011-01-01
We describe a simple dynamical model of a one-dimensional ideal gas and use computer simulations of the model to illustrate two fundamental results of kinetic theory: the Boltzmann transport equation and the Boltzmann "H"-theorem. Although the model is time-reversal invariant, both results predict that the behaviour of the gas is time-asymmetric.…
NASA Astrophysics Data System (ADS)
Chatterjee, Kausik; Roadcap, John R.; Singh, Surendra
2014-11-01
The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson-Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.
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.
Boltzmann-Langevin theory of Coulomb drag
NASA Astrophysics Data System (ADS)
Chen, W.; Andreev, A. V.; Levchenko, A.
2015-06-01
We develop a Boltzmann-Langevin description of the Coulomb drag effect in clean double-layer systems with large interlayer separation d as compared to the average interelectron distance λF. Coulomb drag arises from density fluctuations with spatial scales of order d . At low temperatures, their characteristic frequencies exceed the intralayer equilibration rate of the electron liquid, and Coulomb drag may be treated in the collisionless approximation. As temperature is raised, the electron mean free path becomes short due to electron-electron scattering. This leads to local equilibration of electron liquid, and consequently drag is determined by hydrodynamic density modes. Our theory applies to both the collisionless and the hydrodynamic regimes, and it enables us to describe the crossover between them. We find that drag resistivity exhibits a nonmonotonic temperature dependence with multiple crossovers at distinct energy scales. At the lowest temperatures, Coulomb drag is dominated by the particle-hole continuum, whereas at higher temperatures of the collision-dominated regime it is governed by the plasmon modes. We observe that fast intralayer equilibration mediated by electron-electron collisions ultimately renders a stronger drag effect.
Crystallographic Lattice Boltzmann Method.
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-01-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows. PMID:27251098
Crystallographic Lattice Boltzmann Method
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-01-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows. PMID:27251098
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.
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. PMID:19656523
Collisionless relaxation in beam-plasma systems
Backhaus, Ekaterina Yu.
2001-05-01
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
Lattice Boltzmann model for simulation of magnetohydrodynamics
NASA Technical Reports Server (NTRS)
Chen, Shiyi; Chen, Hudong; Martinez, Daniel; Matthaeus, William
1991-01-01
A numerical method, based on a discrete Boltzmann equation, is presented for solving the equations of magnetohydrodynamics (MHD). The algorithm provides advantages similar to the cellular automaton method in that it is local and easily adapted to parallel computing environments. Because of much lower noise levels and less stringent requirements on lattice size, the method appears to be more competitive with traditional solution methods. Examples show that the model accurately reproduces both linear and nonlinear MHD phenomena.
Boltzmann-Electron Model in Aleph.
Hughes, Thomas Patrick; Hooper, Russell
2014-11-01
We apply the Boltzmann-electron model in the electrostatic, particle-in-cell, finite- element code Aleph to a plasma sheath. By assuming a Boltzmann energy distribution for the electrons, the model eliminates the need to resolve the electron plasma fre- quency, and avoids the numerical "grid instability" that can cause unphysical heating of electrons. This allows much larger timesteps to be used than with kinetic electrons. Ions are treated with the standard PIC algorithm. The Boltzmann-electron model re- quires solution of a nonlinear Poisson equation, for which we use an iterative Newton solver (NOX) from the Trilinos Project. Results for the spatial variation of density and voltage in the plasma sheath agree well with an analytic model
MHD results from a collisionless fluid model
NASA Astrophysics Data System (ADS)
Ramos, J. J.
2002-11-01
A non-conventional closure ansatz for collisionless MHD has been proposed in Ref.[1]. The truncation of the set of fluid moment equations is suggested by a comparison between the standard non-relativistic set and the non-relativistic limit of the relativistic set derived in Ref.[2]. The resulting model is a closed system of evolution equations in conservation form for the particle, momentum and energy densities, and the energy flux, allowing for pressure anisotropy and parallel heat flux. The static equilibrium condition is the same as in the Chew-Goldberger-Low theory, supplemented by the condition that the parallel energy flux be constant along the magnetic field. We study the linear perturbations about such static equilibria to derive the MHD wave dispersion relations in a homogeneous background and the perturbed potential energy associated with a stability energy principle. [1] J.J. Ramos, 2002 International Sherwood Theory Meeting, Rochester, NY, paper 1D25. [2] R.D. Hazeltine and S.M. Mahajan, Ap. J. 567, 1262 (2002).
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. PMID:19792280
Sugimoto, Yu; Kitazumi, Yuki; Shirai, Osamu; Yamamoto, Masahiro; Kano, Kenji
2016-03-31
To understand electrostatic interactions in biomolecules, the bimolecular rate constants (k) between redox enzymes and charged substrates (in this study, redox mediators in the electrode reaction) were evaluated at various ionic strengths (I) for the mediated bioelectrocatalytic reaction. The k value between bilirubin oxidase (BOD) and positively charged mediators increased with I, while that between BOD and negatively charged mediators decreased with I. The opposite trend was observed for the reaction of glucose oxidase (GOD). In the case of noncharged mediators, the k value was independent of I for both BOD and GOD. These results reflect the electrostatic interactions between the enzymes and the mediators. Furthermore, we estimated k/k° (k° being the thermodynamic rate constant) by numerical simulation (finite element method) based on the Poisson-Boltzmann (PB) equation. By considering the charges of individual atoms involved in the amino acids around the substrate binding sites in the enzymes, the simulated k/k° values well reproduced the experimental data. In conclusion, k/k° can be predicted by PB-based simulation as long as the crystal structure of the enzyme and the substrate binding site are known. PMID:26956542
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.
TEMPEST Simulations of Collisionless Damping of Geodesic-Acoustic Mode in Edge Plasma Pedestal
Xu, X Q; Xiong, Z; Nevins, W M; McKee, G R
2007-05-30
The fully nonlinear (full-f) 4D TEMPEST gyrokinetic continuum code produces frequency, collisionless damping of GAM and zonal flow with fully nonlinear Boltzmann electrons for the inverse aspect ratio {epsilon}-scan and the tokamak safety factor q-scan in homogeneous plasmas. The TEMPEST simulation shows that GAM exists in edge plasma pedestal for steep density and temperature gradients, and an initial GAM relaxes to the standard neoclassical residual, rather than Rosenbluth-Hinton residual due to the presence of ion-ion collisions. The enhanced GAM damping explains experimental BES measurements on the edge q scaling of the GAM amplitude.
TEMPEST Simulations of Collisionless Damping of 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 the Geodesic-Acoustic Mode in Edge-Plasma Pedestals
Xu, X. Q.; Xiong, Z.; Nevins, W. M.; Gao, Z.; McKee, G. R.
2008-05-30
The fully nonlinear (full-f) four-dimensional TEMPEST gyrokinetic continuum code correctly produces the frequency and collisionless damping of geodesic-acoustic modes (GAMs) and zonal flow, with fully nonlinear Boltzmann electrons for the inverse aspect ratio {epsilon} scan and the tokamak safety factor q scan in homogeneous plasmas. TEMPEST simulations show that the GAMs exist in the edge pedestal for steep density and temperature gradients in the form of outgoing waves. The enhanced GAM damping may explain experimental beam emission spectroscopy measurements on the edge q scaling of the GAM amplitude.
Tempest Simulations of Collisionless Damping of the Geodesic-Acoustic Mode in Edge-Plasma Pedestals
NASA Astrophysics Data System (ADS)
Xu, X. Q.; Xiong, Z.; Gao, Z.; Nevins, W. M.; McKee, G. R.
2008-05-01
The fully nonlinear (full-f) four-dimensional TEMPEST gyrokinetic continuum code correctly produces the frequency and collisionless damping of geodesic-acoustic modes (GAMs) and zonal flow, with fully nonlinear Boltzmann electrons for the inverse aspect ratio γ scan and the tokamak safety factor q scan in homogeneous plasmas. TEMPEST simulations show that the GAMs exist in the edge pedestal for steep density and temperature gradients in the form of outgoing waves. The enhanced GAM damping may explain experimental beam emission spectroscopy measurements on the edge q scaling of the GAM amplitude.
Lattice Boltzmann 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. PMID:26651812
Lattice Boltzmann Stokesian dynamics
NASA Astrophysics Data System (ADS)
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.
NASA Astrophysics Data System (ADS)
Kulchytskyy, Bohdan; Andriyash, Evgeny; Amin, Mohammed; Melko, Roger
The field of machine learning has been revolutionized by the recent improvements in the training of deep networks. Their architecture is based on a set of stacked layers of simpler modules. One of the most successful building blocks, known as a restricted Boltzmann machine, is an energetic model based on the classical Ising Hamiltonian. In our work, we investigate the benefits of quantum effects on the learning capacity of Boltzmann machines by extending its underlying Hamiltonian with a transverse field. For this purpose, we employ exact and stochastic training procedures on data sets with physical origins.
Navier-Stokes Dynamics by a Discrete Boltzmann Model
NASA Technical Reports Server (NTRS)
Rubinstein, Robet
2010-01-01
This work investigates the possibility of particle-based algorithms for the Navier-Stokes equations and higher order continuum approximations of the Boltzmann equation; such algorithms would generalize the well-known Pullin scheme for the Euler equations. One such method is proposed in the context of a discrete velocity model of the Boltzmann equation. Preliminary results on shock structure are consistent with the expectation that the shock should be much broader than the near discontinuity predicted by the Pullin scheme, yet narrower than the prediction of the Boltzmann equation. We discuss the extension of this essentially deterministic method to a stochastic particle method that, like DSMC, samples the distribution function rather than resolving it completely.
Entropy in the sense of Boltzmann and Poincaré
NASA Astrophysics Data System (ADS)
Vedenyapin, V. V.; Adzhiev, S. Z.
2014-12-01
The H-theorem is proved for generalized equations of chemical kinetics, and important physical examples of such generalizations are considered: a discrete model of the quantum kinetic equations (the Uehling-Uhlenbeck equations) and a quantum Markov process (a quantum random walk). The time means are shown to coincide with the Boltzmann extremals for these equations and for the Liouville equation. Bibliography: 41 titles.
Magnetohydrodynamics for collisionless plasmas from the gyrokinetic perspective
NASA Astrophysics Data System (ADS)
Lee, W. W.
2016-07-01
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)].
Fermion particle production in semiclassical Boltzmann-Vlasov transport theory
Dawson, John F.; Mihaila, Bogdan; Cooper, Fred
2009-07-01
We present numerical solutions of the semiclassical Boltzmann-Vlasov equation for fermion particle-antiparticle production by strong electric fields in boost-invariant coordinates in (1+1) and (3+1) dimensional QED. We compare the Boltzmann-Vlasov results with those of recent quantum field theory calculations and find good agreement. We conclude that extending the Boltzmann-Vlasov approach to the case of QCD should allow us to do a thorough investigation of how backreaction affects recent results on the dependence of the transverse momentum distribution of quarks and antiquarks on a second Casimir invariant of color SU(3)
Nonlinear theory of collisionless trapped ion modes
Hahm, T.S.; Tang, W.M.
1996-03-01
A simplified two field nonlinear model for collisionless trapped-ion-mode turbulence has been derived from nonlinear bounce-averaged drift kinetic equations. The renormalized thermal diffusivity obtained from this analysis exhibits a Bohm-like scaling. A new nonlinearity associated with the neoclassical polarization density is found to introduce an isotope-dependent modification to this Bohm-like diffusivity. The asymptotic balance between the equilibrium variation and the finite banana width induced reduction of the fluctuation potential leads to the result that the radial correlation length decreases with increasing plasma current. Other important conclusions from the present analysis include the predictions that (i) the relative density fluctuation level {delta}n/n{sub 0} is lower than the conventional mixing length estimate, {Delta}r/L{sub n} (ii) the ion temperature fluctuation level {delta}T{sub i}/T{sub i} significantly exceeds the density fluctuation level {delta}n/n{sub 0}; and (iii) the parallel ion velocity fluctuation level {delta}v{sub i}{sub {parallel}}/v{sub Ti} is expected to be negligible.
Dispersion discontinuities of strong collisionless shocks
NASA Technical Reports Server (NTRS)
Coroniti, F. V.
1970-01-01
Linear fluid equations are used to estimate wave train properties of strong collisionless shocks. Fast shocks exhibit several dispersion changes with increasing Mach number. For perpendicular propagation into a finite-beta plasma, an ion cyclotron radius trailing wave train exists only for (M sub F)2 is smaller than 2. Oblique fast shocks have a leading ion inertia wave train if M sub A is smaller than root of M(+)/M(-) cos theta/2 and a trailing electron inertia train if M sub A is greater than root of M(+)/M(-) cos theta/2. If the downstream sound speed exceeds the flow speed, linear wave theory predicts a trailing ion acoustic structure which probably resides within the magnetic shock. For a turbulent shock model in which an effective electron-ion collision frequency exceeds the lower hybrid frequency, ions decouple from the magnetic field; the shock wave train now trails with electron inertia and electron gyroradius lengths. Comparisons of this turbulent model and observations on the earth's bow shock are made.
Turbulent dynamo in a collisionless plasma.
Rincon, François; Califano, Francesco; Schekochihin, Alexander A; Valentini, Francesco
2016-04-12
Magnetic fields pervade the entire universe and affect the formation and evolution of astrophysical systems from cosmological to planetary scales. The generation and dynamical amplification of extragalactic magnetic fields through cosmic times (up to microgauss levels reported in nearby galaxy clusters, near equipartition with kinetic energy of plasma motions, and on scales of at least tens of kiloparsecs) are major puzzles largely unconstrained by observations. A dynamo effect converting kinetic flow energy into magnetic energy is often invoked in that context; however, extragalactic plasmas are weakly collisional (as opposed to magnetohydrodynamic fluids), and whether magnetic field growth and sustainment through an efficient turbulent dynamo instability are possible in such plasmas is not established. Fully kinetic numerical simulations of the Vlasov equation in a 6D-phase space necessary to answer this question have, until recently, remained beyond computational capabilities. Here, we show by means of such simulations that magnetic field amplification by dynamo instability does occur in a stochastically driven, nonrelativistic subsonic flow of initially unmagnetized collisionless plasma. We also find that the dynamo self-accelerates and becomes entangled with kinetic instabilities as magnetization increases. The results suggest that such a plasma dynamo may be realizable in laboratory experiments, support the idea that intracluster medium turbulence may have significantly contributed to the amplification of cluster magnetic fields up to near-equipartition levels on a timescale shorter than the Hubble time, and emphasize the crucial role of multiscale kinetic physics in high-energy astrophysical plasmas. PMID:27035981
Expansion techniques for collisionless stellar dynamical simulations
Meiron, Yohai; Li, Baile; Holley-Bockelmann, Kelly; Spurzem, Rainer
2014-09-10
We present graphics processing unit (GPU) implementations of two fast force calculation methods based on series expansions of the Poisson equation. One method is the self-consistent field (SCF) method, which is a Fourier-like expansion of the density field in some basis set; the other method is the multipole expansion (MEX) method, which is a Taylor-like expansion of the Green's function. MEX, which has been advocated in the past, has not gained as much popularity as SCF. Both are particle-field methods and optimized for collisionless galactic dynamics, but while SCF is a 'pure' expansion, MEX is an expansion in just the angular part; thus, MEX is capable of capturing radial structure easily, while SCF needs a large number of radial terms. We show that despite the expansion bias, these methods are more accurate than direct techniques for the same number of particles. The performance of our GPU code, which we call ETICS, is profiled and compared to a CPU implementation. On the tested GPU hardware, a full force calculation for one million particles took ∼0.1 s (depending on expansion cutoff), making simulations with as many as 10{sup 8} particles fast for a comparatively small number of nodes.
Turbulent dynamo in a collisionless plasma
NASA Astrophysics Data System (ADS)
Rincon, François; Califano, Francesco; Schekochihin, Alexander A.; Valentini, Francesco
2016-04-01
Magnetic fields pervade the entire universe and affect the formation and evolution of astrophysical systems from cosmological to planetary scales. The generation and dynamical amplification of extragalactic magnetic fields through cosmic times (up to microgauss levels reported in nearby galaxy clusters, near equipartition with kinetic energy of plasma motions, and on scales of at least tens of kiloparsecs) are major puzzles largely unconstrained by observations. A dynamo effect converting kinetic flow energy into magnetic energy is often invoked in that context; however, extragalactic plasmas are weakly collisional (as opposed to magnetohydrodynamic fluids), and whether magnetic field growth and sustainment through an efficient turbulent dynamo instability are possible in such plasmas is not established. Fully kinetic numerical simulations of the Vlasov equation in a 6D-phase space necessary to answer this question have, until recently, remained beyond computational capabilities. Here, we show by means of such simulations that magnetic field amplification by dynamo instability does occur in a stochastically driven, nonrelativistic subsonic flow of initially unmagnetized collisionless plasma. We also find that the dynamo self-accelerates and becomes entangled with kinetic instabilities as magnetization increases. The results suggest that such a plasma dynamo may be realizable in laboratory experiments, support the idea that intracluster medium turbulence may have significantly contributed to the amplification of cluster magnetic fields up to near-equipartition levels on a timescale shorter than the Hubble time, and emphasize the crucial role of multiscale kinetic physics in high-energy astrophysical plasmas.
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.
Structural stability of Lattice Boltzmann schemes
NASA Astrophysics Data System (ADS)
David, Claire; Sagaut, Pierre
2016-02-01
The goal of this work is to determine classes of traveling solitary wave solutions for Lattice Boltzmann schemes by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurrence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to Lattice Boltzmann schemes (David and Sagaut 2011; 2009a,b; David et al. 2007).
