Usual Boltzmann Equation Research Articles

On Lattice Boltzmann Methods based on vector-kinetic models for hyperbolic partial differential equations

In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit discretisation of vector-kinetic equation (VKE), we also consider LBE derived by semi-implicit discretisation of VKE and compare the relaxation factors of both. We study the properties such as H-inequality, total variation boundedness and positivity of both the LBEs, and infer that the LBE due to semi-implicit discretisation naturally satisfies all the properties while the LBE due to explicit discretisation requires more restrictive condition on relaxation factor compared to the usual condition obtained from Chapman-Enskog expansion. We also derive the macroscopic finite difference form of the LBEs, and utilise it to establish the consistency of LBEs with the hyperbolic system. Further, we extend this LBM framework to hyperbolic conservation laws with source terms, such that there is no spurious numerical convection due to imbalance between convection and source terms. We also present a D2Q9 model that allows upwinding even along diagonal directions in addition to the usual upwinding along coordinate directions. The different aspects of the results are validated numerically on standard benchmark problems.

The Temperature Dependent Boltzmann Equation with Reservoir of Radiation Photons

Usually, the temperature is introduced by the Boltzmann distribution function under the local equilibrium assumption. In this manuscript, we propose a temperature dependent Boltzmann equation, where the temperature is taken into account by the collision of transport particles and the particles in the reservoir, and we assume the reservoir remain in an equilibrium or local equilibrium state with an external temperature. Different from the usual Boltzmann equation, our temperature dependent Boltzmann equation obtains a new damping force which originates from the collision between transport particle and reservoir and is contrary to the external force applied to the system, meanwhile the relaxation time is also affected by the derivative of the damping force. In our study, the reservoir are consist of the radiations of photons, so the Compton scattering happens between the transport particle and the photon. Finally, we illustrate them by the example of one-dimensional transport process, the damping force and the non-equilibrium temperature defined by us are shown.

Non-Hermite Spinor Boltzmann Equation and Its Hermitization

It is known that the Wigner distribution function in the quantum Boltzmann equation is not positive elsewhere, which causes trouble for its application to the mesoscopic transport problem. Similarly, if we extend the quantum Boltzmann equation to the spinor Boltzmann equation, which includes the spin freedom in the usual Wigner distribution, and the spinor matrix distribution function therein is not Hermite, we must make this non-Hermite spinor Boltzmann equation Hermitization. So in this paper, we propose a spinor Boltzmann equation with Hermite distribution function and then obtain the equations of continuity satisfied by the charge density, charge current, spin accumulation, and spin current. The numerical results for these physical quantities are illustrated by an example of spin-polarized transport scattered by impurities in a system of spintronics. Their differences compared with the usual spinor Boltzmann equation are shown.

Derivation of the zeroth retardation order Boltzmann equation from nonstandard space–time Wigner quantum kinetics

A derivation of the equation of motion satisfied by a prototype of the forward space–time Wigner distribution, that incorporates a Fourier transformation of truncated time shifts as compared with the standard Wigner distribution, is carried out in the case of a many-body system within the formalism of nonequilibrium Green’s functions of quantum field theory. In particular, the single-particle equation found in López and Soler (2016) is recovered when the self-energy interaction effects are neglected. Then, a sinc-corrected quantum kinetic Boltzmann equation is derived that rules the evolution of the standard Wigner transform, which is shown to be reduced to the usual Boltzmann equation when only binary elastic collisions are considered.

Calculation of momentum distribution function of a non-thermal fermionic dark matter

