Classical Boltzmann Equation Research Articles

Discrete Velocity Models for Mixtures Without Nonphysical Collision Invariants

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that DVMs can also have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. For binary mixtures also the concept of supernormal DVMs was introduced, meaning that in addition to the DVM being normal, the restriction of the DVM to any single species also is normal. Here we introduce generalizations of this concept to DVMs for multicomponent mixtures. We also present some general algorithms for constructing such models and give some concrete examples of such constructions. One of our main results is that for any given number of species, and any given rational mass ratios we can construct a supernormal DVM. The DVMs are constructed in such a way that for half-space problems, as the Milne and Kramers problems, but also nonlinear ones, we obtain similar structures as for the classical discrete Boltzmann equation for one species, and therefore we can apply obtained results for the classical Boltzmann equation.

Spectrum Analysis of Some Kinetic Equations

We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with \({\gamma\geqq-2}\). As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate \({t^{-5/4}}\)) as \({t\to\infty}\) to that of the compressible Navier–Stokes equations for initial data around an equilibrium state.

Spectrum Structure and Behaviors of the Vlasov--Maxwell--Boltzmann Systems

The spectrum structures and behaviors of the Vlasov--Maxwell--Boltzmann (VMB) systems for both two species and one species are studied in this paper. The analysis shows the effect of the Lorentz force induced by the electromagnetic field leads to some different structure of spectrum from the classical Boltzmann equation and the closely related VPB system. And the significant difference between the two-species VMB model and one-species VMB model are given. The structure in high frequency illustrates the hyperbolic structure of the Maxwell equation. Furthermore, the long time behaviors and the optimal convergence rates to the equilibrium of the VMB systems for both two species and one species are established based on the spectrum analysis, and in particular the phenomena of the electric field dominating and magnetic field dominating are observed for the one-species VMB system.

Classical Boltzmann equation and high-temperature QED

The equivalence between thermal field theory and the Boltzmann transport equation is investigated at higher orders in the context of quantum electrodynamics. We compare the contributions obtained from the collisionless transport equation with the high temperature limit of the one-loop thermal Green's function. Our approach employs the representation of the thermal Green's functions in terms of forward scattering amplitudes. The general structure of these amplitudes clearly indicates that the physics described by the leading high temperature limit of quantum electrodynamics can be obtained from the Boltzman transport equation. We also present some explicit examples of this interesting equivalence.

Explicit equilibria in bilinear kinetic models for socio-economic interactions

Bilinear kinetic models of Maxwell-Boltzmann type are often used to model socio-economic systems composed by agents that undergo binary interactions, which in general obey to some conserva- tion law. Then, the details of the microscopic interaction are such that an equilibrium solution emerges. At dierence with the classical Boltzmann equation for elastic gas particles, the analytic form of the equilibria is known only in some particular case. In the present note, we present and discuss the situ- ations in which this equilibrium prole can be explicitly found. Applications to dissipative gases and to wealth redistribution models are presented.

Asymptotic-Preserving Exponential Methods for the Quantum Boltzmann Equation with High-Order Accuracy

In this paper we develop high order asymptotic preserving methods for the spatially inhomogeneous quantum Boltzmann equation. We follow the work in Li and Pareschi (J Comput Phys 259:402–420, 2014) where asymptotic preserving exponential Runge–Kutta methods for the classical inhomogeneous Boltzmann equation were constructed. A major difficulty here is related to the non Gaussian steady states characterizing the quantum kinetic behavior. We show that the proposed schemes achieve high-order accuracy uniformly in time for all Planck constants ranging from classical regime to quantum regime, and all Knudsen number ranging from kinetic regime to fluid regime. Computational results are presented for both Bose gas and Fermi gas.

Validity of the Boltzmann equation to describe Donnan equilibrium at the membrane–solution interface

To describe Donnan equilibrium at the membrane–solution interface, the simplest approach uses the classical Boltzmann equation, based on a mean-field description of ions as ideal point charges, in combination with the assumption of fully overlapped electrical double layers in the membrane pores. We test the Boltzmann equation by measurement of the equilibrium counterion and co-ion concentration in densely charged membranes equilibrated with various NaCl solutions (0.01–3M). To obtain a good fit of data it was found necessary to express the membrane charge and ion concentration per volume of aqueous solution phase in the membrane, and to include a volume-exclusion term in the Boltzmann relation. A discrepancy between theory and experiment data is found at low external NaCl concentrations. Similar deviations from the Donnan model have been reported for over half a century, but do not yet have a convincing explanation. Agreement between experiment data and theory at low external NaCl concentrations is obtained when we model the desorption experiment taking into account the role of H+ and OH− ions in closing the charge balance, and postulating the presence in the membrane of a tiny amount of fixed groups with a charge opposite to overall fixed membrane charge.