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.
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.
A Unified Theory of Non-Ideal Gas Lattice Boltzmann Models
NASA Technical Reports Server (NTRS)
Luo, Li-Shi
1998-01-01
A non-ideal gas lattice Boltzmann model is directly derived, in an a priori fashion, from the Enskog equation for dense gases. The model is rigorously obtained by a systematic procedure to discretize the Enskog equation (in the presence of an external force) in both phase space and time. The lattice Boltzmann model derived here is thermodynamically consistent and is free of the defects which exist in previous lattice Boltzmann models for non-ideal gases. The existing lattice Boltzmann models for non-ideal gases are analyzed and compared with the model derived here.
L2-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians
NASA Astrophysics Data System (ADS)
Ha, Seung-Yeal; Xiao, Qinghua; Xiong, Linjie; Zhao, Huijiang
2013-12-01
We present a L2-stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L2-distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and Xiao, Q.-H., "A revisiting to the L2-stability theory of the Boltzmann equation near global Maxwellians," (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., "L2 stability theory of the Boltzmann equation near a global Maxwellian," Arch. Ration. Mech. Anal. 197, 657-688 (2010)] on the L2-stability of the Boltzmann equation to the Boltzmann equation coupled with self-consistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. ["Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space," Commun. Pure Appl. Math. 24, 1497-1546 (2011)] and Guo ["The Vlasov-Maxwell-Boltzmann system near Maxwellians," Invent. Math. 153(3), 593-630 (2003)] satisfy a uniform L2-stability estimate. This is the first result on the L2-stability of the Boltzmann equation coupled with self-consistent field equations in three dimensions.
Collisionless Trapped Electron Mode Turbulence
NASA Astrophysics Data System (ADS)
Lang, Jianying; Chen, Yang; Parker, Scott
2006-10-01
Collisionless Trapped Electron Mode (CTEM) turbulence is a likely canidate for explaining anomolous transport in tokamak discharges that have a strong density gradient relative to the ion temperature gradient. Here, CTEM turbulence is investigated using the Gyrokinetic δf GEM code. GEM is electromagnetic, includes full drift-kinetic electrons, generaly axisymmetric equilbria, collisions and minority species. Here, the flux-tube limit is taken and β is so small that the simulations are essentially electrostatic. Linear theory predicts that the instability occurs at √2ɛRLn>1, which agrees very well with the simulation results. With increasing density gradient, it is observed that the most unstable mode transitions from a CTEM to drift wave mode and the short-wavelength modes are most unstable ( 2 > kρi> 1). Nonlinear simulations are underway to address the parametric dependence of particle and energy transport. The importance of zonal flows for CTEM turbulence, is still not well understood and is under investigation. D. R. Ernst et. al., Phys. Plasma 11 (2004) 2637 T. Dannert and F. Jenko, Phys. Plasma 12 (2005) 072309 R. Gatto et. al., Phys. Plasma 13 (2006) 022306 Y. Chen and S. E. Parker, J. Comput. Phys. 189 (2003) 463 Y. Chen ad S.E. Parker, accepted, to appear in J. Comput. Phys. (2006) J. Wesson (1997) Tokamaks, Oxford Science
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.
Collisional versus Collisionless Dark Matter.
Moore; Gelato; Jenkins; Pearce; Quilis
2000-05-20
We compare the structure and substructure of dark matter halos in model universes dominated by collisional, strongly self-interacting dark matter (SIDM) and collisionless, weakly interacting dark matter (CDM). While SIDM virialized halos are more nearly spherical than CDM halos, they can be rotationally flattened by as much as 20% in their inner regions. Substructure halos suffer ram-pressure truncation and drag, which are more rapid and severe than their gravitational counterparts tidal stripping and dynamical friction. Lensing constraints on the size of galactic halos in clusters are a factor of 2 smaller than predicted by gravitational stripping, and the recent detection of tidal streams of stars escaping from the satellite galaxy Carina suggests that its tidal radius is close to its optical radius of a few hundred parsecs-an order of magnitude smaller than predicted by CDM models but consistent with SIDM models. The orbits of SIDM satellites suffer significant velocity bias, sigmaSIDM&solm0;sigmaCDM=0.85, and are more circular than CDM satellites, betaSIDM approximately 0.5, in agreement with the inferred orbits of the Galaxy's satellites. In the limit of a short mean free path, SIDM halos have singular isothermal density profiles; thus, in its simplest incarnation SIDM, is inconsistent with galactic rotation curves. PMID:10828999
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
On the Solution of a Boltzmann System for Gas Mixtures
NASA Astrophysics Data System (ADS)
Sotirov, Alexander; Yu, Shih-Hsien
2010-02-01
We study the Boltzmann equation for a mixture of two gases in one space dimension with initial condition of one gas near vacuum and the other near a Maxwellian equilibrium state. A qualitative-quantitative mathematical analysis is developed to study this mass diffusion problem based on the Green’s function of the Boltzmann equation for the single species hard sphere collision model in Liu andYu (Commun Pure Appl Math 57:1543-1608, 2004). The cross-species resonance of the mass diffusion and the diffusion-sound wave is investigated. An exponentially sharp global solution is obtained.
Hydromagnetic waves for a collisionless plasma in strong magnetic fields
NASA Astrophysics Data System (ADS)
Duhau, S.; de La Torre, A.
1985-08-01
A system of hydrodynamic equations is used to model the behavior of small-amplitude hydromagnetic waves in order to quantify the effects of the electron thermodynamic variables. The system of equations yields a dispersion relationship which is solved with a linear approximation when small perturbations are introduced into the steady state. The disturbances are expressed as a superposition of small amplitude, plane harmonic waves, which are traced as they propagate through a collisionless heat-conducting plasma. Only the mirror stability criterion is found to change when the electron pressure is considered in a zero heat flux. The phase speed will be symmetric with respect to arising from the presence of the heat flux will strongly couple the slow and fast magnetosonic modes with wavenumber vectors in the positive flux vector directions. The subsequent overstability will be independent of the ion anisotropy.
The microphysics of collisionless shocks
NASA Astrophysics Data System (ADS)
Wilson, Lynn Bruce, III
2010-11-01
Shock waves in interplanetary (IP) space are of considerable interest due to their potential to damage ground based electronic systems and their ability to energize charged particles. The energization of charged particles at IP shocks has the obvious extrapolation to supernova shock waves, which are thought to be a candidate for generating the most energetic particles in the universe. The observations and theory behind collisionless shock wave evolution suggest that IP shocks should, for the most part, be stable structures which require energy dissipation. In a regular fluid, like our atmosphere, energy dissipation is accomplished through binary particle collisions transferring the loss of bulk flow kinetic energy to heat. Plasmas are mostly collisionless fluids, thus requiring other means by which to dissipate energy. The studies herein were performed using wave and particle data primarily from the Wind spacecraft to investigate the microphysics of IP shock energy dissipation mechanisms. Due to their lower Mach numbers, more simplified geometry, and quasi-perpendicular nature, IP shock waves are an excellent laboratory to study wave-particle related dissipation mechanisms. Utilization of multiple data sets from multiple high time resolution instruments on board the Wind spacecraft, we have performed studies on the transition region microphysics of IP shocks. The work began with a statistical study of high frequency (≥ 1 kHz) waveform capture data during 67 IP shocks with Mach numbers ranging from ∼1-6 found ion-acoustic wave amplitudes correlated with the fast mode Mach number and shock strength. The ion-acoustic waves (IAWs) were estimated to produce anomalous resistivities roughly seven orders of magnitude above classical estimates. Another study was an examination of low frequency waves (0.25 Hz < f < 10 Hz) at five quasi-perpendicular IP shocks found the wave modes to be consistent with oblique precusor whistler waves at four of the events. The strongest
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
Gamma-ray bursts and collisionless shocks
NASA Astrophysics Data System (ADS)
Waxman, E.
2006-12-01
Particle acceleration in collisionless shocks is believed to be responsible for the production of cosmic-rays over a wide range of energies, from a few GeV to > 1020 eV, as well as for the non-thermal emission of radiation from a wide variety of high energy astrophysical sources. A theory of collisionless shocks based on first principles does not, however, exist. Observations of γ-ray burst (GRB) 'afterglows' provide a unique opportunity for diagnosing the physics of relativistic collisionless shocks. Most GRBs are believed to be associated with explosions of massive stars. Their 'afterglows', delayed low energy emission following the prompt burst of γ-rays, are well accounted for by a model in which afterglow radiation is due to synchrotron emission of electrons accelerated in relativistic collisionless shock waves driven by the explosion into the surrounding plasma. Within the framework of this model, some striking characteristics of collisionless relativistic shocks are implied. These include the generation of downstream magnetic fields with energy density exceeding that of the upstream field by ~8 orders of magnitude, the survival of this strong field at distances ~1010 skin-depths downstream of the shock and the acceleration of particles to a power-law energy spectrum, d log n/d logɛ ap -2, possibly extending to 1020 eV. I review in this talk the phenomenological considerations, based on which these characteristics are inferred, and the challenges posed to our current models of particle acceleration and magnetic field generation in collisionless shocks. Some recent theoretical results derived based on the assumption of a self-similar shock structure are briefly discussed. Invited review presented at the 33rd annual European Physical Society Conference, Rome, 2006.
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.
Collisionless damping of dust-acoustic waves in a charge varying dusty plasma with nonextensive ions
NASA Astrophysics Data System (ADS)
Amour, Rabia; Tribeche, Mouloud
2014-12-01
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.
Magnetic reconnection in collisionless plasmas - Prescribed fields
NASA Technical Reports Server (NTRS)
Burkhart, G. R.; Drake, J. F.; Chen, J.
1990-01-01
The structure of the dissipation region during magnetic reconnection in collisionless plasma is investigated by examining a prescribed two-dimensional magnetic x line configuration with an imposed inductive electric field E(y). The calculations represent an extension of recent MHD simulations of steady state reconnection (Biskamp, 1986; Lee and Fu, 1986) to the collisionless kinetic regime. It is shown that the structure of the x line reconnection configuration depends on only two parameters: a normalized inductive field and a parameter R which represents the opening angle of the magnetic x lines.
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.
Electrostatic forces in the Poisson-Boltzmann systems
Xiao, Li; Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2013-01-01
Continuum modeling of electrostatic interactions based upon numerical solutions of the Poisson-Boltzmann equation has been widely used in structural and functional analyses of biomolecules. A limitation of the numerical strategies is that it is conceptually difficult to incorporate these types of models into molecular mechanics simulations, mainly because of the issue in assigning atomic forces. In this theoretical study, we first derived the Maxwell stress tensor for molecular systems obeying the full nonlinear Poisson-Boltzmann equation. We further derived formulations of analytical electrostatic forces given the Maxwell stress tensor and discussed the relations of the formulations with those published in the literature. We showed that the formulations derived from the Maxwell stress tensor require a weaker condition for its validity, applicable to nonlinear Poisson-Boltzmann systems with a finite number of singularities such as atomic point charges and the existence of discontinuous dielectric as in the widely used classical piece-wise constant dielectric models. PMID:24028101
The Poisson-Boltzmann model for tRNA
Gruziel, Magdalena; Grochowski, Pawel; Trylska, Joanna
2008-01-01
Using tRNA molecule as an example, we evaluate the applicability of the Poisson-Boltzmann model to highly charged systems such as nucleic acids. Particularly, we describe the effect of explicit crystallographic divalent ions and water molecules, ionic strength of the solvent, and the linear approximation to the Poisson-Boltzmann equation on the electrostatic potential and electrostatic free energy. We calculate and compare typical similarity indices and measures, such as Hodgkin index and root mean square deviation. Finally, we introduce a modification to the nonlinear Poisson-Boltzmann equation, which accounts in a simple way for the finite size of mobile ions, by applying a cutoff in the concentration formula for ionic distribution at regions of high electrostatic potentials. We test the influence of this ionic concentration cutoff on the electrostatic properties of tRNA. PMID:18432617
Stability analysis of lattice Boltzmann methods
Sterling, J.D.; Chen, Shiyi
1996-01-01
The lattice Boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered. Although the equation is a derivative of lattice gas automata, it may be interpreted as a Lagrangian finite-difference method for the numerical simulation of the discrete-velocity Boltzmann equation that makes use of a BGK collision operator. As a result, it is not surprising that numericaI instability of lattice Boltzmann methods have been frequently encountered by researchers. We present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constant-density uniform mean flow. The linear stability depends on the following parameters: the distribution of the mass at a site between the different discrete speeds, the BGK relaxation time, the mean velocity, and the wave-number of the perturbations. This parameter space is too large to compute the complete stability characteristics. We report some stability results for a subset of the parameter space for a 7-velocity hexagonal lattice, a 9-velocity square lattice, and a 15-velocity cubic lattice. Results common to all three lattices are (1) the BGK relaxation time {tau} must be greater than 1/2 corresponding to positive shear viscosity, (2) there exists a maximum stable mean velocity for fixed values of theother parameters, and (3) as {tau} is increased from 1/2 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger {tau}.
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.
Lattice Boltzmann model for numerical relativity.
Ilseven, E; Mendoza, M
2016-02-01
In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems. PMID:26986435
Lattice Boltzmann model for numerical relativity
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.
Collisionless reconnection in two-dimensional magnetotail equilibria
NASA Technical Reports Server (NTRS)
Pritchett, P. L.; Coroniti, F. V.; Pellat, R.; Karimabadi, H.
1991-01-01
A two-dimensional particle simulation model based on the Darwin approximation to Maxwell's equations for studying collisionless reconnection in the magnetotail has been developed. Simulations of the pure ion tearing mode in a thin current sheet with normal B(z) field component demonstrate that in this limit this mode grows more slowly than expected based on previous analytic estimates. The saturation level of the tearing instability greatly surpasses estimates based on a simple trapping argument. The effect of the normal field component on the evolution of the tearing instability is considered. It is found that a normal field of even a few percent on axis strongly inhibits the growth of the instability.
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. PMID:18276427
Lattice Boltzmann formulation for Braginskii magnetohydrodynamics
NASA Astrophysics Data System (ADS)
Dellar, Paul
2012-03-01
We present a lattice Boltzmann formulation of the Braginskii magnetohydrodynamic equations that describe large-scale motions in strongly magnetised plasmas. Fluid quantities, density, velocity and stress, are represented by a finite set of distribution functions associated with particles moving on a square or cubic lattice. Equilibrium distributions are constructed from Hermite moment expansions, so slowly varying solutions of the discrete kinetic equation exactly satisfy the Navier--Stokes or MHD momentum equations. Electromagnetic quantities are represented by a second kinetic equation for a set of vector-valued distribution functions. Maxwell's equations and the resistive MHD induction equation may be recovered from slowly varying solutions using different scalings. The resulting algorithm, comprising only local operations at grid points and data copying between adjacent points, readily lends itself to large-scale parallel computations. We modify the collision operator to apply different relaxation times to components of the stress parallel and perpendicular to the local magnetic field, simulating a form of the Braginskii MHD equations encountered in astrophysics. Large shears develop in simulations where the fluid velocity perpendicular to the field lines reverses.
Lattice Boltzmann method for weakly ionized isothermal plasmas
Li Huayu; Ki, Hyungson
2007-12-15
In this paper, a lattice Boltzmann method (LBM) for weakly ionized isothermal plasmas is presented by introducing a rescaling scheme for the Boltzmann transport equation. Without using this rescaling, we found that the nondimensional relaxation time used in the LBM is too large and the LBM does not produce physically realistic results. The developed model was applied to the electrostatic wave problem and the diffusion process of singly ionized helium plasmas with a 1-3% degree of ionization under an electric field. The obtained results agree well with theoretical values.
The iterative Monte Carlo technique for collisionless plasma flow to a surface
Pitcher, C.S.
1993-03-01
A new technique for modelling the boundary plasma of magnetic fusion devices is described. The technique represents a natural extension of existing Monte Carlo codes, which are presently constrained to have the plasma background specified by either measurements or predictions from plasma fluid codes. The new approach, the Iterative Monte Carlo (IMC) technique, self-consistently determines the ambipolar electric field in the plasma by feeding back into the simulation the evolving plasma density using the Boltzmann relation. The IMC technique is applied, for demonstrative purposes, to the problem of collisionless one-dimensional plasma flow to a surface. Such a problem has previously been solved exactly using kinetic approaches in the published literature using two different particle source functions. Good agreement between the IMC results and the exact solutions is obtained.
Multi-Species Thermal Lattice Boltzmann Models
NASA Astrophysics Data System (ADS)
Wah, Darren; Vahala, George; Vahala, Linda; Pavlo, Pavol; Carter, Jonathan
1998-11-01
Thermal Lattice Boltzmann models (TLBM) are ideal for simulating nonlinear macroscopic conservation systems because of their inherent parallelizeability (nearly all operations are purely local). The TLBM solves a linear BGK-like kinetic equation so that the standard nonlinear convective terms in the standard fluid codes are now replaced by a simple shift operator (linear advection) at the kinetic level. Here we extend our previous TLBM to handle a two-species system, utilizing the models of Morse (1964),Greene (1973) and Kotelnikov & Montgomery (1997). Each kinetic equation now has 2 BGK-like relaxation terms : the first is due to self-collisions and the other is due to different- species collisions. The relaxation rates used are appropriate for electron-ion collisions. Certain constraints can be imposed on the relaxed distribution functions so that the cross-species momentum and energy evolutions relax at the rate determined from the full nonlinear Boltzmann integral collision operator. Ionization and recombination processes will also be examined. Both hexagonal and octagonal lattices are studied.
