Spinor Boltzmann Equation with Two Momenta at the Fermi Level

Based on the formalism of Keldysh's nonequilibrium Green function, we establish a two momenta spinor Boltzmann equation for longitudinal scalar distribution function and transverse vector distribution function. The longitudinal charge currents, transverse spin currents and the continuity equations satisfied by them are then studied, it indicates that both the charge currents and spin currents decay oscillately along with position, which is due to the momenta integral over the Fermi surface. We also compare our charge currents and spin currents with the corresponding results of one momentum spinor Boltzmann equation, the differences are obvious.     

Exact Eternal Solutions of the Boltzmann Equation

We construct two families of self-similar solutions of the Boltzmann equation in an explicit form. They turn out to be eternal and positive. They do not possess finite energy. Asymptotic properties of the solutions are also studied.     

The Relativistic Boltzmann Equation and Two Times

We discuss a covariant relativistic Boltzmann equation which describes the evolution of a system of particles in spacetime evolving with a universal invariant parameter τ. The observed time t of Einstein and Maxwell, in the presence of interaction, is not necessarily a monotonic function of τ. If t(τ) increases with τ, the worldline may be associated with a normal particle, but if it is decreasing in τ, it is observed in the laboratory as an antiparticle. This paper discusses the implications for entropy evolution in this relativistic framework. It is shown that if an ensemble of particles and antiparticles, converge in a region of pair annihilation, the entropy of the antiparticle beam may decreaase in time.     

Estimating the Solutions of the Boltzmann Equation

We discuss some possible estimates of the solutions of the Boltzmann equation, which might permit a progress in the theory of existence of weak solutions.     

Global Weak Solutions of the Boltzmann Equation

A new definition of the concept of weak solution of the nonlinear Boltzmann equation is introduced. It is proved that, without any truncation on the collision kernel, the Boltzmann equation in the one-dimensional case has a global weak solution in this sense. Global conservation of energy follows.     

Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions

We consider some questions related to the self-similar asymptotics in the kinetic theory of both elastic and inelastic particles. In the second case we have in mind granular materials, when the model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of the colliding particles. We first discuss how to define the n-dimensional (n = 1,2,...) inelastic Maxwell model and its connection with the more basic Boltzmann equation for inelastic hard spheres. Then we consider both elastic and inelastic Maxwell models from a unified viewpoint. We prove the existence of (positive in the inelastic case) self-similar solutions with finite energy and investigate their role in large time asymptotics. It is proved that a recent conjecture by Ernst and Brito devoted to high energy tails for inelastic Maxwell particles is true for a certain class of initial data which includes Maxwellians. We also prove that the self-similar asymptotics for high energies is typical for some classes of solutions of the classical (elastic) Boltzmann equation for Maxwell molecules. New classes of (not necessarily positive) finite-energy eternal solutions of this equation are also studied.     

On the Convergence of the Boltzmann Equation for Semiconductors Toward the Energy Transport Model

The diffusion limit of the Boltzmann equation of semiconductors is analyzed. The dominant collisions are the elastic collisions on one hand and the electron–electron collisions with the Pauli exclusion terms on the other hand. Under a nondegeneracy hypothesis on the distribution function, a lower bound of the entropy dissipation rate of the leading term of the Boltzmann kernel for semiconductors in terms of a distance to the space of Fermi–Dirac functions is proved. This estimate and a mean compactness lemma are used to prove the convergence of the solution of the Boltzmann equation to a solution of the energy transport model.     

Half-Space Problems for the Boltzmann Equation: A Survey

This paper reviews recent mathematical results on the half-space problem for the Boltzmann equation. The case of a phase transition is discussed in detail.     

应用于非线性热传导方程的格子玻尔兹曼方法

将格子玻尔兹曼方法应用于非线性热传导方程的求解,详细推导一种新的Lattice Boltzmann模型,并给出新方法所对应的多尺度方案和宏观量形式.导热系数与温度之间满足多项式函数关系,计算中模拟了不同的参数情况,并与线性热传导方程的理论解进行比较.新的Lattice Boltzmann方法展现出极大的灵活性和普适性,具有很好的应用前景.     

Weak Solutions of the Boltzmann Equation Without Angle Cutoff

The definition of the concept of weak solution of the nonlinear Boltzmann equation, recently introduced by the author, is used to prove that, without any cutoff in the collision kernel, the Boltzmann equation for Maxwell molecules in the one-dimensional case has a global weak solution in this sense. Global conservation of energy follows.     

Self-Similar Solutions of the Boltzmann Equation for Non-Maxwell Molecules

We show that the method previously used by the authors to obtain self-similar, eternal solutions of the space-homogeneous Boltzmann equation for Maxwell molecules yields different results when extended to other power-law potentials (including hard spheres). In particular, self-similar solutions cease to exist for a positive time for hard potentials. In the case of soft potentials, the solutions exist for all potive times, but are not eternal.     