A Milne problem from a Bose condensate with excitations

This paper deals with a half-space linearized problem for the distribution function of the excitations in a Bose gas close to equilibrium. Existence and uniqueness of the solution, as well as its asymptotic properties are proven for a given energy flow. The problem differs from the ones for the classical Boltzmann and related equations, where the hydrodynamic mass flow along the half-line is constant. Here it is no more constant. Instead we use the energy flow which is constant, but no more hydrodynamic.

Generalized Boltzmann solution for non-equilibrium flows and the computation of flowfields of binary gas mixture

Hypersonic flows about space vehicles produce flowfields in thermodynamic non-equilibrium with the local Knudsen numbers Kn which may lie in all the three regimes: continuum, transition and rarefied. Continuum flows can be modeled accurately by solving the Navier–Stokes (NS) equations; however, the flows in transition and rarefied regimes require a kinetic approach such as the direct simulation Monte Carlo (DSMC) method or the solution of the Boltzmann equation. The Boltzmann equation and the general solution approach, using the splitting method, will be introduced in this paper. Details of the method used for solving both the classical Boltzmann equation (CBE) and the generalized Boltzmann equation (GBE) are also provided. The gas mixture discussed in this paper may consist of both monoatomic and diatomic gases. In particular, the method is applied to simulate two of the three primary constituents of air (N2, O2, and Ar) in a binary mixture at 1:1 density ratio at Mach 2 and 5, with gases in translational, rotational and vibrational non-equilibrium.

The Boltzmann equation from quantum field theory

We show from first principles the emergence of classical Boltzmann equations from relativistic nonequilibrium quantum field theory as described by the Kadanoff–Baym equations. Our method applies to a generic quantum field, coupled to a collection of background fields and sources, in a homogeneous and isotropic spacetime. The analysis is based on analytical solutions to the full Kadanoff–Baym equations, using the WKB approximation. This is in contrast to previous derivations of kinetic equations that rely on similar physical assumptions, but obtain approximate equations of motion from a gradient expansion in momentum space. We show that the system follows a generalized Boltzmann equation whenever the WKB approximation holds. The generalized Boltzmann equation, which includes off-shell transport, is valid far from equilibrium and in a time dependent background, such as the expanding universe.

Inhomogeneous Boltzmann equations: Distance, asymptotics and comparison of the classical and quantum cases

The notion of distance between a global Maxwellian function and an arbitrary solution f (with the same total density ρ at the fixed moment t) of Boltzmann equation is introduced. In this way we essentially generalize the important Kullback–Leibler distance, which was used before. Namely, we generalize it for the spatially inhomogeneous case. An extremal problem to find a solution of the Boltzmann equation, such that dist{M,f} is minimal in the class of solutions with the fixed values of energy and of n moments, is solved. The cases of the classical and quantum (for Fermi and Bose particles) Boltzmann equations are studied and compared. The asymptotics and stability of solutions of the Boltzmann equations are also considered.

Direct simulation Monte Carlo method for cold-atom dynamics: Classical Boltzmann equation in the quantum collision regime

In this paper we develop a direct simulation Monte Carlo (DSMC) method for simulating highly nonequilibrium dynamics of nearly degenerate ultra-cold gases. We show that our method can simulate the high-energy collision of two thermal clouds in the regime observed in experiments [Thomas, et al. Phys. Rev. Lett. \textbf{93}, 173201 (2004)], which requires the inclusion of beyond s-wave scattering. We also consider the long-time dynamics of this system, demonstrating that this would be a practical experimental scenario for testing the Boltzmann equation and studying rethermalization.

A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation

We present a class of asymptotic-preserving (AP) schemes for the nonhomogeneous Fokker–Planck–Landau (nFPL) equation. Filbet and Jin [16] designed a class of AP schemes for the classical Boltzmann equation, by penalization with the BGK operator, so they become efficient in the fluid dynamic regime. We generalize their idea to the nFPL equation, with a different penalization operator, the Fokker–Planck operator that can be inverted by the conjugate-gradient method. We compare the effects of different penalization operators, and conclude that the Fokker–Planck (FP) operator is a good choice. Such schemes overcome the stiffness of the collision operator in the fluid regime, and can capture the fluid dynamic limit without numerically resolving the small Knudsen number. Numerical experiments demonstrate that the schemes possess the AP property for general initial data, with numerical accuracy uniformly in the Knudsen number.