Supermagnetosonic Jets behind a Collisionless Quasiparallel Shock
Hietala, H.; Vainio, R.; Laitinen, T. V.; Vaivads, A.; Andreeova, K.; Palmroth, M.; Pulkkinen, T. I.; Koskinen, H. E. J.; Lucek, E. A.; Reme, H.
2009-12-11
The downstream region of a collisionless quasiparallel shock is structured containing bulk flows with high kinetic energy density from a previously unidentified source. We present Cluster multispacecraft measurements of this type of supermagnetosonic jet as well as of a weak secondary shock front within the sheath, that allow us to propose the following generation mechanism for the jets: The local curvature variations inherent to quasiparallel shocks can create fast, deflected jets accompanied by density variations in the downstream region. If the speed of the jet is super(magneto)sonic in the reference frame of the obstacle, a second shock front forms in the sheath closer to the obstacle. Our results can be applied to collisionless quasiparallel shocks in many plasma environments.
Boltzmann's H theorem for systems with frictional dissipation.
Bizarro, João P S
2011-03-01
By use of Boltzmann's equation to describe an ensemble of particles under the influence of a friction force, Boltzmann's H theorem is refined to explicitly include frictional dissipation, the accompanying fluctuations being modeled via an added diffusive, Fokker-Planck term. If the friction force per particle mass is proportional to velocity, as is the case with viscous drag with a friction coefficient γ, Boltzmann's H theorem for the time rate of change of the quantity H reads dH/dt ≤ γ. The classical formulation stating that H can never increase is thus replaced by the statement that H cannot increase at a rate higher than γ, a general result but of particular relevance when fluctuations are negligible and the system is far from equilibrium. When the particles are not far from thermal equilibrium, an alternative, more suitable expression emerges which can be written in the form of a Clausius inequality. PMID:21517545
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
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.
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 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.
Temperature based Restricted Boltzmann Machines
NASA Astrophysics Data System (ADS)
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-01
Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view.
Boltzmann's Approach to Statistical Mechanics
NASA Astrophysics Data System (ADS)
Goldstein, Sheldon
In the last quarter of the nineteenth century, Ludwig Boltzmann explained how irreversible macroscopic laws, in particular the second law of thermodynamics, originate in the time-reversible laws of microscopic physics. Boltzmann's analysis, the essence of which I shall review here, is basically correct. The most famous criticisms of Boltzmann's later work on the subject have little merit. Most twentieth century innovations - such as the identification of the state of a physical system with a probability distribution \\varrho on its phase space, of its thermodynamic entropy with the Gibbs entropy of \\varrho, and the invocation of the notions of ergodicity and mixing for the justification of the foundations of statistical mechanics - are thoroughly misguided.
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 model for Coulomb-driven flows in dielectric liquids
NASA Astrophysics Data System (ADS)
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work.
Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids.
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work. PMID:26986441
NASA Technical Reports Server (NTRS)
Collins, William
1989-01-01
The dispersion equation of Barnes (1966) is used to study the dissipation of asymptotic wave packets generated by localized periodic sources. The solutions of the equation are linear waves, damped by Landau and transit-time processes, in a collisionless warm plasma. For the case of an ideal MHD system, most of the waves emitted from a source are shown to cancel asympotically through destructive interference. The modes transporting significant flux to asymptotic distances are found to be Alfven waves and fast waves with theta (the angle between the magnetic field and the characteristics of the far-field waves) of about 0 and about pi/2.
Zermelo, Boltzmann, and the recurrence paradox
NASA Astrophysics Data System (ADS)
Steckline, Vincent S.
1983-10-01
The papers exchanged by Ludwig Boltzmann and Ernst Zermelo concerning the recurrence paradox are summarized. The historical context of the paradox, Zermelo's proof of the paradox, his opinions of its consequences, Boltzmann's reply, and the ensuing discussion are described.
NASA Astrophysics Data System (ADS)
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.
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.
Thin-shell instability in collisionless plasma
NASA Astrophysics Data System (ADS)
Dieckmann, M. E.; Ahmed, H.; Doria, D.; Sarri, G.; Walder, R.; Folini, D.; Bret, A.; Ynnerman, A.; Borghesi, M.
2015-09-01
Thin-shell instability is one process which can generate entangled structures in astrophysical plasma on collisional (fluid) scales. It is driven by a spatially varying imbalance between the ram pressure of the inflowing upstream plasma and the downstream's thermal pressure at a nonplanar shock. Here we show by means of a particle-in-cell simulation that an analog process can destabilize a thin shell formed by two interpenetrating, unmagnetized, and collisionless plasma clouds. The amplitude of the shell's spatial modulation grows and saturates after about ten inverse proton plasma frequencies, when the shell consists of connected piecewise linear patches.
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.
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.
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.
Collisionless Relaxation in Non-Neutral Plasmas
Levin, Yan; Pakter, Renato; Teles, Tarcisio N.
2008-02-01
A theoretical framework is presented which allows us to quantitatively predict the final stationary state achieved by a non-neutral plasma during a process of collisionless relaxation. As a specific application, the theory is used to study relaxation of charged-particle beams. It is shown that a fully matched beam relaxes to the Lynden-Bell distribution. However, when a mismatch is present and the beam oscillates, parametric resonances lead to a core-halo phase separation. The approach developed accounts for both the density and the velocity distributions in the final stationary state.
New expression for collisionless magnetic reconnection rate
Klimas, Alex
2015-04-15
For 2D, symmetric, anti-parallel, collisionless magnetic reconnection, new expressions for the reconnection rate in the electron diffusion region are introduced. It is shown that these expressions can be derived in just a few simple steps from a physically intuitive starting point; the derivations are given in their entirety, and the validity of each step is confirmed. The predictions of these expressions are compared to the results of several long-duration, open-boundary particle-in-cell reconnection simulations to demonstrate excellent agreement.
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.
A twenty-moment model for collisionless guide field reconnection
NASA Astrophysics Data System (ADS)
Ng, Jonathan; Hakim, Ammar; Bhattacharjee, Amitava
2015-11-01
The integration of kinetic effects in fluid models is an important problem in global simulations of the Earth's magnetosphere and space weather modelling. Here we introduce a new fluid model and closure for collisionless magnetic reconnection and more general applications. It has recently been shown that electron pressure anisotropy is important in setting the structure of the reconnection region, and a closure based on the drift kinetic equation using a distribution of trapped and passing particles has been derived. We extend the model and present a general expression for moments of the distribution function. By evolving the heat flux tensor and closing at the fourth velocity moment, we obtain a self-consistent set of fluid equations, which includes the evolution of the off-diagonal elements of the pressure tensor. The model is implemented in a two-fluid code and the results are compared to PIC simulations of guide field reconnection. This work was supported by NSF Grant No. AGS-1338944, DOE Contract DE-AC02-09CH11466.
Lattice Uehling-Uhlenbeck Boltzmann-Bhatnagar-Gross-Krook hydrodynamics of quantum gases
NASA Astrophysics Data System (ADS)
Yang, Jaw-Yen; Hung, Li-Hsin
2009-05-01
We present a semiclassical lattice Boltzmann method based on quantum kinetic theory. The method is directly derived by projecting the Uehling-Uhlenbeck Boltzmann-Bhatnagar-Gross-Krook equations onto the tensor Hermite polynomials following Grad’s moment expansion method. The intrinsic discrete nodes of the Gauss-Hermite quadrature provide the natural lattice velocities for the semiclassical lattice Boltzmann method. Gases of particles of arbitrary statistics can be considered. Simulation of one-dimensional compressible gas flow and two-dimensional hydrodynamic flows are shown. The results indicate distinct characteristics of the effects of quantum statistics.
Polar Coordinate Lattice Boltzmann Kinetic Modeling of Detonation Phenomena
NASA Astrophysics Data System (ADS)
Lin, Chuan-Dong; Xu, Ai-Guo; Zhang, Guang-Cai; Li, Ying-Jun
2014-11-01
A novel polar coordinate lattice Boltzmann kinetic model for detonation phenomena is presented and applied to investigate typical implosion and explosion processes. In this model, the change of discrete distribution function due to local chemical reaction is dynamically coupled into the modified lattice Boltzmann equation which could recover the Navier—Stokes equations, including contribution of chemical reaction, via the Chapman—Enskog expansion. For the numerical investigations, the main focuses are the nonequilibrium behaviors in these processes. The system at the disc center is always in its thermodynamic equilibrium in the highly symmetric case. The internal kinetic energies in different degrees of freedom around the detonation front do not coincide. The dependence of the reaction rate on the pressure, influences of the shock strength and reaction rate on the departure amplitude of the system from its local thermodynamic equilibrium are probed.
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.
Physical symmetry and lattice symmetry in the lattice Boltzmann method
Cao, N.; Chen, S.; Jin, S.; Martinez, D.
1997-01-01
The lattice Boltzmann method (LBM) is regarded as a specific finite difference discretization for the kinetic equation of the discrete velocity distribution function. We argue that for finite sets of discrete velocity models, such as LBM, the physical symmetry is necessary for obtaining the correct macroscopic Navier-Stokes equations. In contrast, the lattice symmetry and the Lagrangian nature of the scheme, which is often used in the lattice gas automaton method and the existing lattice Boltzmann methods and directly associated with the property of particle dynamics, is not necessary for recovering the correct macroscopic dynamics. By relaxing the lattice symmetry constraint and introducing other numerical discretization, one can also obtain correct hydrodynamics. In addition, numerical simulations for applications, such as nonuniform meshes and thermohydrodynamics can be easily carried out and numerical stability can be ensured by the Courant-Friedricks-Lewey condition and using the semi-implicit collision scheme. {copyright} {ital 1997} {ital The American Physical Society}
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.
NASA Astrophysics Data System (ADS)
Gao, Dong-Ning; Qi, Xin; Hong, Xue-Ren; Yang, Xue; Duan, Wen-Shan; Yang, Lei; Yang
2014-06-01
Numerical and theoretical investigations are carried out for the stability of the dust acoustic waves (DAWs) under the transverse perturbation in a two-ion temperature magnetized and collisionless dusty plasma. The Zakharov-Kuznetsov (ZK) equation, modified ZK equation, and Extended ZK (EZK) equation of the DAWs are given by using the reductive perturbation technique. The cut-off frequency is obtained by applying higher-order transverse perturbations to the soliton solution of the EZK equation. The propagation velocity of solitary waves, the real cut-off frequency, as well as the growth rate of the higher-order perturbation to the solitary wave are obtained.
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
Assessment of linear finite-difference Poisson-Boltzmann solvers.
Wang, Jun; Luo, Ray
2010-06-01
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
Fully relativistic lattice Boltzmann algorithm
Romatschke, P.; Mendoza, M.; Succi, S.
2011-09-15
Starting from the Maxwell-Juettner equilibrium distribution, we develop a relativistic lattice Boltzmann (LB) algorithm capable of handling ultrarelativistic systems with flat, but expanding, spacetimes. The algorithm is validated through simulations of a quark-gluon plasma, yielding excellent agreement with hydrodynamic simulations. The present scheme opens the possibility of transferring the recognized computational advantages of lattice kinetic theory to the context of both weakly and ultrarelativistic systems.
Localized enhancements of energetic particles at oblique collisionless shocks
NASA Astrophysics Data System (ADS)
Fraschetti, F.; Giacalone, J.
2015-04-01
We investigate the spatial distribution of charged particles accelerated by non-relativistic oblique fast collisionless shocks using three-dimensional test-particle simulations. We find that the density of low-energy particles exhibits a localized enhancement at the shock, resembling the `spike' measured at interplanetary shocks. In contrast to previous results based on numerical solutions to the focused transport equation, we find a shock spike for any magnetic obliquity, from quasi-perpendicular to parallel. We compare the pitch-angle distribution with respect to the local magnetic field and the momentum distribution far downstream and very near the shock within the spike; our findings are compatible with predictions from the scatter-free shock drift acceleration limit in these regions. The enhancement of low-energy particles measured by Voyager 1 at solar termination shock is comparable with our profiles. Our simulations allow for predictions of suprathermal protons at interplanetary shocks within 10 solar radii to be tested by Solar Probe mission. They also have implications for the interpretation of ions accelerated at supernova remnant shocks.
Landau, Case, van Kampen and Collisionless Fluid Closures
NASA Astrophysics Data System (ADS)
Joseph, Ilon
2015-11-01
Landau damping represents a fundamental paradox within plasma physics. The equations of motion of classical particles and fields are symmetric under time-reversal; yet, the open system formed by integration over velocity space is not invariant and damping results from phase-mixing. Here, it is shown that the Case-van Kampen theorem can be extended to magnetized plasmas: the linear eigenfunctions provide a complete representation of the particle distribution function and exponentially damped and growing eigenmodes must appear in pairs. The numerical Case-van Kampen transformation can performed efficiently in Fourier velocity space and allows fast timescales in the evolution of the system to be treated using exponential integration. On the other hand, fluid moments require integration over velocity space, and, thus, representation of Landau damping requires explicit introduction of the arrow of time through a collisionless damping operator. This operator captures linear phenomena at the cost of damping nonlinear phenomena such as the plasma echo. Numerical comparisons of these two rather different representations will be presented. LLNL-ABS-674917 prepared by LLNL under Contract DE-AC52-07NA27344.
Collisionless 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.
Collisionless Reconnection in Global Modeling of Magnetospheric Dynamics
NASA Astrophysics Data System (ADS)
Kuznetsova, M. M.; Hesse, M.; Rastaetter, L.; Gombosi, T.; de Zeeuw, D.; Toth, G.
2006-12-01
Recent advances in small-scale kinetic modeling of magnetic reconnection significantly improved our understanding of physical mechanisms controlling the dissipation in the vicinity of the reconnection site in collisionless plasma. However the progress in studies of small-scale geometries was not very helpful for large scale simulations. Global magnetosphere simulations usually include non-ideal processes in terms of numerical dissipation and/or ad hoc anomalous resistivity. To understand the role of magnetic reconnection in global evolution of magnetosphere and to place spacecraft observations into global context it is desirable to perform global simulations with physically motivated model of dissipation that are capable to reproduce reconnection rates observed in kinetic models. In our efforts to bridge the gap between small scale kinetic modeling and global simulations we introduced an approach that allows to quantify the interaction between large-scale global magnetospheric dynamics and microphysical processes in diffusion regions near reconnection sites. We utilized the global MHD code BATSRUS and incorporate primary mechanism controlling the dissipation in the vicinity of the reconnection site in terms of non-gyrotropic corrections to the induction equation. We demonstrated that nongyrotropic effects can significantly alter the global magnetosphere evolution. Our approach allowed for the first time to model loading/unloading cycle in response to steady southward IMF driving. We will extend our approach to cases with nonzero IMF By and analyze the effects of solar wind parameters and ionospheric conductance on reconnection rate and global magnetosphere dynamics.
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.
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.
Theory and simulation of collisionless parallel shocks
NASA Technical Reports Server (NTRS)
Quest, K. B.
1988-01-01
This paper presents a self-consistent theoretical model for collisionless parallel shock structure, based on the hypothesis that shock dissipation and heating can be provided by electromagnetic ion beam-driven instabilities. It is shown that shock formation and plasma heating can result from parallel propagating electromagnetic ion beam-driven instabilities for a wide range of Mach numbers and upstream plasma conditions. The theoretical predictions are compared with recently published observations of quasi-parallel interplanetary shocks. It was found that low Mach number interplanetary shock observations were consistent with the explanation that group-standing waves are providing the dissipation; two high Mach number observations confirmed the theoretically predicted rapid thermalization across the shock.
Weak collisionless shocks in laser-plasmas
NASA Astrophysics Data System (ADS)
Cairns, R. A.; Bingham, R.; Trines, R. G. M.; Norreys, P.
2015-04-01
We obtain a theory describing laminar shock-like structures in a collisionless plasma and examine the parameter limits, in terms of the ion sound Mach number and the electron/ion temperature ratio, within which these structures exist. The essential feature is the inclusion of finite ion temperature with the result that some ions are reflected from a potential ramp. This destroys the symmetry between upstream and downstream regions that would otherwise give the well-known ion solitary wave solution. We have shown earlier (Cairns et al 2014 Phys. Plasmas 21 022112) that such structures may be relevant to problems such as the existence of strong, localized electric fields observed in laser compressed pellets and laser acceleration of ions. Here we present results on the way in which these structures may produce species separation in fusion targets and suggest that it may be possible to use shock ion acceleration for fast ignition.
Temperature based Restricted Boltzmann Machines
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-01
Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view. PMID:26758235
The microphysics of collisionless shock waves.