Self-Similar Solutions of the Boltzmann Equation and Their Applications

We consider a class of solutions of the Boltzmann equation with infinite energy. Using the Fourier-transformed Boltzmann equation, we prove the existence of a wide class of solutions of this kind. They fall into subclasses, labelled by a parameter a, and are shown to be asymptotic (in a very precise sense) to the self-similar one with the same value of a (and the same mass). Specializing to the case of a Maxwell-isotropic cross section, we give evidence to the effect that the only self-similar closed form solutions are the BKW mode and the two solutions recently found by the authors. All the self-similar solutions discussed in this paper are eternal, i.e., they exist for –<t<, which shows that a recent conjecture cannot be extended to solutions with infinite energy. Eternal solutions with finite moments of all orders, and different from a Maxwellian, are also studied. It is shown that these solutions cannot be positive. Moreover all such solutions (partly negative) must be asymptotically (for large negative times) close to the exact eternal solution of BKW type.     

On the Rate of Entropy Production for the Boltzmann Equation

We show that there exists a wide class of distribution functions (with moments of any order as close to their equilibrium values as we like) which can provide an abnormally low rate of entropy production. The result is valid for the Boltzmann equation with any cross section (|V|, ) satisfying a mild restriction. The functions are constructed in an explicit form and we discuss some applications of our results.     

Global Existence Proof for Relativistic Boltzmann Equation with Hard Interactions

By combining the DiPerna and Lions techniques for the nonrelativistic Boltzmann equation and the Dudyński and Ekiel-Jeżewska device of the causality of the relativistic Boltzmann equation, it is shown that there exists a global mild solution to the Cauchy problem for the relativistic Boltzmann equation with the assumptions of the relativistic scattering cross section including some relativistic hard interactions and the initial data satisfying finite mass, energy and entropy. This is in fact an extension of the result of Dudyński and Ekiel-Jeżewska to the case of the relativistic Boltzmann equation with hard interactions. This work was supported by NSFC 10271121 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the Ministry of Education of China, and sponsored by joint grants of NSFC 10511120278/10611120371 and RFBR 04-02-39026.     

On the Boltzmann Equation for Fermi–Dirac Particles with Very Soft Potentials: Averaging Compactness of Weak Solutions

The paper considers macroscopic behavior of a Fermi–Dirac particle system. We prove the L ¹-compactness of velocity averages of weak solutions of the Boltzmann equation for Fermi–Dirac particles in a periodic box with the collision kernel b(cos θ)|ρ−ρ _*|^γ, which corresponds to very soft potentials: −5 < γ ≤ −3 with a weak angular cutoff: ∫₀ ^π b(cos θ)sin ³θ dθ < ∞. Our proof for the averaging compactness is based on the entropy inequality, Hausdorff–Young inequality, the L ^∞-bounds of the solutions, and a specific property of the value-range of the exponent γ. Once such an averaging compactness is proven, the proof of the existence of weak solutions will be relatively easy.     

A Weak Formulation of the Boltzmann Equation Based on the Fourier Transform

In this article we present an alternative formulation of the spatially homogeneous Boltzmann equation. Rewriting the weak form of the equation with shifted test functions and using Fourier techniques, it turns out that the transformed problem contains only a three-fold integral. Explicit formulas for the transformed collision kernel are presented in the case of VHS models for hard and soft potentials. For isotropic Maxwellian molecules, a classical result by Bobylev is recovered, too.     

Nonlinear conductivity and entropy in the two-body Boltzmann gas

We find exact solutions of the two-particle Boltzmann equation for hard disks and hard spheres diffusing isothermally in an external field. The corresponding transport coefficient, one-particle current divided by field strength, decreases as the field increases. This nonlinear dependence of the current on the field and the corresponding nonlinear dependence of the distribution function on the current are compared to the predictions of single-time information theory. Our exact steady-state distribution function, from Boltzmann's equation, is quite different from the approximate information-theory analog. The approximate theory badly underestimates the nonlinear decrease of entropy with current. The exact two-particle solutions we find here should prove useful in testing and improving theories of steady-state and transient distribution functions far from equilibrium.     

Solutions to the Boltzmann Equation in the Boussinesq Regime

We consider a gas in a horizontal slab in which the top and bottom walls are kept at different temperatures. The system is described by the Boltzmann equation (BE) with Maxwellian boundary conditions specifying the wall temperatures. We study the behavior of the system when the Knudsen number is small and the temperature difference between the walls as well as the velocity field is of order , while the gravitational force is of order ². We prove that there exists a solution to the BE for which is near a global Maxwellian, and whose moments are close, up to order ², to the density, velocity and temperature obtained from the smooth solution of the Oberbeck–Boussinesq equations assumed to exist for .     

The Development of the Green's Function for the Boltzmann Equation

We review the particle-like and wave-like property of the Boltzmann equation. This property leads to a sequence of developments on the mathematical theory of the Green's function for the Boltzmann equation.