The most widely studied scenario in dark matter phenomenology is the thermal WIMP scenario. Inspite of numerous efforts to detect WIMP, till now we have no direct evidence for it. A possible explanation for this non-observation of dark matter could be because of its very feeble interaction strength and hence, failing to thermalise with the rest of the cosmic soup. In other words, the dark matter might be of non-thermal origin where the relic density is obtained by the so-called freeze-in mechanism. Furthermore, if this non-thermal dark matter is itself produced substantially from the decay of another non-thermal mother particle, then their distribution functions may differ in both size and shape from the usual equilibrium distribution function. In this work, we have studied such a non-thermal (fermionic) dark matter scenario in the light of a new type of U(1)B−L model. The U(1)B−L model is interesting, since, besides being anomaly free, it can give rise to neutrino mass by Type II see-saw mechanism. Moreover, as we will show, it can accommodate a non-thermal fermionic dark matter as well. Starting from the collision terms, we have calculated the momentum distribution function for the dark matter by solving a coupled system of Boltzmann equations. We then used it to calculate the final relic abundance, as well as other relevant physical quantities. We have also compared our result with that obtained from solving the usual Boltzmann (or rate) equations directly in terms of comoving number density, Y. Our findings suggest that the latter approximation is valid only in cases where the system under study is close to equilibrium, and hence should be used with caution.

Microscopic description of quantum Lorentz gas and extension of the Boltzmann equation to entire space-time scale.

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.

Classical non-Markovian Boltzmann equation

The modeling of particle transport involves anomalous diffusion, ⟨x2(t) ⟩ ∝ tα with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.

On the method of activating Brownian motion for application of the lattice Boltzmann method to magnetic particle dispersions

We discuss the feasibility of several methods for activating the Brownian motion of dispersed particles in the lattice Boltzmann method by considering aggregation phenomena in a dispersion composed of magnetic particles. The methods treated in the present study are a method based on fluctuation hydrodynamics (HF-type), in which the fluctuation term is combined with the basic equations of the usual lattice Boltzmann equations, and a method based on Brownian dynamics (hybrid-type), in which the random forces of the Brownian dynamics are simply added to the equations of motion of magnetic particles. The validity of the present results is discussed by a comparison with the results of Monte Carlo simulations. The HF-type method gives rise to a relatively emphasized activation of the Brownian motion in contrast to the hybrid-type method, which yields a weaker activation of such motion. In order to improve these Brownian-motion-activating techniques, the velocity-scaling method was introduced into both techniques. The modified HF-type method shows significantly good agreement with Monte Carlo results, implying that the translational and rotational Brownian motion of magnetic particles is activated with sufficient accuracy. In contrast, the implementation of the hybrid-type method is found to be unable to reproduce aggregate structures within the bounds that may be considered physically reasonable.

格子ボルツマン法の磁性粒子分散系への適用のためのブラウン運動誘起法の検討(流体工学,流体機械)

We have discussed the feasibility of several methods for activating the Brownian motion of dispersed particles in the lattice Boltzmann method by considering aggregation phenomena in a dispersion composed of magnetic particles. The methods to be treated in the present study are a technique based on the fluctuation hydrodynamics (flucuation-hydro-type), in which the fluctuation term is combined into the basic equations of the usual lattice Boltzmann equations, and another technique based on the Brownian dynamics method (hybrid-type), in which the random forces in the Brownian dynamics are simply added to the equations of motion of magnetic particles. The validity of the present results has been discussed by comparing those with the results of Monte Carlo simulations. The fluctuation-hydro-type method gives rise to a relatively emphasized activation of the Brownian motion, and, in contrast, the hybrid-type method yields a weaker activation of such motion. In order to improve these Brownian-motion-activating techniques, the velocity scaling method has been introduced to both these techniques. The modified fluctuation-hydro-type method has been seen to give rise to significantly good agreement with Monte Carlo results. That is, in this method, the translational and rotational Brownian motion of magnetic particles can be activated with sufficient accuracy. In contrast, the hybrid-type method with such a modification cannot reproduce aggregate structures of magnetic particles on a physically accurate level.

Computer experiments on the relaxation of collisionless plasmas

The relaxation of a collisionless plasma is not regulated by the usual collisional Boltzmann equation and itsrelated H-theorem. The interactions are long range and the collisions are not instantaneous. A directsimulation approach is presented for measuring by computer experiment what the relaxeddistribution is. The conclusion of the analysis is that the relaxed distribution includes botha low energy component that is well described using the usual Boltzmann distribution anda high energy tail described using a power law. The results of the simulation study arecompared directly with the model recently proposed by Kaniadakis (2002 Phys. Rev. E 66056125). The observed cumulative distribution function is well reproduced by thetheory.