Marcowith, A; Bret, A; Bykov, A; Dieckman, M E; Drury, L O'C; Lembège, B; Lemoine, M; Morlino, G; Murphy, G; Pelletier, G; Plotnikov, I; Reville, B; Riquelme, M; Sironi, L; Novo, A Stockem
2016-04-01
Collisionless shocks, that is shocks mediated by electromagnetic processes, are customary in space physics and in astrophysics. They are to be found in a great variety of objects and environments: magnetospheric and heliospheric shocks, supernova remnants, pulsar winds and their nebulæ, active galactic nuclei, gamma-ray bursts and clusters of galaxies shock waves. Collisionless shock microphysics enters at different stages of shock formation, shock dynamics and particle energization and/or acceleration. It turns out that the shock phenomenon is a multi-scale non-linear problem in time and space. It is complexified by the impact due to high-energy cosmic rays in astrophysical environments. This review adresses the physics of shock formation, shock dynamics and particle acceleration based on a close examination of available multi-wavelength or in situ observations, analytical and numerical developments. A particular emphasis is made on the different instabilities triggered during the shock formation and in association with particle acceleration processes with regards to the properties of the background upstream medium. It appears that among the most important parameters the background magnetic field through the magnetization and its obliquity is the dominant one. The shock velocity that can reach relativistic speeds has also a strong impact over the development of the micro-instabilities and the fate of particle acceleration. Recent developments of laboratory shock experiments has started to bring some new insights in the physics of space plasma and astrophysical shock waves. A special section is dedicated to new laser plasma experiments probing shock physics. PMID:27007555
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
NASA Astrophysics Data System (ADS)
Marcowith, A.; Bret, A.; Bykov, A.; Dieckman, M. E.; O'C Drury, L.; Lembège, B.; Lemoine, M.; Morlino, G.; Murphy, G.; Pelletier, G.; Plotnikov, I.; Reville, B.; Riquelme, M.; Sironi, L.; Stockem Novo, A.
2016-04-01
Collisionless shocks, that is shocks mediated by electromagnetic processes, are customary in space physics and in astrophysics. They are to be found in a great variety of objects and environments: magnetospheric and heliospheric shocks, supernova remnants, pulsar winds and their nebulæ, active galactic nuclei, gamma-ray bursts and clusters of galaxies shock waves. Collisionless shock microphysics enters at different stages of shock formation, shock dynamics and particle energization and/or acceleration. It turns out that the shock phenomenon is a multi-scale non-linear problem in time and space. It is complexified by the impact due to high-energy cosmic rays in astrophysical environments. This review adresses the physics of shock formation, shock dynamics and particle acceleration based on a close examination of available multi-wavelength or in situ observations, analytical and numerical developments. A particular emphasis is made on the different instabilities triggered during the shock formation and in association with particle acceleration processes with regards to the properties of the background upstream medium. It appears that among the most important parameters the background magnetic field through the magnetization and its obliquity is the dominant one. The shock velocity that can reach relativistic speeds has also a strong impact over the development of the micro-instabilities and the fate of particle acceleration. Recent developments of laboratory shock experiments has started to bring some new insights in the physics of space plasma and astrophysical shock waves. A special section is dedicated to new laser plasma experiments probing shock physics.
Mishra, S. K.; Sharma, A.
2013-07-15
We investigate the spatiotemporal focusing dynamics of a Gaussian laser pulse in preformed collisionless plasma subjected to an axial nonuniformity in the plasma density. In order to follow up the pulse dynamics, a nonlinear Schrödinger wave equation characterizing the beam spot size in space and time frame has been derived and solved numerically to investigate the propagation characteristics as the pulse advances in the plasma. The effect of inhomogeneity on focusing length and ponderomotive electron heating have been analyzed and illustrated graphically. It is seen that ponderomotive heating is quite sensitive to the inhomogeneity parameters and the energy gain by electrons can be optimized by suitable choice of parameters.
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)
Topological Interactions in a Boltzmann-Type Framework
NASA Astrophysics Data System (ADS)
Blanchet, Adrien; Degond, Pierre
2016-04-01
We consider a finite number of particles characterised by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of another particle, the leader. The follower chooses its leader according to the proximity rank of the latter with respect to the former. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit equation is akin to the Boltzmann equation. However, it exhibits a spatial non-locality instead of the classical non-locality in velocity space. This result relies on the approximation properties of Bernstein polynomials. We illustrate the dynamics with numerical simulations.
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.
Boltzmann-Fokker-Planck calculations using standard discrete-ordinates codes
Morel, J.E.
1987-01-01
The Boltzmann-Fokker-Planck (BFP) equation can be used to describe both neutral and charged-particle transport. Over the past several years, the author and several collaborators have developed methods for representing Fokker-Planck operators with standard multigroup-Legendre cross-section data. When these data are input to a standard S/sub n/ code such as ONETRAN, the code actually solves the Boltzmann-Fokker-Planck equation rather than the Boltzmann equation. This is achieved wihout any modification to the S/sub n/ codes. Because BFP calculations can be more demanding from a numerical viewpoint than standard neutronics calculations, we have found it useful to implement new quadrature methods ad convergence acceleration methods in the standard discrete-ordinates code, ONETRAN. We discuss our BFP cross-section representation techniques, our improved quadrature and acceleration techniques, and present results from BFP coupled electron-photon transport calculations performed with ONETRAN. 19 refs., 7 figs.
Master Equation for a Quantum Particle in a Gas
Hornberger, Klaus
2006-08-11
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
Kinetic model for the collisionless sheath of a collisional plasma
NASA Astrophysics Data System (ADS)
Tang, Xian-Zhu; Guo, Zehua
2016-08-01
Collisional plasmas typically have mean-free-path still much greater than the Debye length, so the sheath is mostly collisionless. Once the plasma density, temperature, and flow are specified at the sheath entrance, the profile variation of electron and ion density, temperature, flow speed, and conductive heat fluxes inside the sheath is set by collisionless dynamics, and can be predicted by an analytical kinetic model distribution. These predictions are contrasted here with direct kinetic simulations, showing good agreement.
Global scale-invariant dissipation in collisionless plasma turbulence.
Kiyani, K H; Chapman, S C; Khotyaintsev, Yu V; Dunlop, M W; Sahraoui, F
2009-08-14
A higher-order multiscale analysis of the dissipation range of collisionless plasma turbulence is presented using in situ high-frequency magnetic field measurements from the Cluster spacecraft in a stationary interval of fast ambient solar wind. The observations, spanning five decades in temporal scales, show a crossover from multifractal intermittent turbulence in the inertial range to non-Gaussian monoscaling in the dissipation range. This presents a strong observational constraint on theories of dissipation mechanisms in turbulent collisionless plasmas. PMID:19792654
The Lattice Boltzmann Method applied to neutron transport
Erasmus, B.; Van Heerden, F. A.
2013-07-01
In this paper the applicability of the Lattice Boltzmann Method to neutron transport is investigated. One of the main features of the Lattice Boltzmann method is the simultaneous discretization of the phase space of the problem, whereby particles are restricted to move on a lattice. An iterative solution of the operator form of the neutron transport equation is presented here, with the first collision source as the starting point of the iteration scheme. A full description of the discretization scheme is given, along with the quadrature set used for the angular discretization. An angular refinement scheme is introduced to increase the angular coverage of the problem phase space and to mitigate lattice ray effects. The method is applied to a model problem to investigate its applicability to neutron transport and the results are compared to a reference solution calculated, using MCNP. (authors)
A comparison between simulation and poisson-boltzmann fields
NASA Astrophysics Data System (ADS)
Pettitt, B. Montgomery; Valdeavella, C. V.
1999-11-01
The electrostatic potentials from molecular dynamics (MD) trajectories and Poisson-Boltzmann calculations on a tetra peptide are compared to understand the validity of the resulting free energy surface. The Tuftsin peptide with sequence, Thr-Lys-Pro-Arg, in water is used for the comparison. The results obtained from the analysis of the MD trajectories for the total electrostatic potential at points on a grid using the Ewald technique are compared with the solution to the Poisson-Boltzmann (PB) equation averaged over the same set of configurations. The latter was solved using an optimal set of dielectric constant parameters. Structural averaging of the field over the MD simulation was examined in the context of the PB results. The detailed spatial variation of the electrostatic potential on the molecular surface are not qualitatively reproducible from MD to PB. Implications of using such field calculations and the implied free energies are discussed.
Modified lattice Boltzmann method for compressible fluid simulations.
Hinton, F L; Rosenbluth, M N; Wong, S K; Lin-Liu, Y R; Miller, R L
2001-06-01
A modified lattice Boltzmann algorithm is shown to have much better stability to growing temperature perturbations, when compared with the standard lattice Boltzmann algorithm. The damping rates of long-wavelength waves, which determine stability, are derived using a collisional equilibrium distribution function which has the property that the Euler equations are obtained exactly in the limit of zero time step. Using this equilibrium distribution function, we show that our algorithm has inherent positive hyperviscosity and hyperdiffusivity, for very small values of viscosity and thermal diffusivity, which are lacking in the standard algorithm. Short-wavelength modes are shown to be stable for temperatures greater than a lower limit. Results from a computer code are used to compare these algorithms, and to confirm the damping rate predictions made analytically. Finite amplitude sound waves in the simulated fluid steepen, as expected from gas dynamic theory. PMID:11415085
Multiscale dynamics based on kinetic simulation of collisionless magnetic reconnection
NASA Astrophysics Data System (ADS)
Fujimoto, Keizo; Takamoto, Makoto
2016-07-01
Magnetic reconnection is a natural energy converter which allows explosive energy release of the magnetic field energy into plasma kinetic energy. The reconnection processes inherently involve multi-scale process. The breaking of the field lines takes place predominantly in a small region called the diffusion region formed near the x-line, while the fast plasma jets resulting from reconnection extend to a distance far beyond the ion kinetic scales from the x-line. There has been a significant gap in understanding of macro-scale and micro-scale processes. The macro-scale model of reconnection has been developed using the magnetohydrodynamics (MHD) equations, while the micro-scale processes around the x-line have been based on kinetic equations including the ion and electron inertia. The problem is that these two kinds of model have significant discrepancies. It has been believed without any guarantee that the microscopic model near the x-line would connect to the macroscopic model far downstream of the x-line. In order to bridge the gap between the macro and micro-scale processes, we have performed large-scale particle-in-cell simulations with the adaptive mesh refinement. The simulation results suggest that the microscopic processes around the x-line do not connect to the previous MHD model even in the region far downstream of the x-line. The slow mode shocks and the associated plasma acceleration do not appear at the exhaust boundary of kinetic reconnection. Instead, the ions are accelerated due to the Speiser motion in the current layer extending to a distance beyond the kinetic scales. The different acceleration mechanisms between the ions and electrons lead to the Hall current system in broad area of the exhaust. Therefore, the previous MHD model could be inappropriate for collisionless magnetic reconnection. Ref. K. Fujimoto & M. Takamoto, Phys. Plasmas, 23, 012903 (2016).
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.
Spectrum structure and behavior of the Vlasov-Maxwell-Boltzmann system without angular cutoff
NASA Astrophysics Data System (ADS)
Huang, Yongting
2016-02-01
The spectrum structure and behavior of the Vlasov-Maxwell-Boltzmann (VMB) system with physical angular non-cutoff intermolecular collisions are studied in this paper. The analysis shows the effect of the Lorentz force induced by the electro-magnetic field leads to some different spectrum structure from the non-cutoff Boltzmann equation. The spectrum structure in high frequency, quite different from the VMB system with angular cutoff assumption, also illustrates the hyperbolic structure of the Maxwell equation. Furthermore, the large time behavior and optimal convergence rates to the equilibrium of the non-cutoff VMB system are established on the spectrum analysis.
Lattice Boltzmann methods in Geosciences
NASA Astrophysics Data System (ADS)
Huber, Christian; Parmigiani, Andrea; Su, Yanqing
2014-05-01
Numerical models often offer the only possible approach to study the complex non-linear dynamics of geodynamical processes that are difficult or impossible to scale for laboratory experiments. The development of improved computer resources has allowed the emergence of large-scale parallel computations in Earth Sciences. These resources have lead to an increasing complexity in models where a greater number of adjustable parameters arise. Although the increasing number of free parameters offers a greater flexibility to fit satisfyingly the set of available constraints (e.g. geochemical, structural) it also provides new challenges in terms of the size of the parameter space and non-uniqueness of model solutions. Another significant challenge associated with state-of-the-art models is that their complexity is in general associated with the addition of parameterizations of the unresolved (small) scale processes. This trend calls for the development of complementary high-performance models to constrain the physics at small-scales where mass, momentum and energy exchanges at interfaces between different phases control the dynamics in heterogeneous media. We argue that more attention should be devoted to the development of multiphase numerical modeling at the granular (pore) scale to investigate the dynamical behavior of heterogeneous media and the emergence of feedbacks that influence the response of these media at much greater scales. The lattice Boltzmann method is a paradigm that emerged almost three decades ago. It is based on kinetic theory and follows a bottom-up approach that contrast the top-down strategy of standard methods such as Finite Volumes, FEM and Finite Differences. Lattice Boltzmann is ideally suited to handle the complex dynamics of multiphase systems at small spatial scales and is very efficient for parallel programing. In this presentation, we discuss the development of different lattice Boltzmann models developed in our group over the last years
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.
Electrostatic Potential Across Supercritical Collisionless Shock
NASA Astrophysics Data System (ADS)
Khotyaintsev, Y. V.; Vaivads, A.; Krasnoselskikh, V.
2013-12-01
Using multi-spacecraft Cluster observations of the Earth's bow shock we estimate the electrostatic potential across a supercritical quasi-perpendicular collisionless shock. We find the potential values of the order of several kV, which correspond roughly to the kinetic energy of the inflowing solar wind. It is expected that the plasma flow in the de Hoffmann-Teller frame is field-aligned, and thus the potential computed in this frame is the parallel potential experienced by both ions and electrons. Contrary to this expectation, we show that most of the potential computed in the de Hoffmann-Teller frame is contributed by sub-proton scale Hall electric field, E~JxB/ne, which exists due to decoupling of electron and ion motions at such small scales (ions are demagnetized, and electrons are still well magnetized), and therefore the electron motion in such field is perpendicular to B. In order to calculate the parallel potential drop experienced by electrons, one needs to go to the 'electron' Hoffmann-Teller frame at small scales, in which the JxB/ne field is zero. In this 'electron' frame we find much smaller values of the potential drop across the shock of the order of 100 eV, which is comparable to the change of electron temperature across the shock, and is in agreement with theoretical estimates.
Student Understanding of the Boltzmann Factor
ERIC Educational Resources Information Center
Smith, Trevor I.; Mountcastle, Donald B.; Thompson, John R.
2015-01-01
We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data…
Recent advances in lattice Boltzmann methods
Chen, S.; Doolen, G.D.; He, X.; Nie, X.; Zhang, R.
1998-12-31
In this paper, the authors briefly present the basic principles of lattice Boltzmann method and summarize recent advances of the method, including the application of the lattice Boltzmann method for fluid flows in MEMS and simulation of the multiphase mixing and turbulence.
NASA Astrophysics Data System (ADS)
Patacchini, Leonardo; Hutchinson, Ian H.
2007-11-01
The interaction of a spherical object with a collisionless plasma under weakly magnetized conditions is investigated by means of the PIC code SCEPTIC [1]. The key features of this 2D3v electrostatic ion kinetic code are a spherical geometry accurately resolving the collector's edge, and a Boltzmann treatment of the electrons, whose current is calculated using a recently developed empirical formula accounting for their magnetization [2]. By asymmetrically reducing the ion and electron fluxes to the collector, the magnetic field (B) has a strong influence on the floating potential (φf). The non monotonic dependence of φf on B is documented for a wide range of plasma parameters relevant to probes and dust particles. The magnetic field is also shown to reduce the ion focusing effects present in an unmagnetized plasma when the drift velocity is non negligible, thus the electrostatic part of the ion drag force. This effect is compared with the variation of the electron-ion Coulomb collision frequency with the local magnetic field.[1] I.H. Hutchinson PPCF 47, 71-87 (2005)[2] L. Patacchini et al. Phys. Plasma 14, 062111 (2007)
Collisionless current collection by a spherical particle in a flowing, weakly magnetized plasma
NASA Astrophysics Data System (ADS)
Patacchini, Leonardo; Hutchinson, Ian H.
2007-04-01
Collisionless-plasma current collection by a spherical object such as a dust particle or probe under weakly magnetized conditions (Larmor radius larger than particle radius) is an important, long-studied, but analytically intractable problem. We solve it computationally by means of the hybrid Boltzmann/Particle-in-Cell code SCEPTIC[1] for a wide range of parameters, with finite Debye length. In addition to reducing the ion current, the magnetic field is shown to cancel the ion focusing effects present in an unmagnetized plasma when the drift velocity is comparable to the sound speed. Thus, the magnetic field prevents such phenomena as the reversal of angular flux density asymmetry (greater collection on the downstream side) or the local maximum of the drag force. The floating potential dependence on the ratio of the probe radius to Larmor radius is computed using a newly developed empirical formula for the electron current, which is also documented. [1] I.H. Hutchinson, Plasma Phys. Control. Fusion 45 (2003)
Procassini, R.J.; Birdsall, C.K.; Morse, E.C. )
1990-12-01
A fully kinetic particle-in-cell (PIC) model is used to self-consistently determine the steady-state potential profile in a collisionless plasma that contacts a floating, absorbing boundary. To balance the flow of particles to the wall, a distributed source region is used to inject particles into the one-dimensional system. The effect of the particle source distribution function on the source region and collector sheath potential drops, and particle velocity distributions is investigated. The ion source functions proposed by Emmert {ital et} {ital al}. (Phys. Fluids {bold 23}, 803 (1980)) and Bissell and Johnson (Phys. Fluids {bold 30}, 779 (1987)) (and various combinations of these) are used for the injection of both ions {ital and} electrons. The values of the potential drops obtained from the PIC simulations are compared to those from the theories of Emmert {ital et} {ital al}., Bissell and Johnson, and Scheuer and Emmert (Phys. Fluids {bold 31}, 3645 (1988)), all of which assume that the electron density is related to the plasma potential via the Boltzmann relation. The values of the source region and total potential drop are found to depend on the choice of the electron source function, as well as the ion source function. The question of an infinite electric field at the plasma--sheath interface, which arises in the analyses of Bissell and Johnson and Scheuer and Emmert, is also addressed.