Finite density aspects of leptogenesis

Leptogenesis takes place in the early universe at high temperatures and densities and a deviation from equilibrium in the decay of heavy Majorana neutrinos is a fundamental requirement for the generation of the asymmetry. The equations, commonly used for its description, are largely based on classical Boltzmann equations (BEs) while the source of CP-violation is a quantum interference phenomenon. In view of this clash, it is desirable to study such processes in terms of non-equilibrium quantum field theory. On the other hand, it is simpler to solve BEs rather than the corresponding quantum field theoretical ones. Therefore, we derive modified BEs from first principles in the Kadanoff-Baym (KB) formalism. The results, found for a simple toy model, can be applied to popular phenomenological scenarios by analogy. This approach uncovers structural differences of the corrected equations and leads to different results for the form of the finite density contributions to the CP-violating parameter. In the case of degenerate heavy neutrino masses, corresponding to the popular scenario of resonant leptogenesis, it allows to explicitly distinguish between regimes where BEs are applicable or inapplicable.

Wash-out in N2-dominated leptogenesis

We study the wash-out of a cosmological baryon asymmetry produced via leptogenesis by subsequent interactions. Therefore we focus on a scenario in which a lepton asymmetry is established in the out-of-equilibrium decays of the next-to-lightest right-handed neutrino. We apply the full classical Boltzmann equations without the assumption of kinetic equilibrium and including all quantum statistical factors to calculate the wash-out of the lepton asymmetry by interactions of the lightest right-handed state. We include scattering processes with top quarks in our analysis. This is of particular interest since the wash-out is enhanced by scatterings and the use of mode equations with quantum statistical distribution functions. In this way we provide a restriction on the parameter space for this scenario.

Dipole oscillations in fermionic mixtures

We study dipole oscillations in a general fermionic mixture: starting from the Boltzmann equation, we classify the different solutions in the parameter space through the number of real eigenvalues of the small oscillations matrix. We discuss how this number can be computed using the Sturm algorithm and its relation with the properties of the Laplace transform of the experimental quantities. After considering two components in harmonic potentials having different trapping frequencies, we study dipole oscillations in three-component mixtures. Explicit computations are done for realistic experimental setups using the classical Boltzmann equation without intra-species interactions. A brief discussion of the application of this classification to general collective oscillations is also presented.

Discrete velocity models of the Boltzmann equation and conservation laws

We consider in this paper the general problem of construction and classification of normal, i.e. without spurious invariants, discrete velocity models (DVMs)of the classical (elastic) Boltzmann equation. We explain in detail how this problem can be solved and present a complete classification of (i.e. we present all distinct) normal plane DVMswith relatively small number $n$ of velocities ($n\leq 10$). Some results for models with larger number of velocities are also presented.

Quantum linear Boltzmann equation with finite intercollision time

Inconsistencies are pointed out in the usual quantum versions of the classical linear Boltzmann equation constructed for a quantized test particle in a gas. These are related to the incorrect formal treatment of momentum decoherence. We prove that ideal collisions with the molecules would result in complete momentum decoherence, the persistence of coherence is only due to the finite intercollision time. A corresponding quantum linear Boltzmann equation is proposed.

Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space

We study the hypocoercivity property for some kinetic equations in the whole space and obtain the optimal convergence rates of solutions to the equilibrium state in some function spaces. The analysis relies on the basic energy method and the compensating function introduced by Kawashima to the classical Boltzmann equation and developed by Glassey and Strauss in the relativistic setting. It is also motivated by the recent work (Duan et al., 2008 [8]) on the Boltzmann equation by combining the spectrum analysis and energy method. The advantage of the method introduced in this paper is that it can be applied to some complicated system whose detailed spectrum is not known. In fact, only some estimates through the Fourier transform on the conservative transport operator and the dissipation of the linearized operator on the subspace orthogonal to the collision invariants are needed.

Electron–phonon interaction in nanowires: A Monte Carlo study of the effect of the field

The femtosecond dynamics of highly non-equilibrium, confined carriers is analyzed within a Monte Carlo approach. The physical process considered corresponds to a locally excited or injected into a semiconductor nanowire distribution of heated carriers, which evolve under the action of an applied electric field. The carriers are cooled down by dissipation processes caused by phonons. The process is described by a quantum-kinetic equation which generalizes the classical Boltzmann equation with respect to two classical assumptions, namely for temporal and spatial locality of the carrier–phonon interaction. We investigate the effect of the field on the electron–phonon interaction—the intra-collisional field effect (ICFE). A Monte Carlo method for simulation of the considered process has been utilized. Simulation results for carrier evolution in a GaAs nanowire are obtained and analyzed for phenomena related to the ICFE.