Landau–Pomeranchuk–Migdal effect in a quark–gluon plasma and the Boltzmann equation

We show how the Landau-Pomeranchuk-Migdal effect on photon production rates in a quark-gluon plasma can be derived via the usual Boltzmann equation. To do this, we first derive the electromagnetic polarization tensor using linear response theory, and then formulate the Boltzmann equation including the collisions mediated by soft gluon exchanges. We then identify the resulting expression for the production rate with that obtained by the field-theoretic formalism recently proposed by Arnold, Moore and Yaffe. To illustrate the LPM effect we solve the Boltzmann equation in the diffusion approximation.

Mechanism for carrier velocity saturation in pure organic crystals

We present a possible mechanism for the saturation of the velocity of photoinjected charge carriers in pure organic crystals, the analysis having been motivated by observations in pentacene. The mechanism is based on the strong wave vector dependence of the damping coefficient appearing in the appropriate effective mass equation. Our primary goal is to explore this mechanism in general terms rather than to apply it to a specific system such as pentacene. Our analysis is based on a Fokker-Planck equation treatment. We indicate how the damping coeffcient is related to relaxation times in usual Boltzmann equations and how it can be computed from Fermi golden rule transition rates.

Generalized T-Matrix Approximation in Quantum Kinetic Equations

The correct description of relaxation processes in strongly correlated plasmas requires quantumkinetic equations which go beyond the usual Boltzm ann equation. Here, the Kadanoff– Baym equations (KBE) in the frame of real-time Green’s functions have a high degree of generality. We present a possibility to solve the KBE in T–matrix approximation by solving the Lippmann–Schwinger equation using a separable potential ansatz. Results for the T–matrix are shown for a hydrogen plasma for several screening parameters.

Stochastic dynamics of correlations in quantum field theory: From the Schwinger-Dyson to Boltzmann-Langevin equation

The aim of this paper is twofold: to probe the statistical mechanical properties of interacting quantum fields, and to provide a field theoretical justification for a stochastic source term in the Boltzmann equation. We start with the formulation of quantum field theory in terms of the set of Schwinger-Dyson equations for the correlation functions, which we describe by a closed-time-path master $(n=\ensuremath{\infty}\mathrm{PI})$ effective action. When the hierarchy is simply truncated to a certain order, one obtains the usual closed system of correlation functions up to that order, and from the $n\mathrm{PI}$ effective action, a set of time-reversal invariant equations of motion. (This is the Dyson equation, the quantum field theoretical parallel of the collisionless Boltzmann equation.) But when the effect of the higher order correlation functions is included through a causal factorization condition (such as the molecular chaos assumption in Boltzmann's theory) called slaving, the dynamics of the lower order correlations shows dissipative features, as familiar in the usual (dissipative yet noiseless) Boltzmann equation, the field-theoretical version of which being the dissipative Dyson equations. We show that a fluctuation-dissipation relation should exist for such effectively open systems, and use this fact to show that a stochastic term, which explicitly introduces quantum fluctuations in the lower order correlation functions, necessarily accompanies the dissipative term. This leads to a stochastic Dyson equation, which is the quantum field theoretic parallel of the classical Boltzmann-Langevin equation, encompassing both the dissipative and stochastic dynamics of correlation functions.

Quantum kinetics and thermalization in a particle bath model.

We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and non-Markovian quantum kinetics. There are two different cases that are studied in detail: (i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and (ii) a stable renormalized "particle" state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. For the case of quasiparticles, the exact occupation number asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes whereas in the stable particle case, the distribution of particles does not thermalize with the bath. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian, and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.