Entropy Production in Collisionless Systems. II. Arbitrary Phase-space Occupation Numbers
NASA Astrophysics Data System (ADS)
Barnes, Eric I.; Williams, Liliya L. R.
2012-04-01
We present an analysis of two thermodynamic techniques for determining equilibria of self-gravitating systems. One is the Lynden-Bell (LB) entropy maximization analysis that introduced violent relaxation. Since we do not use the Stirling approximation, which is invalid at small occupation numbers, our systems have finite mass, unlike LB's isothermal spheres. (Instead of Stirling, we utilize a very accurate smooth approximation for ln x!.) The second analysis extends entropy production extremization to self-gravitating systems, also without the use of the Stirling approximation. In addition to the LB statistical family characterized by the exclusion principle in phase space, and designed to treat collisionless systems, we also apply the two approaches to the Maxwell-Boltzmann (MB) families, which have no exclusion principle and hence represent collisional systems. We implicitly assume that all of the phase space is equally accessible. We derive entropy production expressions for both families and give the extremum conditions for entropy production. Surprisingly, our analysis indicates that extremizing entropy production rate results in systems that have maximum entropy, in both LB and MB statistics. In other words, both thermodynamic approaches lead to the same equilibrium structures.
ENTROPY PRODUCTION IN COLLISIONLESS SYSTEMS. II. ARBITRARY PHASE-SPACE OCCUPATION NUMBERS
Barnes, Eric I.; Williams, Liliya L. R. E-mail: llrw@astro.umn.edu
2012-04-01
We present an analysis of two thermodynamic techniques for determining equilibria of self-gravitating systems. One is the Lynden-Bell (LB) entropy maximization analysis that introduced violent relaxation. Since we do not use the Stirling approximation, which is invalid at small occupation numbers, our systems have finite mass, unlike LB's isothermal spheres. (Instead of Stirling, we utilize a very accurate smooth approximation for ln x{exclamation_point}.) The second analysis extends entropy production extremization to self-gravitating systems, also without the use of the Stirling approximation. In addition to the LB statistical family characterized by the exclusion principle in phase space, and designed to treat collisionless systems, we also apply the two approaches to the Maxwell-Boltzmann (MB) families, which have no exclusion principle and hence represent collisional systems. We implicitly assume that all of the phase space is equally accessible. We derive entropy production expressions for both families and give the extremum conditions for entropy production. Surprisingly, our analysis indicates that extremizing entropy production rate results in systems that have maximum entropy, in both LB and MB statistics. In other words, both thermodynamic approaches lead to the same equilibrium structures.
Shearing Box Simulations of the MRI in a Collisionless Plasma
Sharma, Prateek; Hammett, Gregory, W.; Quataert, Eliot; Stone, James, M.
2005-08-31
We describe local shearing box simulations of turbulence driven by the magnetorotational instability (MRI) in a collisionless plasma. Collisionless effects may be important in radiatively inefficient accretion flows, such as near the black hole in the Galactic Center. The MHD version of ZEUS is modified to evolve an anisotropic pressure tensor. A fluid closure approximation is used to calculate heat conduction along magnetic field lines. The anisotropic pressure tensor provides a qualitatively new mechanism for transporting angular momentum in accretion flows (in addition to the Maxwell and Reynolds stresses). We estimate limits on the pressure anisotropy due to pitch angle scattering by kinetic instabilities. Such instabilities provide an effective ''collision'' rate in a collisionless plasma and lead to more MHD-like dynamics. We find that the MRI leads to efficient growth of the magnetic field in a collisionless plasma, with saturation amplitudes comparable to those in MHD. In the saturated state, the anisotropic stress is comparable to the Maxwell stress, implying that the rate of angular momentum transport may be moderately enhanced in a collisionless plasma.
NASA Astrophysics Data System (ADS)
Cai, Zhenning; Torrilhon, Manuel
2015-08-01
A sequence of approximate linear collision models for hard-sphere and inverse-power-law gases is introduced. These models are obtained by expanding the linearized Boltzmann collision operator into series, and a practical algorithm is proposed for evaluating the coefficients in the series. The sequence is proven to be convergent to the linearized Boltzmann operator, and it established a connection between the Shakhov model and the linearized collision model. The convergence is demonstrated by solving the spatially homogeneous Boltzmann equation. By observing the magnitudes of the coefficients, simpler models are developed through removing small entries in the coefficient matrices.
Three-dimensional lattice Boltzmann model for compressible flows.
Sun, Chenghai; Hsu, Andrew T
2003-07-01
A three-dimensional compressible lattice Boltzmann model is formulated on a cubic lattice. A very large particle-velocity set is incorporated in order to enable a greater variation in the mean velocity. Meanwhile, the support set of the equilibrium distribution has only six directions. Therefore, this model can efficiently handle flows over a wide range of Mach numbers and capture shock waves. Due to the simple form of the equilibrium distribution, the fourth-order velocity tensors are not involved in the formulation. Unlike the standard lattice Boltzmann model, no special treatment is required for the homogeneity of fourth-order velocity tensors on square lattices. The Navier-Stokes equations were recovered, using the Chapman-Enskog method from the Bhatnagar-Gross-Krook (BGK) lattice Boltzmann equation. The second-order discretization error of the fluctuation velocity in the macroscopic conservation equation was eliminated by means of a modified collision invariant. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Since the present scheme deals only with the equilibrium distribution that depends only on fluid density, velocity, and internal energy, boundary conditions on curved wall are easily implemented by an extrapolation of macroscopic variables. To verify the scheme for inviscid flows, we have successfully simulated a three-dimensional shock-wave propagation in a box and a normal shock of Mach number 10 over a wedge. As an application to viscous flows, we have simulated a flat plate boundary layer flow, flow over a cylinder, and a transonic flow over a NACA0012 airfoil cascade. PMID:12935242
NASA Astrophysics Data System (ADS)
Hahn, Oliver; Angulo, Raul E.
2016-01-01
N-body simulations are essential for understanding the formation and evolution of structure in the Universe. However, the discrete nature of these simulations affects their accuracy when modelling collisionless systems. We introduce a new approach to simulate the gravitational evolution of cold collisionless fluids by solving the Vlasov-Poisson equations in terms of adaptively refineable `Lagrangian phase-space elements'. These geometrical elements are piecewise smooth maps between Lagrangian space and Eulerian phase-space and approximate the continuum structure of the distribution function. They allow for dynamical adaptive splitting to accurately follow the evolution even in regions of very strong mixing. We discuss in detail various one-, two- and three-dimensional test problems to demonstrate the performance of our method. Its advantages compared to N-body algorithms are: (i) explicit tracking of the fine-grained distribution function, (ii) natural representation of caustics, (iii) intrinsically smooth gravitational potential fields, thus (iv) eliminating the need for any type of ad hoc force softening. We show the potential of our method by simulating structure formation in a warm dark matter scenario. We discuss how spurious collisionality and large-scale discreteness noise of N-body methods are both strongly suppressed, which eliminates the artificial fragmentation of filaments. Therefore, we argue that our new approach improves on the N-body method when simulating self-gravitating cold and collisionless fluids, and is the first method that allows us to explicitly follow the fine-grained evolution in six-dimensional phase-space.
Recent advances in collisionless magnetic reconnection
NASA Astrophysics Data System (ADS)
Porcelli, F.; Borgogno, D.; Califano, F.; Grasso, D.; Ottaviani, M.; Pegoraro, F.
2002-12-01
One of the recurring problems in magnetic reconnection is the identification of the appropriate generalized Ohm's law. In weakly collisional plasmas with a strong magnetic guide field component, a fluid model may be adopted, where electron inertia and the electron pressure gradient play important roles. In the absence of collisions, electron inertia provides the mechanism for magnetic field-line breaking. Electron compressibility alters significantly the structure of the reconnection region and allows for faster reconnection rates, which are consistent with the fast relaxation times of sawtooth oscillations in tokamak plasmas. The Hall term may also become important when the guide field is weak. The very possibility of nonlinear, irreversible magnetic reconnection in the absence of dissipation is addressed. We show that in a collisionless plasma, magnetic islands can grow and reach a saturated state in a coarse-grained sense. Magnetic energy is transferred to kinetic energy in smaller and smaller spatial scale lengths through a phase mixing process. The same model is then applied to the interpretation of driven reconnection events in the vicinity of a magnetic X-line observed in the VTF experiment at MIT. The reconnection is driven by externally induced plasma flows in a background magnetic configuration that has a hyperbolic null in the reconnection plane and a magnetic guide field component perpendicular to that plane. In the limit where the guide field is strong, assuming the external drive to be sufficiently weak for a linear approximation to hold, a dynamic evolution of the system is obtained which does not reach a stationary state. The reconnection process develops in two phases: an initial phase, whose characteristic rate is a fraction of the Alfvén frequency, and a later one, whose rate is determined by the electron collision frequency.
An Infinite Restricted Boltzmann Machine.
Côté, Marc-Alexandre; Larochelle, Hugo
2016-07-01
We present a mathematical construction for the restricted Boltzmann machine (RBM) that does not require specifying the number of hidden units. In fact, the hidden layer size is adaptive and can grow during training. This is obtained by first extending the RBM to be sensitive to the ordering of its hidden units. Then, with a carefully chosen definition of the energy function, we show that the limit of infinitely many hidden units is well defined. As with RBM, approximate maximum likelihood training can be performed, resulting in an algorithm that naturally and adaptively adds trained hidden units during learning. We empirically study the behavior of this infinite RBM, showing that its performance is competitive to that of the RBM, while not requiring the tuning of a hidden layer size. PMID:27171012
Boltzmann: The Genius of Disorder
NASA Astrophysics Data System (ADS)
Mussardo, G.; Merlone, A.
2010-07-01
The tragedy and greatness of the contribution of Ludwig Boltzmann cannot be understood without taking into account for the relevant scientific developments that took place in the nineteenth century, one of the most eventful periods in the history of science. The kinetic theory opened a new theoretical perspective in understanding natural phenomena. The introduction of new categories of order and disorder changed radically the point of view of those physicists that accepted Boltzmann’s thesis and led, at the same time, to strong opposition to the Viennese Scientist. In this article, we present the academic situation, scientific theories, and disputes involving the Boltzmann’s theories. A short introduction on the birth of the atomistic theories opens the article, while a view on the evolution of the concept of temperature and the definition of its unit quantity closes it.
A Semianalytical Ion Current Model for Radio Frequency Driven Collisionless Sheaths
NASA Technical Reports Server (NTRS)
Bose, Deepak; Govindan, T. R.; Meyyappan, M.; Arnold, Jim (Technical Monitor)
2001-01-01
We propose a semianalytical ion dynamics model for a collisionless radio frequency biased sheath. The model uses bulk plasma conditions and electrode boundary condition to predict ion impact energy distribution and electrical properties of the sheath. The proposed model accounts for ion inertia and ion current modulation at bias frequencies that are of the same order of magnitude as the ion plasma frequency. A relaxation equation for ion current oscillations is derived which is coupled with a damped potential equation in order to model ion inertia effects. We find that inclusion of ion current modulation in the sheath model shows marked improvements in the predictions of sheath electrical properties and ion energy distribution function.
Collisionless tearing in a field-reversed sheet pinch assuming nonparallel propagation
NASA Technical Reports Server (NTRS)
Quest, K. B.; Coroniti, F. V.
1985-01-01
The problem of collisionless linear tearing is examined assuming a wave vector with a component normal to the equilibrium field. The geometry is defined and the general form of the linear dispersion equation is calculated. The linear theory results when k is parallel to B are reviewed, and Ampere's law is calculated for the external adiabatic region when k times B does not equal zero, using two-fluid theory. A solution is obtained for the approximate form of the perturbed currents and vector potential assuming quasi-parallel k. The resonant current contributions within the singular layer are calculated, obtaining an estimate of the dispersion equation. The form of the adiabatic currents within the singular layer is calculated, showing that an x-z current system persists even in the limit k perpendicular to B goes to zero. Finally, the perturbed vector potential solutions across the singular layer are matched to obtain the shape of the complete eigenfunction.
Thermodynamic-consistent lattice Boltzmann model for nonideal fluids
NASA Astrophysics Data System (ADS)
Wen, Binghai; Qin, Zhangrong; Zhang, Chaoying; Fang, Haiping
2015-11-01
A lattice Boltzmann model to simulate phase separation and two-phase flow is proposed. The nonideal force in multiphase flow is directly computed from the free energy. Thermodynamic consistency and Galilean invariance are theoretically analyzed and numerically verified. Remarkably, the theoretical simplicity endues the model with the advantages of high efficiency and easy implementation. We also find that it can work well together with various equations of state in order to simulate different kinds of multiphase flows. Importantly, it has a tunable parameter κ, which can be used to reduce the effect of spurious current and adjust the surface tension to meet the requirements of researches.
An implicit Lagrangian lattice Boltzmann method for the compressible flows
NASA Astrophysics Data System (ADS)
Yan, Guangwu; Dong, Yinfeng; Liu, Yanhong
2006-08-01
In this paper, we propose a new Lagrangian lattice Boltzmann method (LBM) for simulating the compressible flows. The new scheme simulates fluid flows based on the displacement distribution functions. The compressible flows, such as shock waves and contact discontinuities are modelled by using Lagrangian LBM. In this model, we select the element in the Lagrangian coordinate to satisfy the basic fluid laws. This model is a simpler version than the corresponding Eulerian coordinates, because the convection term of the Euler equations disappears. The numerical simulations conform to classical results.
Static contact angle in lattice Boltzmann models of immiscible fluids.
Latva-Kokko, M; Rothman, Daniel H
2005-10-01
We study numerically the capillary rise between two horizontal plates and in a rectangular tube, using a lattice Boltzmann (LB) method. We derive an equation for the static fluid-solid contact angle as a function of the wetting tendency of the walls and test its validity. We show that the generalized Laplace law with two independent radii of curvature is followed in capillary rise in rectangular tubes. Our method removes the history dependence of the fluid-solid contact angle that had been present in earlier LB schemes. PMID:16383561
Two relaxation time lattice Boltzmann model for rarefied gas flows
NASA Astrophysics Data System (ADS)
Esfahani, Javad Abolfazli; Norouzi, Ali
2014-01-01
In this paper, the lattice Boltzmann equation (LBE) with two relaxation times (TRT) is implemented in order to study gaseous flow through a long micro/nano-channel. A new relation is introduced for the reflection factor in the bounce-back/specular reflection (BSR) boundary condition based on the analytical solution of the Navier-Stokes equations. The focus of the present study is on comparing TRT with the other LBE models called multiple relaxation times (MRT) and single relaxation time (SRT) in simulation of rarefied gas flows. After a stability analysis for the TRT and SRT models, the numerical results are presented and validated by the analytical solution of the Navier-Stokes equations with slip boundary condition, direct simulation of Monte Carlo (DSMC) and information preservation (IP) method. The effect of various gases on flow behavior is also investigated by using the variable hard sphere (VHS) model through the symmetrical relaxation time.
Collisionless Damping of Laser Wakes in Plasma Channels
Li, X.; Shvets, G.
1998-08-01
Excitation of accelerating modes in transversely inhomogeneous plasma channels is considered as an initial value problem. Discrete eigenmodes are supported by plasma channels with sharp density gradients. These eigenmodes are collisionlessly damped as the gradients are smoothed. Using collisionless Landau damping as the analogy, the existence and damping of these "quasi-modes" is studied by constructing and analytically continuing the causal Green's function of wake excitation into the lower half of the complex frequency plane. Electromagnetic nature of the plasma wakes in the channel makes their excitation nonlocal. This results in the algebraic decay of the fields with time due to phase-mixing of plasma oscillations with spatially-varying fequencies. Characteristic decay rate is given by the mixing time, which corresponds to the dephasing of two plasma fluid elements separated by the collisionless skin depth. For wide channels the exact expressions for the field evolution are derived. Implications for electron acceleration in plasma channels are discussed.
Accessing the new collisionless reconnection regime in laboratory experiment
NASA Astrophysics Data System (ADS)
Olson, Joseph; Egedal, Jan; Greess, Samuel; Wallace, John; Clark, Michael; Forest, Cary
2015-11-01
The Terrestrial Reconnection Experiment (TREX), the largest dedicated reconnection experiment to date, is currently in operation at the Wisconsin Plasma Astrophysics Laboratory (WiPAL). In its inaugural run, TREX demonstrated its ability to operate in what has traditionally been called the collisionless reconnection regime by observing the out-of-plane magnetic field characteristic of Hall reconnection. Additionally, TREX is projected to access even more collisionless parameters in which electron pressure anisotropy develops, greatly influencing the dynamics of the reconnection process beyond two fluid effects. For example, spacecraft observations and kinetic simulations show that large-scale current layers are driven by this pressure anisotropy. In the last year, TREX has undergone upgrades to its plasma heating, reconnection drive, and diagnostic suite in order to study these features exclusive to truly collisionless reconnection. Preliminary results from the newly optimized experimental runs will be presented. Supported in part by DoE grant DE-SC0010463.
Collisionless Weibel shocks: Full formation mechanism and timing
Bret, A.; Stockem, A.; Narayan, R.; Silva, L. O.
2014-07-15
Collisionless shocks in plasmas play an important role in space physics (Earth's bow shock) and astrophysics (supernova remnants, relativistic jets, gamma-ray bursts, high energy cosmic rays). While the formation of a fluid shock through the steepening of a large amplitude sound wave has been understood for long, there is currently no detailed picture of the mechanism responsible for the formation of a collisionless shock. We unravel the physical mechanism at work and show that an electromagnetic Weibel shock always forms when two relativistic collisionless, initially unmagnetized, plasma shells encounter. The predicted shock formation time is in good agreement with 2D and 3D particle-in-cell simulations of counterstreaming pair plasmas. By predicting the shock formation time, experimental setups aiming at producing such shocks can be optimised to favourable conditions.
Low Frequency Waves at and Upstream of Collisionless Shocks
NASA Astrophysics Data System (ADS)
Wilson, L. B.
2016-02-01
This chapter focuses on the range of low frequency electromagnetic modes observed at and upstream of collisionless shocks in the heliosphere. It discusses a specific class of whistler mode wave observed immediately upstream of collisionless shock ramps, called a whistler precursor. Though these modes have been (and are often) observed upstream of quasi-parallel shocks, the authors limit their discussion to those observed upstream of quasi-perpendicular shocks. The chapter discusses the various ion velocity distributions observed at and upstream of collisionless shocks. It also introduces some terminology and relevant instabilities for ion foreshock waves. The chapter discusses the most common ultra-low frequency (ULF) wave types, their properties, and their free energy sources. It discusses modes that are mostly Alfvénic (i.e., mostly transverse but can be compressive) in nature.
Second-order accurate kinetic schemes for the ultra-relativistic Euler equations
NASA Astrophysics Data System (ADS)
Kunik, Matthias; Qamar, Shamsul; Warnecke, Gerald
2003-12-01
A second-order accurate kinetic scheme for the numerical solution of the relativistic Euler equations is presented. These equations describe the flow of a perfect fluid in terms of the particle density n, the spatial part of the four-velocity u and the pressure p. The kinetic scheme, is based on the well-known fact that the relativistic Euler equations are the moments of the relativistic Boltzmann equation of the kinetic theory of gases when the distribution function is a relativistic Maxwellian. The kinetic scheme consists of two phases, the convection phase (free-flight) and collision phase. The velocity distribution function at the end of the free-flight is the solution of the collisionless transport equation. The collision phase instantaneously relaxes the distribution to the local Maxwellian distribution. The fluid dynamic variables of density, velocity, and internal energy are obtained as moments of the velocity distribution function at the end of the free-flight phase. The scheme presented here is an explicit method and unconditionally stable. The conservation laws of mass, momentum and energy as well as the entropy inequality are everywhere exactly satisfied by the solution of the kinetic scheme. The scheme also satisfies positivity and L1-stability. The scheme can be easily made into a total variation diminishing method for the distribution function through a suitable choice of the interpolation strategy. In the numerical case studies the results obtained from the first- and second-order kinetic schemes are compared with the first- and second-order upwind and central schemes. We also calculate the experimental order of convergence and numerical L1-stability of the scheme for smooth initial data.
Reprint of : The Boltzmann--Langevin approach: A simple quantum-mechanical derivation
NASA Astrophysics Data System (ADS)
Nagaev, K. E.
2016-08-01
We present a simple quantum-mechanical derivation of correlation function of Langevin sources in the semiclassical Boltzmann-Langevin equation. The specific case of electron-phonon scattering is considered. It is shown that the assumption of weak scattering leads to the Poisson nature of the scattering fluxes.
The Boltzmann-Langevin approach: A simple quantum-mechanical derivation
NASA Astrophysics Data System (ADS)
Nagaev, K. E.
2015-11-01
We present a simple quantum-mechanical derivation of correlation function of Langevin sources in the semiclassical Boltzmann-Langevin equation. The specific case of electron-phonon scattering is considered. It is shown that the assumption of weak scattering leads to the Poisson nature of the scattering fluxes.
Reassessing the single relaxation time Lattice Boltzmann method for the simulation of Darcy’s flows
NASA Astrophysics Data System (ADS)
Prestininzi, Pietro; Montessori, Andrea; La Rocca, Michele; Succi, Sauro
2016-09-01
It is shown that the single relaxation time (SRT) version of the Lattice Boltzmann (LB) equation permits to compute the permeability of Darcy’s flows in porous media within a few percent accuracy. This stands in contrast with previous claims of inaccuracy, which we relate to the lack of recognition of the physical dependence of the permeability on the Knudsen number.
PARTICLE ACCELERATION DURING MAGNETOROTATIONAL INSTABILITY IN A COLLISIONLESS ACCRETION DISK
Hoshino, Masahiro
2013-08-20
Particle acceleration during the magnetorotational instability (MRI) in a collisionless accretion disk was investigated by using a particle-in-cell simulation. We discuss the important role that magnetic reconnection plays not only on the saturation of MRI but also on the relativistic particle generation. The plasma pressure anisotropy of p > p{sub ||} induced by the action of MRI dynamo leads to rapid growth in magnetic reconnection, resulting in the fast generation of nonthermal particles with a hard power-law spectrum. This efficient particle acceleration mechanism involved in a collisionless accretion disk may be a possible model to explain the origin of high-energy particles observed around massive black holes.
Nonlinearly interacting trapped particle solitons in collisionless plasmas
NASA Astrophysics Data System (ADS)
Mandal, Debraj; Sharma, Devendra
2016-02-01
The formulation of collective waves in collisionless plasmas is complicated by the kinetic effects produced by the resonant particles, capable of responding to the smallest of the amplitude disturbance. The dispersive plasma manifests this response by generating coherent nonlinear structures associated with phase-space vortices, or holes, at very small amplitudes. The nonlinear interaction between solitary electron phase-space holes is studied in the electron acoustic regime of a collisionless plasma using Vlasov simulations. Evolution of the analytic trapped particle solitary solutions is examined, observing them propagate stably, preserve their identity across strong mutual interactions in adiabatic processes, and display close correspondence with observable processes in nature.
Collisionless kinetic regimes for quasi-stationary axisymmetric accretion disc plasmas
Cremaschini, C.; Tessarotto, M.
2012-08-15
This paper is concerned with the kinetic treatment of quasi-stationary axisymmetric collisionless accretion disc plasmas. The conditions of validity of the kinetic description for non-relativistic magnetized and gravitationally bound plasmas of this type are discussed. A classification of the possible collisionless plasma regimes which can arise in these systems is proposed, which can apply to accretion discs around both stellar-mass compact objects and galactic-center black holes. Two different classifications are determined, which are referred to, respectively, as energy-based and magnetic field-based classifications. Different regimes are pointed out for each plasma species, depending both on the relative magnitudes of kinetic and potential energies and the magnitude of the magnetic field. It is shown that in all cases, there can be quasi-stationary Maxwellian-like solutions of the Vlasov equation. The perturbative approach outlined here permits unique analytical determination of the functional form for the distribution function consistent, in each kinetic regime, with the explicit inclusion of finite Larmor radius-diamagnetic and/or energy-correction effects.
Cremaschini, Claudio; Miller, John C.; Tessarotto, Massimo
2011-06-15
A kinetic treatment is developed for collisionless magnetized plasmas occurring in high-temperature, low-density astrophysical accretion disks, such as are thought to be present in some radiatively inefficient accretion flows onto black holes. Quasi-stationary configurations are investigated, within the framework of a Vlasov-Maxwell description. The plasma is taken to be axisymmetric and subject to the action of slowly time-varying gravitational and electromagnetic fields. The magnetic field is assumed to be characterized by a family of locally nested but open magnetic surfaces. The slow collisionless dynamics of these plasmas is investigated, yielding a reduced gyrokinetic Vlasov equation for the kinetic distribution function. For doing this, an asymptotic quasi-stationary solution is first determined, represented by a generalized bi-Maxwellian distribution expressed in terms of the relevant adiabatic invariants. The existence of the solution is shown to depend on having suitable kinetic constraints and conditions leading to particle trapping phenomena. With this solution, one can treat temperature anisotropy, toroidal and poloidal flow velocities, and finite Larmor-radius effects. An asymptotic expansion for the distribution function permits analytic evaluation of all the relevant fluid fields. Basic theoretical features of the solution and their astrophysical implications are discussed. As an application, the possibility of describing the dynamics of slowly time-varying accretion flows and the self-generation of magnetic field by means of a ''kinetic dynamo effect'' are discussed. Both effects are shown to be related to intrinsically kinetic physical mechanisms.
Ergodicity, ensembles, irreversibility in Boltzmann and beyond
NASA Astrophysics Data System (ADS)
Gallavotti, Giovanni
1995-03-01
The contents of a not too well-known paper by Boltzmann are critically examined. The etymology of the word ergodic and its implications are discussed. A connection with the modern theory of Ruelle is attempted.
Chiavazzo, Eliodoro; Karlin, Iliya V.; Gorban, Alexander N.; Boulouchos, Konstantinos
2010-10-15
A new framework of simulation of reactive flows is proposed based on a coupling between accurate reduced reaction mechanism and the lattice Boltzmann representation of the flow phenomena. The model reduction is developed in the setting of slow invariant manifold construction, and the simplest lattice Boltzmann equation is used in order to work out the procedure of coupling of the reduced model with the flow solver. Practical details of constructing slow invariant manifolds of a reaction system under various thermodynamic conditions are reported. The proposed method is validated with the two-dimensional simulation of a premixed counterflow flame in the hydrogen-air mixture. (author)
A multispeed Discrete Boltzmann Model for transcritical 2D shallow water flows
NASA Astrophysics Data System (ADS)
La Rocca, Michele; Montessori, Andrea; Prestininzi, Pietro; Succi, Sauro
2015-03-01
In this work a Discrete Boltzmann Model for the solution of transcritical 2D shallow water flows is presented and validated. In order to provide the model with transcritical capabilities, a particular multispeed velocity set has been employed for the discretization of the Boltzmann equation. It is shown that this particular set naturally yields a simple and closed procedure to determine higher order equilibrium distribution functions needed to simulate transcritical flow. The model is validated through several classical benchmarks and is proven to correctly and accurately simulate both 1D and 2D transitions between the two flow regimes.
Dielectric Boundary Forces in Numerical Poisson-Boltzmann Methods: Theory and Numerical Strategies.
Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2011-10-01
Continuum modeling of electrostatic interactions based upon the numerical solutions of the Poisson-Boltzmann equation has been widely adopted in biomolecular applications. To extend their applications to molecular dynamics and energy minimization, robust and efficient methodologies to compute solvation forces must be developed. In this study, we have first reviewed the theory for the computation of dielectric boundary forces based on the definition of the Maxwell stress tensor. This is followed by a new formulation of the dielectric boundary force suitable for the finite-difference Poisson-Boltzmann methods. We have validated the new formulation with idealized analytical systems and realistic molecular systems. PMID:22125339
Dielectric boundary force in numerical Poisson-Boltzmann methods: Theory and numerical strategies
NASA Astrophysics Data System (ADS)
Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2011-10-01
Continuum modeling of electrostatic interactions based upon the numerical solutions of the Poisson-Boltzmann equation has been widely adopted in biomolecular applications. To extend their applications to molecular dynamics and energy minimization, robust and efficient methodologies to compute solvation forces must be developed. In this study, we have first reviewed the theory for the computation of dielectric boundary force based on the definition of the Maxwell stress tensor. This is followed by a new formulation of the dielectric boundary force suitable for the finite-difference Poisson-Boltzmann methods. We have validated the new formulation with idealized analytical systems and realistic molecular systems.
Lattice Boltzmann simulation of a fluid flow around a triangular unit of three isothermal cylinders
NASA Astrophysics Data System (ADS)
Alinejad, J.
2016-01-01
The lattice Boltzmann method is employed to simulate heat transfer in the flow past three arrangements of elliptical and circular cylinders under an isothermal boundary condition. The lattice Boltzmann equations and the Bhatnagar-Gross-Krook model are used to simulate two-dimensional forced convection at 30 ≤ Re ≤ 100 and Pr = 0.71. Pressure distributions, isotherms, and streamlines are obtained. Vortex shedding maps are observed in detail for several cases. The present results are in good agreement with available experimental and numerical data.
Student understanding of the Boltzmann factor
NASA Astrophysics Data System (ADS)
Smith, Trevor I.; Mountcastle, Donald B.; Thompson, John R.
2015-12-01
[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations of student discussions about the Boltzmann factor and its derivation during the tutorial development process. This additional information informed modifications that improved students' abilities to complete the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. These data also show an increase in students' appreciation of the origin and significance of the Boltzmann factor during the student discussions. Our findings provide evidence that working in groups to better understand the physical origins of the canonical probability distribution helps students gain a better understanding of when the Boltzmann factor is applicable and how to use it appropriately in answering relevant questions.
Alternating minimization and Boltzmann machine learning.
Byrne, W
1992-01-01
Training a Boltzmann machine with hidden units is appropriately treated in information geometry using the information divergence and the technique of alternating minimization. The resulting algorithm is shown to be closely related to gradient descent Boltzmann machine learning rules, and the close relationship of both to the EM algorithm is described. An iterative proportional fitting procedure for training machines without hidden units is described and incorporated into the alternating minimization algorithm. PMID:18276461
The Q,G-method in physics of collisionless plasma
Kuznetsov, V. I. Ender, A. Ya.
2015-03-15
An analytic method for analyzing nonstationary processes in collisionless one-dimensional bounded plasma—the Q,G-method—is proposed. Formulas for the distribution function of charged particles escaping from the boundary and moving without collisions in a nonstationary electric field of arbitrary form are obtained. Specific features of the distribution function are revealed and analyzed.
Scaled Laboratory Collisionless Shock Experiments in the Large Plasma Device
NASA Astrophysics Data System (ADS)
Clark, S. E.; Schaeffer, D.; Everson, E.; Bondarenko, A.; Winske, D.; Constantin, C.; Niemann, C.
2013-12-01
Collisionless shocks in space plasmas have been investigated since the fifties and are typically studied via in-situ satellite observations, which are limited due to the large structure of collisionless shocks in space environments relative to the satellite observation platform. Scaled, repeatable experiments in the Large Plasma Device (LAPD) at UCLA provide a test bed for studying collisionless shocks in the laboratory, where questions of ion and electron heating and acceleration can be addressed and examined in detail. The experiments are performed by ablating a graphite or plastic target using the Raptor kilojoule-class laser facility at UCLA. The laser provides an on-target energy in the range of 100-500 J that drives a super-Alfvénic (MA > 1) debris plasma across a background magnetic field (200-800 G) into the ambient, magnetized LAPD plasma. Typical plasma parameters in the LAPD consist of a H+ or He+ ambient plasma with a core column (diameter > 20 cm ) density ni ~ 1013 cm-3 and electron temperature Te ~ 10 eV embedded in a larger plasma discharge (diameter ~ 80 cm) of density ni ~ 1012 cm-3 and Te ~ 5 eV. The ambient ion temperature is Ti ~ 1 eV. Experimental results from the latest collisionless shock campaign will be presented and compared with two dimensional hybrid simulations of the experiment. Fielded diagnostics include Thomson scattering, ion spectroscopy, magnetic flux probes, Langmuir probes, and microwave reflectometry.
Maxwell-Vlasov equations as a continuous Hamiltonian system
Morrison, P.J.
1980-11-01
The well-known Maxwell-Vlasov equations that describe a collisionless plasma are cast into Hamiltonian form. The dynamical variables are the physical although noncanonical variables E, B, and f. We present a Poisson bracket which acts on these variables and the energy functional to produce the equations of motion.
Maxwell-Vlasov equations as a continuous Hamiltonian system
Morrison, P.J.
1980-09-01
The well-known Maxwell-Vlasov equations that describe a collisionless plasma are cast into Hamiltonian form. The dynamical variables are the physical although noncanonical variables E, B and f. We present a Poisson bracket which acts on these variables and the energy functional to produce the equations of motion.
Wang, Liang Germaschewski, K.; Hakim, Ammar H.; Bhattacharjee, A.
2015-01-15
We introduce an extensible multi-fluid moment model in the context of collisionless magnetic reconnection. This model evolves full Maxwell equations and simultaneously moments of the Vlasov-Maxwell equation for each species in the plasma. Effects like electron inertia and pressure gradient are self-consistently embedded in the resulting multi-fluid moment equations, without the need to explicitly solving a generalized Ohm's law. Two limits of the multi-fluid moment model are discussed, namely, the five-moment limit that evolves a scalar pressures for each species and the ten-moment limit that evolves the full anisotropic, non-gyrotropic pressure tensor for each species. We first demonstrate analytically and numerically that the five-moment model reduces to the widely used Hall magnetohydrodynamics (Hall MHD) model under the assumptions of vanishing electron inertia, infinite speed of light, and quasi-neutrality. Then, we compare ten-moment and fully kinetic particle-in-cell (PIC) simulations of a large scale Harris sheet reconnection problem, where the ten-moment equations are closed with a local linear collisionless approximation for the heat flux. The ten-moment simulation gives reasonable agreement with the PIC results regarding the structures and magnitudes of the electron flows, the polarities and magnitudes of elements of the electron pressure tensor, and the decomposition of the generalized Ohm's law. Possible ways to improve the simple local closure towards a nonlocal fully three-dimensional closure are also discussed.
NASA Astrophysics Data System (ADS)
Wang, Liang; Hakim, Ammar H.; Bhattacharjee, A.; Germaschewski, K.