Quantum kinetic equation and cosmic pair annihilation

Pair annihilation of heavy stable particle that occurs in the early universe is investigated, and quantum kinetic equation for the momentum distribution of the annihilating particle is derived, using the influence functional method. A bosonic field theory model is used to describe the pair annihilation in the presence of decay product particles making up a thermal environment. A crossing symmetric Hartree approximation that determines self-consistently the equilibrium distribution is developed for an otherwise intractable theory. The time evolution equation and its Markovian approximation is derived, to give a generalized Boltzmann equation including off-shell effects. The narrow width approximation to an energy integral in this equation gives the usual Boltzmann equation in a thermal bath of light particles. The off-shell effect is a correction to the Boltzmann equation at high temperatures, but is dominant at low temperatures. The effect changes the equilibrium distribution from the familiar $1/(e^{\omega_{k}/T} - 1)$ to a modified one given by a Gibbs formula. Integrated over momenta, the particle number density becomes roughly of order (coupling) $\times \sqrt{T/M}\cdot T^{3}$ at low temperatures for the S-wave annihilation. The relic mass density in the present universe is insensitive to the coupling strength in a large range of the mass and the coupling parameters, and scales with the WIMP mass as (\approx 6 \times 10^{4} eV cm^{-3} (M/GeV)^{4/3}). The bound from the closure density gives an upper WIMP mass bound roughly of order 1 $GeV$ in the present model.

Quantum kinetics and thermalization in an exactly solvable model

We study the dynamics of relaxation and thermalization in an exactly solvable model with the goal of understanding the effects of off-shell processes. The focus is to compare the exact evolution of the distribution function with different approximations to the relaxational dynamics: Boltzmann, non-Markovian and Markovian quantum kinetics. The time evolution of the distribution function is evaluated exactly using two methods: time evolution of an initially prepared density matrix and by solving the Heisenberg equations of motion. There are two different cases that are studied in detail: i) no stable particle states below threshold of the bath and a quasiparticle resonance above it and ii) a stable discrete exact `particle' state below threshold. For the case of quasiparticles in the continuum (resonances) the exact quasiparticle distribution asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes. In the case ii), the distribution of particles does not thermalize with the bath. We study the kinetics of thermalization and relaxation by deriving a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian and Boltzmann approximations are compared to the exact result of the model. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.

Transverse Transport Processes of Charged Particle Systems in a Strong Magnetic Field (II) Some Discussions about the Boltzmann Equation

In this article, it is pointed out that in a strong magnetic field the usual Boltzmann Equation with Lorentz force term is inconsistent with the fundamental requirements of the general theory of non-equilibrium statistical mechanics. Furthermore, if a collision term corresponding to particles in different gyro-orbits is used, we find that all the transport coefficients are exactly zero.General expressions for the transport coefficients in classical systems are then obtained by introducing the macroscopic velocities[3]. A comparison between our results and those of the usual Boltzmann Equation indicates that in the case of short range force, the lowest order terms are the same and the results of the Boltzmann Equation become increasingly worse as more higher order terms are kept.

Electron transport in semiconductors in the presence of impact ionization

The development of impact ionization in semiconductors has been considered. A transport equation for the electron distribution function in the presence of impact ionization has been derived. It has been found that the collison integral of this equation is nonlinear with respect to the distribution function. The relation between the solutions of this equation and the usual Boltzmann equation have been determined. A Monte Carlo method for numerical calculations in the presence of impact ionization has been developed. The results of numerical calculations carried out using a simplified model of indium antimonide are presented.

Kinetic theory of hydrodynamic flows. I. The generalized normal solution method and its application to the drag on a sphere

We consider the flow of a dilute gas around a macroscopic heavy object. The state of the gas is described by an extended Boltzmann equation where the interactions between the gas molecules and the object are taken into account in computing the rate of change of the distribution function of the gas. We then show that the extended Boltzmann is equivalent to the usual Boltzmann equation, supplemented by boundary conditions imposed on the distribution function at the surface of the object. The remainder of the paper is devoted to a study of the solution of the extended Boltzmann equation in the case that the mean free path of a gas molecule is small compared to some characteristic dimension of the macroscopic object. We show that the Chapman-Enskog normal solution of the ordinary Boltzmann equation is not in general a solution of the extended equation near the surface of the object and must be supplemented by a boundary layer term. We then introduce a projection operator method which allows us to decompose the solution of the extended equation into a normal solution part and a boundary layer part when the gas flow is sufficiently slow. As a specific example of the method we consider the flow around a sphere, and derive the Stokes-Boussinesq form for the frequency-dependent force on the sphere for arbitrary slip coefficient. This derivation is the first one that starts from the Boltzmann equation for a general dilute gas and incorporates the effect of the boundary layer on the drag force.