2015-01-01
We introduce an extensible multi-fluid moment model in the context of collisionless magnetic reconnection. This model evolves full Maxwell equations and simultaneously moments of the Vlasov-Maxwell equation for each species in the plasma. Effects like electron inertia and pressure gradient are self-consistently embedded in the resulting multi-fluid moment equations, without the need to explicitly solving a generalized Ohm's law. Two limits of the multi-fluid moment model are discussed, namely, the five-moment limit that evolves a scalar pressures for each species and the ten-moment limit that evolves the full anisotropic, non-gyrotropic pressure tensor for each species. We first demonstrate analytically and numerically that the five-moment model reduces to the widely used Hall magnetohydrodynamics (Hall MHD) model under the assumptions of vanishing electron inertia, infinite speed of light, and quasi-neutrality. Then, we compare ten-moment and fully kinetic particle-in-cell (PIC) simulations of a large scale Harris sheet reconnection problem, where the ten-moment equations are closed with a local linear collisionless approximation for the heat flux. The ten-moment simulation gives reasonable agreement with the PIC results regarding the structures and magnitudes of the electron flows, the polarities and magnitudes of elements of the electron pressure tensor, and the decomposition of the generalized Ohm's law. Possible ways to improve the simple local closure towards a nonlocal fully three-dimensional closure are also discussed.
Linearized lattice Boltzmann method for micro- and nanoscale flow and heat transfer
NASA Astrophysics Data System (ADS)
Shi, Yong; Yap, Ying Wan; Sader, John E.
2015-07-01
Ability to characterize the heat transfer in flowing gases is important for a wide range of applications involving micro- and nanoscale devices. Gas flows away from the continuum limit can be captured using the Boltzmann equation, whose analytical solution poses a formidable challenge. An efficient and accurate numerical simulation of the Boltzmann equation is thus highly desirable. In this article, the linearized Boltzmann Bhatnagar-Gross-Krook equation is used to develop a hierarchy of thermal lattice Boltzmann (LB) models based on half-space Gaussian-Hermite (GH) quadrature ranging from low to high algebraic precision, using double distribution functions. Simplified versions of the LB models in the continuum limit are also derived, and are shown to be consistent with existing thermal LB models for noncontinuum heat transfer reported in the literature. Accuracy of the proposed LB hierarchy is assessed by simulating thermal Couette flows for a wide range of Knudsen numbers. Effects of the underlying quadrature schemes (half-space GH vs full-space GH) and continuum-limit simplifications on computational accuracy are also elaborated. The numerical findings in this article provide direct evidence of improved computational capability of the proposed LB models for modeling noncontinuum flows and heat transfer at small length scales.
Linearized lattice Boltzmann method for micro- and nanoscale flow and heat transfer.
Shi, Yong; Yap, Ying Wan; Sader, John E
2015-07-01
Ability to characterize the heat transfer in flowing gases is important for a wide range of applications involving micro- and nanoscale devices. Gas flows away from the continuum limit can be captured using the Boltzmann equation, whose analytical solution poses a formidable challenge. An efficient and accurate numerical simulation of the Boltzmann equation is thus highly desirable. In this article, the linearized Boltzmann Bhatnagar-Gross-Krook equation is used to develop a hierarchy of thermal lattice Boltzmann (LB) models based on half-space Gaussian-Hermite (GH) quadrature ranging from low to high algebraic precision, using double distribution functions. Simplified versions of the LB models in the continuum limit are also derived, and are shown to be consistent with existing thermal LB models for noncontinuum heat transfer reported in the literature. Accuracy of the proposed LB hierarchy is assessed by simulating thermal Couette flows for a wide range of Knudsen numbers. Effects of the underlying quadrature schemes (half-space GH vs full-space GH) and continuum-limit simplifications on computational accuracy are also elaborated. The numerical findings in this article provide direct evidence of improved computational capability of the proposed LB models for modeling noncontinuum flows and heat transfer at small length scales. PMID:26274307
A multiple-relaxation-time lattice Boltzmann method for high-speed compressible flows
NASA Astrophysics Data System (ADS)
Li, Kai; Zhong, Cheng-Wen
2015-05-01
This paper presents a coupling compressible model of the lattice Boltzmann method. In this model, the multiple-relaxation-time lattice Boltzmann scheme is used for the evolution of density distribution functions, whereas the modified single-relaxation-time (SRT) lattice Boltzmann scheme is applied for the evolution of potential energy distribution functions. The governing equations are discretized with the third-order Monotone Upwind Schemes for scalar conservation laws finite volume scheme. The choice of relaxation coefficients is discussed simply. Through the numerical simulations, it is found that compressible flows with strong shocks can be well simulated by present model. The numerical results agree well with the reference results and are better than that of the SRT version. Project supported by the Innovation Fund for Aerospace Science and Technology of China (Grant No. 2009200066) and the Aeronautical Science Fund of China (Grant No. 20111453012).
Comment on ‘A low-uncertainty measurement of the Boltzmann constant’
NASA Astrophysics Data System (ADS)
Macnaughton, Donald B.
2016-02-01
The International Committee for Weights and Measures has projected a major revision of the International System of Units in which all the base units will be defined by fixing the values of certain fundamental constants of nature. To assist, de Podesta et al recently experimentally obtained a precise new estimate of the Boltzmann constant. This estimate is proposed as a basis for the redefinition of the unit of temperature, the kelvin. The present paper reports a reanalysis of de Podesta et al’s data that reveals systematic non-random patterns in the residuals of the key fitted model equation. These patterns violate the assumptions underlying the analysis and thus they raise questions about the validity of de Podesta et al’s estimate of the Boltzmann constant. An approach is discussed to address these issues, which should lead to an accurate estimate of the Boltzmann constant with a lower uncertainty.
Topology optimization in thermal-fluid flow using the lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Yaji, Kentaro; Yamada, Takayuki; Yoshino, Masato; Matsumoto, Toshiro; Izui, Kazuhiro; Nishiwaki, Shinji
2016-02-01
This paper proposes a topology optimization method for thermal-fluid flow problems using the lattice Boltzmann method (LBM). The design sensitivities are derived based on the adjoint lattice Boltzmann method (ALBM), whose basic idea is that the adjoint problem is first formulated using a continuous adjoint approach, and the adjoint problem is then solved using the LBM. In this paper, the discrete velocity Boltzmann equation, in which only the particle velocities are discretized, is introduced to the ALBM to deal with the various boundary conditions in the LBM. The novel sensitivity analysis is applied in two flow channel topology optimization problems: 1) a pressure drop minimization problem, and 2) a heat exchange maximization problem. Several numerical examples are provided to confirm the utility of the proposed method.
Phase segregation via Vlasov-Boltzmann particle dynamics
Bastea, S
1999-01-19
In order to better understand and model the phase segregation of binary fluids we opted for a mesoscopic description that proves to be simplifying both conceptually and computationally. The system that we studied is a mixture of two kinds of particles. All particles interact with each other through strong short-range interactions modeled by hard spheres with the same mass and diameter. There is also a smooth long-range repulsion between particles of different kinds. At low overall densities and weak enough repulsion the natural dynamical description for this system is given in terms of two coupled, energy and momentum conserving Vlasov- Boltzmann equations, making it what we call a dynamical mean-field model. The computational scheme that we used is a combination of direct sim- ulation Monte Carlo (DSMC) and particle-in-the-cell (PIC) evolution, that inherits the efficiency and robustness of these two algorithms. The DSMC is a stochastic algorithm due to Bird that consistently incorporates the as- sumptions behind the Boltzmann equation into the particle dynamics. The method is essentially the following: the physical space is divided into a net- work of cells containing typically tens of particles and the free flow of the particles over a small time interval {Delta}t is followed by representative collisions among pairs of particles sharing the same cell. The typical linear dimension of a cell is a fraction of the mean free path between collisions. The PIC method for integrating the equations of motion was first used to deal with the l/r potential in plasma physics. It takes advantage of the simple form of the Vlasov potential, which is a product in Fourier space, by calculating the densities on a grid through some weighting, then the potentials and forces on the same grid, and finally interpolating the forces at the position of each particle. These two methods can be naturally brought together by replacing the free flow of the DSMC procedure by motion in the
Progress in developing Poisson-Boltzmann equation solvers
Li, Chuan; Li, Lin; Petukh, Marharyta; Alexov, Emil
2013-01-01
This review outlines the recent progress made in developing more accurate and efficient solutions to model electrostatics in systems comprised of bio-macromolecules and nano-objects, the last one referring to objects that do not have biological function themselves but nowadays are frequently used in biophysical and medical approaches in conjunction with bio-macromolecules. The problem of modeling macromolecular electrostatics is reviewed from two different angles: as a mathematical task provided the specific definition of the system to be modeled and as a physical problem aiming to better capture the phenomena occurring in the real experiments. In addition, specific attention is paid to methods to extend the capabilities of the existing solvers to model large systems toward applications of calculations of the electrostatic potential and energies in molecular motors, mitochondria complex, photosynthetic machinery and systems involving large nano-objects. PMID:24199185
Progress in developing Poisson-Boltzmann equation solvers.
Li, Chuan; Li, Lin; Petukh, Marharyta; Alexov, Emil
2013-03-01
This review outlines the recent progress made in developing more accurate and efficient solutions to model electrostatics in systems comprised of bio-macromolecules and nano-objects, the last one referring to objects that do not have biological function themselves but nowadays are frequently used in biophysical and medical approaches in conjunction with bio-macromolecules. The problem of modeling macromolecular electrostatics is reviewed from two different angles: as a mathematical task provided the specific definition of the system to be modeled and as a physical problem aiming to better capture the phenomena occurring in the real experiments. In addition, specific attention is paid to methods to extend the capabilities of the existing solvers to model large systems toward applications of calculations of the electrostatic potential and energies in molecular motors, mitochondria complex, photosynthetic machinery and systems involving large nano-objects. PMID:24199185
Coupling Boltzmann and Navier-Stokes equations by friction
Bourgat, J.F.; Le Tallec, P. |; Tidriri, M.D.
1996-09-01
The aim of this paper is to introduce and validate a coupled Navier-Stokes Boltzman approach for the calculation of hypersonic rarefied flows around maneuvering vehicles. The proposed strategy uses locally a kinetic model in the boundary layer coupled through wall friction forces to a global Navier-Stokes solver. Different numerical experiments illustrate the potentialities of the method. 29 refs., 24 figs.
On kinetic dissipation in collisionless turbulent plasmas
NASA Astrophysics Data System (ADS)
Parashar, Tulasi Nandan
Plasma turbulence is a phenomenon that is present in astrophysical as well as terrestrial plasmas. The earth is embedded in a turbulent plasma, emitting from the sun, called the solar wind. It is important to understand the nature of this plasma in order to understand space weather. A critical unsolved problem is that of the source of dissipation in turbulent plasmas. It is believed to play a central role in the heating of the solar corona which in turn drives the solar wind. The solar wind itself is observed to be highly turbulent and hotter than predicted through adiabatic expansion models. Turbulence and its associated dissipation have been studied extensively through the use of MHD models. However, the solar wind and large regions of the solar corona have very low collisionality, which calls into question the use of simple viscosity and resistivity in most MHD models. A kinetic treatment is needed for a better understanding of turbulent dissipation. This thesis studies the dissipation of collisionless turbulence using direct numerical hybrid simulations of turbulent plasmas. Hybrid simulations use kinetic ions and fluid electrons. Having full kinetic ion physics, the dissipation in these simulations at the ion scales is self consistent and requires no assumptions. We study decaying as well as quasi steady state systems (driven magnetically). Initial studies of the Orszag-Tang vortex [Orszag, JFM, 1979] (which is an initial condition that quickly generates decaying strong turbulence) showed preferential perpendicular heating of protons (with T_perp /T_|| > 1). An energy budget analysis showed that in the turbulent regime, almost all the dissipation occurs through magnetic interactions. We study the energy budget of waves using the k - o spectra (energy in the wavenumber-frequency space). The k - o spectra of this study and subsequent studies of driven turbulent plasmas do not show any significant power in the linear wave modes of the system. This suggests that
Lattice Boltzmann simulations of a viscoelastic shear-thinning fluid
NASA Astrophysics Data System (ADS)
Papenkort, S.; Voigtmann, Th.
2015-07-01
We present a hybrid lattice Boltzmann algorithm for the simulation of flow glass-forming fluids, characterized by slow structural relaxation, at the level of the Navier-Stokes equation. The fluid is described in terms of a nonlinear integral constitutive equation, relating the stress tensor locally to the history of flow. As an application, we present results for an integral nonlinear Maxwell model that combines the effects of (linear) viscoelasticity and (nonlinear) shear thinning. We discuss the transient dynamics of velocities, shear stresses, and normal stress differences in planar pressure-driven channel flow, after switching on (startup) and off (cessation) of the driving pressure. This transient dynamics depends nontrivially on the channel width due to an interplay between hydrodynamic momentum diffusion and slow structural relaxation.
Lattice Boltzmann simulations of a viscoelastic shear-thinning fluid.
Papenkort, S; Voigtmann, Th
2015-07-28
We present a hybrid lattice Boltzmann algorithm for the simulation of flow glass-forming fluids, characterized by slow structural relaxation, at the level of the Navier-Stokes equation. The fluid is described in terms of a nonlinear integral constitutive equation, relating the stress tensor locally to the history of flow. As an application, we present results for an integral nonlinear Maxwell model that combines the effects of (linear) viscoelasticity and (nonlinear) shear thinning. We discuss the transient dynamics of velocities, shear stresses, and normal stress differences in planar pressure-driven channel flow, after switching on (startup) and off (cessation) of the driving pressure. This transient dynamics depends nontrivially on the channel width due to an interplay between hydrodynamic momentum diffusion and slow structural relaxation. PMID:26233150
Lattice Boltzmann Method for Spacecraft Propellant Slosh Simulation
NASA Technical Reports Server (NTRS)
Orr, Jeb S.; Powers, Joseph F.; Yang, Hong Q.
2015-01-01
A scalable computational approach to the simulation of propellant tank sloshing dynamics in microgravity is presented. In this work, we use the lattice Boltzmann equation (LBE) to approximate the behavior of two-phase, single-component isothermal flows at very low Bond numbers. Through the use of a non-ideal gas equation of state and a modified multiple relaxation time (MRT) collision operator, the proposed method can simulate thermodynamically consistent phase transitions at temperatures and density ratios consistent with typical spacecraft cryogenic propellants, for example, liquid oxygen. Determination of the tank forces and moments relies upon the global momentum conservation of the fluid domain, and a parametric wall wetting model allows tuning of the free surface contact angle. Development of the interface is implicit and no interface tracking approach is required. Numerical examples illustrate the method's application to predicting bulk fluid motion including lateral propellant slosh in low-g conditions.
Lattice Boltzmann Method for Spacecraft Propellant Slosh Simulation
NASA Technical Reports Server (NTRS)
Orr, Jeb S.; Powers, Joseph F.; Yang, Hong Q
2015-01-01
A scalable computational approach to the simulation of propellant tank sloshing dynamics in microgravity is presented. In this work, we use the lattice Boltzmann equation (LBE) to approximate the behavior of two-phase, single-component isothermal flows at very low Bond numbers. Through the use of a non-ideal gas equation of state and a modified multiple relaxation time (MRT) collision operator, the proposed method can simulate thermodynamically consistent phase transitions at temperatures and density ratios consistent with typical spacecraft cryogenic propellants, for example, liquid oxygen. Determination of the tank forces and moments is based upon a novel approach that relies on the global momentum conservation of the closed fluid domain, and a parametric wall wetting model allows tuning of the free surface contact angle. Development of the interface is implicit and no interface tracking approach is required. A numerical example illustrates the method's application to prediction of bulk fluid behavior during a spacecraft ullage settling maneuver.
Beyond Poisson-Boltzmann: Numerical Sampling of Charge Density Fluctuations.
Poitevin, Frédéric; Delarue, Marc; Orland, Henri
2016-07-01
We present a method aimed at sampling charge density fluctuations in Coulomb systems. The derivation follows from a functional integral representation of the partition function in terms of charge density fluctuations. Starting from the mean-field solution given by the Poisson-Boltzmann equation, an original approach is proposed to numerically sample fluctuations around it, through the propagation of a Langevin-like stochastic partial differential equation (SPDE). The diffusion tensor of the SPDE can be chosen so as to avoid the numerical complexity linked to long-range Coulomb interactions, effectively rendering the theory completely local. A finite-volume implementation of the SPDE is described, and the approach is illustrated with preliminary results on the study of a system made of two like-charge ions immersed in a bath of counterions. PMID:27075231
Del Sarto, D.; Califano, F.; Pegoraro, F.
2005-01-01
The nonlinear phase of a magnetic field line reconnection instability in a collisionless two-dimensional cold plasma is investigated in the Hall dominated regime, described by the electron-magnetohydrodynamic equations, which corresponds to the frequency range of whistler waves. It is found that the regular pattern of current density layers that forms in the initial nonlinear phase of the reconnection instability is destroyed by the onset of a Kelvin-Helmholtz-type instability and the formation of current jets that develop into vortex rings. These processes can be interpreted in terms of a Hasegawa-Mima-type regime inside the magnetic island and lead to the creation of magnetic vortices. It is shown that electron compressibility, which is related to charge separation, tends to stabilize these processes.
Noncoplanar magnetic fields at collisionless shocks - A test of a new approach
NASA Technical Reports Server (NTRS)
Gosling, J. T.; Winske, D.; Thomsen, M. F.
1988-01-01
The expressions derived by Jones and Ellison (1987) (JE) for the spatial integral of the noncoplanar field component in a fast-mode collisionless shock and for the magnitudes of the magnetic-field rotation and potential drop difference are evaluated on the basis of ISEE 1 and 2 observations in the earth bow shock and numerical simulations using the hybrid code of Leroy et al. (1981 and 1982). The data and simulation results are presented graphically and shown to be consistent with the JE hypothesis, that the rotation and drop difference are due to unequal electron and ion masses. The rotations and potential drop differences predicted by the JE equations, however, are found to be accurate only for subcritical shocks at low Mach numbers; the underestimation of the values for hypercritical shocks is attributed to neglect of the ion current from reflected gyrating ions.
NASA Technical Reports Server (NTRS)
Matsuda, Y.
1974-01-01
A low-noise plasma simulation model is developed and applied to a series of linear and nonlinear problems associated with electrostatic wave propagation in a one-dimensional, collisionless, Maxwellian plasma, in the absence of magnetic field. It is demonstrated that use of the hybrid simulation model allows economical studies to be carried out in both the linear and nonlinear regimes with better quantitative results, for comparable computing time, than can be obtained by conventional particle simulation models, or direct solution of the Vlasov equation. The characteristics of the hybrid simulation model itself are first investigated, and it is shown to be capable of verifying the theoretical linear dispersion relation at wave energy levels as low as .000001 of the plasma thermal energy. Having established the validity of the hybrid simulation model, it is then used to study the nonlinear dynamics of monochromatic wave, sideband instability due to trapped particles, and satellite growth.
On kinetic instabilities in collisionless ultra-relativistic streaming cold electron-proton plasma
NASA Astrophysics Data System (ADS)
Vereshchagin, Gregory; Chechetkin, V. M.; Dyachenko, V. F.; Ginzburg, S. L.; Fimin, N. N.; Ruffini, Remo; Siutsou, Ivan
2009-05-01
We consider cold collisionless electron-proton plasma, moving in the vacuum with large bulk Lorentz factor. In order to describe such a system numerical integration of Vlasov-Maxwell equations is performed by a 3-dimensional Eulerian code. The plasma is shown to experience kinetic instabilities, leading to generation of stochastic electro-magnetic fields. The motion of electrons and protons randomizes in these stochastic fields, leading to a thermal-like spread in the distribution function of electrons and protons. Both electrons and protons, and the electromagnetic field reach equipartition on a timescale L/c, where L is the characteristic size in the problem, c is the speed of light. We discuss particle distributions in coordinate and momentum space, as well as the structure of electromagnetic fields. Consequences of the considered phenomenon for astrophysical sources are quite general and include, in particular, gamma-ray bursts.
Accuracy of non-Newtonian Lattice Boltzmann simulations
NASA Astrophysics Data System (ADS)
Conrad, Daniel; Schneider, Andreas; Böhle, Martin
2015-11-01
This work deals with the accuracy of non-Newtonian Lattice Boltzmann simulations. Previous work for Newtonian fluids indicate that, depending on the numerical value of the dimensionless collision frequency Ω, additional artificial viscosity is introduced, which negatively influences the accuracy. Since the non-Newtonian fluid behavior is incorporated through appropriate modeling of the dimensionless collision frequency, a Ω dependent error EΩ is introduced and its influence on the overall error is investigated. Here, simulations with the SRT and the MRT model are carried out for power-law fluids in order to numerically investigate the accuracy of non-Newtonian Lattice Boltzmann simulations. A goal of this accuracy analysis is to derive a recommendation for an optimal choice of the time step size and the simulation Mach number, respectively. For the non-Newtonian case, an error estimate for EΩ in the form of a functional is derived on the basis of a series expansion of the Lattice Boltzmann equation. This functional can be solved analytically for the case of the Hagen-Poiseuille channel flow of non-Newtonian fluids. With the help of the error functional, the prediction of the global error minimum of the velocity field is excellent in regions where the EΩ error is the dominant source of error. With an optimal simulation Mach number, the simulation is about one order of magnitude more accurate. Additionally, for both collision models a detailed study of the convergence behavior of the method in the non-Newtonian case is conducted. The results show that the simulation Mach number has a major impact on the convergence rate and second order accuracy is not preserved for every choice of the simulation Mach number.
NASA Astrophysics Data System (ADS)
Gulati, Mamta; Saini, Tarun Deep
2016-04-01
The short-wave asymptotics (WKB) of spiral density waves in self-gravitating stellar discs is well suited for the study of the dynamics of tightly-wound wavepackets. But the textbook WKB theory is not well adapted to the study of the linear eigenmodes in a collisionless self-gravitating disc because of the transcendental nature of the dispersion relation. We present a modified WKB theory of spiral density waves, for collisionless discs in the epicyclic limit, in which the perturbed gravitational potential is related to the perturbed surface density by the Poisson integral in Kalnaj's logarithmic spiral form. An integral equation is obtained for the surface density perturbation, which is seen to also reduce to the standard WKB dispersion relation. Although our formulation is general and applies to all discs, we present our analysis only for nearly Keplerian, low-mass, self-gravitating discs revolving around massive central objects, and derive an integral equation governing the slow precessional modes of such discs. For a prograde disc, the integral kernel turns out be real and symmetric, implying that all slow modes are stable. We apply the slow mode integral equation to two unperturbed disc profiles, the Jalali-Tremaine annular discs, and the Kuzmin disc. We determine eigenvalues and eigenfunctions for both m = 1 and m = 2 slow modes for these profiles and discuss their properties. Our results compare well with those of Jalali-Tremaine.
NASA Astrophysics Data System (ADS)
Gulati, Mamta; Saini, Tarun Deep
2016-07-01
The short-wave asymptotics (WKB) of spiral density waves in self-gravitating stellar discs is well suited for the study of the dynamics of tightly-wound wavepackets. But the textbook WKB theory is not well adapted to the study of the linear eigenmodes in a collisionless self-gravitating disc because of the transcendental nature of the dispersion relation. We present a modified WKB theory of spiral density waves, for collisionless discs in the epicyclic limit, in which the perturbed gravitational potential is related to the perturbed surface density by the Poisson integral in Kalnaj's logarithmic spiral form. An integral equation is obtained for the surface density perturbation, which is seen to also reduce to the standard WKB dispersion relation. Although our formulation is general and applies to all discs, we present our analysis only for nearly Keplerian, low-mass, self-gravitating discs revolving around massive central objects, and derive an integral equation governing the slow precessional modes of such discs. For a prograde disc, the integral kernel turns out be real and symmetric, implying that all slow modes are stable. We apply the slow mode integral equation to two unperturbed disc profiles, the Jalali-Tremaine annular discs, and the Kuzmin disc. We determine eigenvalues and eigenfunctions for both m = 1 and m = 2 slow modes for these profiles and discuss their properties. Our results compare well with those of Jalali-Tremaine.
The ion polytropic coefficient in a collisionless sheath containing hot ions
NASA Astrophysics Data System (ADS)
Lin, Binbin; Xiang, Nong; Ou, Jing
2016-08-01
The fluid approach has been widely used to study plasma sheath dynamics. For a sheath containing hot ions whose temperature is greater than the electron's, how to truncate the fluid hierarchy chain equations while retaining to the fullest extent of the kinetic effects is always a difficult problem. In this paper, a one-dimensional, collisionless sheath containing hot ions is studied via particle-in-cell simulations. By analyzing the ion energy equation and taking the kinetic effects into account, we have shown that the ion polytropic coefficient in the vicinity of the sheath edge is approximately constant so that the state equation with the modified polytropic coefficient can be used to close the hierarchy chain of the ion fluid equations. The value of the polytropic coefficient strongly depends on the hot ion temperature and its concentration in the plasma. The semi-analytical model is given to interpret the simulation results. As an application, the kinetic effects on the ion saturation current density in the probe theory are discussed.
Non-Boltzmann Modeling for Air Shock-Layer Radiation at Lunar-Return Conditions
NASA Technical Reports Server (NTRS)
Johnston, Christopher O.; Hollis, Brian R.; Sutton, Kenneth
2008-01-01
This paper investigates the non-Boltzmann modeling of the radiating atomic and molecular electronic states present in lunar-return shock-layers. The Master Equation is derived for a general atom or molecule while accounting for a variety of excitation and de-excitation mechanisms. A new set of electronic-impact excitation rates is compiled for N, O, and N2+, which are the main radiating species for most lunar-return shock-layers. Based on these new rates, a novel approach of curve-fitting the non-Boltzmann populations of the radiating atomic and molecular states is developed. This new approach provides a simple and accurate method for calculating the atomic and molecular non-Boltzmann populations while avoiding the matrix inversion procedure required for the detailed solution of the Master Equation. The radiative flux values predicted by the present detailed non-Boltzmann model and the approximate curve-fitting approach are shown to agree within 5% for the Fire 1634 s case.
Revised lattice Boltzmann model for traffic flow with equilibrium traffic pressure
NASA Astrophysics Data System (ADS)
Shi, Wei; Lu, Wei-Zhen; Xue, Yu; He, Hong-Di
2016-02-01
A revised lattice Boltzmann model concerning the equilibrium traffic pressure is proposed in this study to tackle the phase transition phenomena of traffic flow system. The traditional lattice Boltzmann model has limitation to investigate the complex traffic phase transitions due to its difficulty for modeling the equilibrium velocity distribution. Concerning this drawback, the equilibrium traffic pressure is taken into account to derive the equilibrium velocity distribution in the revised lattice Boltzmann model. In the proposed model, a three-dimensional velocity-space is assumed to determine the equilibrium velocity distribution functions and an alternative, new derivative approach is introduced to deduct the macroscopic equations with the first-order accuracy level from the lattice Boltzmann model. Based on the linear stability theory, the stability conditions of the corresponding macroscopic equations can be obtained. The outputs indicate that the stability curve is divided into three regions, i.e., the stable region, the neutral stability region, and the unstable region. In the stable region, small disturbance appears in the initial uniform flow and will vanish after long term evolution, while in the unstable region, the disturbance will be enlarged and finally leads to the traffic system entering the congested state. In the neutral stability region, small disturbance does not vanish with time and maintains its amplitude in the traffic system. Conclusively, the stability of traffic system is found to be enhanced as the equilibrium traffic pressure increases. Finally, the numerical outputs of the proposed model are found to be consistent with the recognized, theoretical results.
Nonlinear collisionless damping of Weibel turbulence in relativistic blast waves
NASA Astrophysics Data System (ADS)
Lemoine, Martin
2015-01-01
The Weibel/filamentation instability is known to play a key role in the physics of weakly magnetized collisionless shock waves. From the point of view of high energy astrophysics, this instability also plays a crucial role because its development in the shock precursor populates the downstream with a small-scale magneto-static turbulence which shapes the acceleration and radiative processes of suprathermal particles. The present work discusses the physics of the dissipation of this Weibel-generated turbulence downstream of relativistic collisionless shock waves. It calculates explicitly the first-order nonlinear terms associated to the diffusive nature of the particle trajectories. These corrections are found to systematically increase the damping rate, assuming that the scattering length remains larger than the coherence length of the magnetic fluctuations. The relevance of such corrections is discussed in a broader astrophysical perspective, in particular regarding the physics of the external relativistic shock wave of a gamma-ray burst.
Evolution of velocity dispersion along cold collisionless flows
Banik, Nilanjan; Sikivie, Pierre
2015-11-17
We found that the infall of cold dark matter onto a galaxy produces cold collisionless flows and caustics in its halo. If a signal is found in the cavity detector of dark matter axions, the flows will be readily apparent as peaks in the energy spectrum of photons from axion conversion, allowing the densities, velocity vectors and velocity dispersions of the flows to be determined. We also discuss the evolution of velocity dispersion along cold collisionless flows in one and two dimensions. A technique is presented for obtaining the leading behaviour of the velocity dispersion near caustics. The results are used to derive an upper limit on the energy dispersion of the Big Flow from the sharpness of its nearby caustic, and a prediction for the dispersions in its velocity components.
Magnetized collisionless shock studies using high velocity plasmoids
NASA Astrophysics Data System (ADS)
Weber, Thomas; Intrator, Thomas; Gao, Kevin
2012-10-01
Magnetized collisionless shocks are ubiquitous throughout the cosmos and are observed to accelerate particles to relativistic velocities, amplify magnetic fields, transport energy, and create non-thermal distributions. They exhibit transitional scale lengths much shorter than the collisional mean free path and are mediated by collective interactions rather than Coulomb collisions. The Magnetized Shock Experiment (MSX) leverages advances in Field Reversed Configuration (FRC) plasmoid formation and acceleration to produce highly supersonic and super-Alfvènic supercritical shocks with pre-existing magnetic field at perpendicular, parallel or oblique angles to the direction of propagation. Adjustable shock speed, density, and magnetic field provide unique access to a range of parameter space relevant to a variety of naturally occurring shocks. This effort examines experimentally, analytically, and numerically the physics of collisionless shock formation, structure, and kinetic effects in a laboratory setting and draw comparisons between experimental data and astronomical observations. Approved for Public Release: LA-UR-12-22886
Collisionless "thermalization" in the sheath of an argon discharge
NASA Astrophysics Data System (ADS)
Coulette, David; Manfredi, Giovanni
2015-04-01
We performed kinetic Vlasov simulations of the plasma-wall transition for a low-pressure argon discharge without external magnetic fields, using the same plasma parameters as in the experiments of Claire et al. [Phys. Plasmas 13, 062103 (2006)]. Experimentally, it was found that the ion velocity distribution function is highly asymmetric in the presheath, but, surprisingly, becomes again close to Maxwellian inside the sheath. Here, we show that this "thermalization" can be explained by purely collisionless effects that are akin to the velocity bunching phenomenon observed in charged particles beams. Such collisionless thermalization is also observed in the presheath region close to the sheath entrance, although it is much weaker there and in practice probably swamped by collisional processes (standard or enhanced by instabilities).
The role of microturbulence on collisionless reconnection. [in magnetospheric plasmas
NASA Technical Reports Server (NTRS)
Papadopoulos, K.
1980-01-01
The linear, non-linear and anomalous transport properties associated with various microinstabilities driven by cross field currents in reconnecting geometries are reviewed. An assessment of their role in collisionless tearing based on analytic theory, computer simulations and experimental evidence, supports the dominant role of lower hybrid waves. The relevance of microturbulence on macroscopic stationary and time dependent models of merging is presented. It is concluded that a fluid-numerical simulation approach that includes (at each space and time step) the effects of anomalous transport in a self consistent manner, similar to the one used for laboratory collisionless shocks, represents the best method for studying and modeling the details of the reconnection process.
Collisionless shock experiments with lasers and observation of Weibel instabilitiesa)
NASA Astrophysics Data System (ADS)
Park, H.-S.; Huntington, C. M.; Fiuza, F.; Drake, R. P.; Froula, D. H.; Gregori, G.; Koenig, M.; Kugland, N. L.; Kuranz, C. C.; Lamb, D. Q.; Levy, M. C.; Li, C. K.; Meinecke, J.; Morita, T.; Petrasso, R. D.; Pollock, B. B.; Remington, B. A.; Rinderknecht, H. G.; Rosenberg, M.; Ross, J. S.; Ryutov, D. D.; Sakawa, Y.; Spitkovsky, A.; Takabe, H.; Turnbull, D. P.; Tzeferacos, P.; Weber, S. V.; Zylstra, A. B.
2015-05-01
Astrophysical collisionless shocks are common in the universe, occurring in supernova remnants, gamma ray bursts, and protostellar jets. They appear in colliding plasma flows when the mean free path for ion-ion collisions is much larger than the system size. It is believed that such shocks could be mediated via the electromagnetic Weibel instability in astrophysical environments without pre-existing magnetic fields. Here, we present laboratory experiments using high-power lasers and investigate the dynamics of high-Mach-number collisionless shock formation in two interpenetrating plasma streams. Our recent proton-probe experiments on Omega show the characteristic filamentary structures of the Weibel instability that are electromagnetic in nature with an inferred magnetization level as high as ˜1% [C. M. Huntington et al., "Observation of magnetic field generation via the weibel instability in interpenetrating plasma flows," Nat. Phys. 11, 173-176 (2015)]. These results imply that electromagnetic instabilities are significant in the interaction of astrophysical conditions.
Simulations of Relativistic Collisionless Shocks: Shock Structure and Particle Acceleration
Spitkovsky, Anatoly; /KIPAC, Menlo Park
2006-04-10
We discuss 3D simulations of relativistic collisionless shocks in electron-positron pair plasmas using the particle-in-cell (PIC) method. The shock structure is mainly controlled by the shock's magnetization (''sigma'' parameter). We demonstrate how the structure of the shock varies as a function of sigma for perpendicular shocks. At low magnetizations the shock is mediated mainly by the Weibel instability which generates transient magnetic fields that can exceed the initial field. At larger magnetizations the shock is dominated by magnetic reflections. We demonstrate where the transition occurs and argue that it is impossible to have very low magnetization collisionless shocks in nature (in more than one spatial dimension). We further discuss the acceleration properties of these shocks, and show that higher magnetization perpendicular shocks do not efficiently accelerate nonthermal particles in 3D. Among other astrophysical applications, this may pose a restriction on the structure and composition of gamma-ray bursts and pulsar wind outflows.
Evolution of velocity dispersion along cold collisionless flows
NASA Astrophysics Data System (ADS)
Banik, Nilanjan; Sikivie, Pierre
2016-05-01
The infall of cold dark matter onto a galaxy produces cold collisionless flows and caustics in its halo. If a signal is found in the cavity detector of dark matter axions, the flows will be readily apparent as peaks in the energy spectrum of photons from axion conversion, allowing the densities, velocity vectors and velocity dispersions of the flows to be determined. We discuss the evolution of velocity dispersion along cold collisionless flows in one and two dimensions. A technique is presented for obtaining the leading behavior of the velocity dispersion near caustics. The results are used to derive an upper limit on the energy dispersion of the big flow from the sharpness of its nearby caustic and a prediction for the dispersions in its velocity components.