Exact solutions of the nonlinear Boltzmann equation

A review is given of research activities since 1976 on the nonlinear Boltzmann equation and related equations of Boltzmann type, in which several rediscoveries have been made and several conjectures have been disproved. Subjects are (i) the BKW solution of the Boltzmann equation for Maxwell molecules, first discovered by Krupp in 1967, and the Krook-Wu conjecture concerning the universal significance of the BKW solution for the large(v, t) behavior of the velocity distribution functionf (v, t); (ii) moment equations and polynomial expansions off (v, t); (iii) model Boltzmann equation for a spatially uniform system of very hard particles, that can be solved in closed form for general initial conditions; (iv) for Maxwell and non-Maxwell-type molecules there exist solutions of the nonlinear Boltzmann equation with algebraic decrease at ; connections with nonuniqueness and violation of conservation laws; (v) conjectured super-H-theorem and the BKW solution; (vi) exactly soluble one-dimensional Boltzmann equation with spatial dependence.Reference due to C. Cercignani.     

Exact solution of the linearized Boltzmann equation in p dimensions

The solution of the transport in p-dimensional space is given for a system of convex boundary surfaces with arbitrary shape and finite diameter. It satisfies homogeneous or non-homogeneous Dirichlet boundary conditions. The equation for the determination of the point spectrum of the Boltzmann operator is also given.     

Exact solution of the non-linear Boltzmann equation for Maxwell models

The exact solution of the Boltzmann equation, obtained recently for Maxwell molecules, holds for various models with arbitrary dimensionality. Furthermore, simple non-linear equations are derived for the ordinary and the Sonine moments, which can be solved subsequentially.     

The linear Boltzmann equation properties and solutions

This work presents results on the transport equation obtained recently. Properties and distribution functions obeying the Boltzmann equation in one and many-dimensional spaces are derived and discussed. New polynomials have been found that under certain conditions represent solutions of the linear transport equation. A number of structural and spectral theorems have been demonstrated permitting a better understanding of the equation. It is shown that the streaming operator, $\text{Ω·Δ + σ(v)Σ}{}_{\mathrm{tot}}\text{(v)}$, maps a class of functions {$\text{;ψ(}\text{x, v}\text{)}$} of two variables onto another class of functions, {$\text{ψ(}\text{x}\text{)}$}, depending only on one variable. The distribution functions obtained here satisfy rigorously homogeneous or inhomogeneous Dirichlet conditions on the boundary surface of the convex system for any order of the polynomial representation. This is obtained by using the new polynomials which are characterised by particular structural properties. In many-dimensional cases the polynomials become operators with tempered distributional character. Numerical evaluations are given in the one dimensional case for both isotropic and anisotropic scattering. An application of the theory is also given for the heterogeneous system of plane geometry. This paper is organized in three parts. Part A gives an introduction to the present method and indicates the way leading from the original to the linearised Boltzmann equation. Other solution methods are comparatively discussed. Part B is dealing with the one-dimensional equation and the eigenvalues and eigenfunctions are found for various physical conditions. Degenerate kernels have been used throughout. Both the critical and non-critical problems have been solved and results are presented in form of graphs and tables. Part C proceeds to the examination of some problems in spaces of dimension p > 1. The main advantages of the structural approach are that (i) the solution is found in an elementary way completely analytically. (ii) The boundary conditions are rigorously satisfied. (iii) The heterogeneous system is very easily solved. (iv) The eigenvalues are found algebraically.     

On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics

Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data fromL ²(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics.     

Exact solutions to the time-dependent Lorentz gas Boltzmann Equation: The approach to hydrodynamics

New exact solutions to the time-dependent Lorentz gas Boltzmann equation are presented for two classes of nonequilibrium initial value problems: thedecay of localized disturbances and theresponse to applied electric fields. These exact results are used to gain some insight into the crossover of the nonequilibrium state from the early-timekinetic regime to the late-timehydrodynamic regime.     

Exact moment solution of the Boltzmann equation for uniform shear flow

The Ikenberry-Truesdell exact solution to the Boltzmann equation for Maxwell molecules is revisited. This solution refers to a state characterized by a linear profile of the velocity flow and spatially uniform density and temperature. The solution is extended to include explicit expressions for the fourth-degree moments. It is shown that if the shear rate is larger than a certain critical value, the fourth-degree moments do not reach stationary values, even when the temperature is kept constant. The explicit shear-rate dependence of the moments below this critical value are obtained.     

Exact solution of the Boltzmann equation in the homogeneous color conductivity problem

An exact solution of the Boltzmann equation for a binary mixture of colored Maxwell molecules is found. The solution corresponds to a nonequilibrium homogeneous steady state created by a nonconservative external force. Explicit expressions for the moments of the distribution function are obtained. By using information theory, an approximate velocity distribution function is constructed, which is exact in the limits of small and large field strengths. Comparison is made between the exact energy flux and the one obtained from the information theory distribution.     

A perturbation solution to generalized linear Boltzmann equation

A perturbation method to solve the generalized linear Boltzmann equation is presented for the calculation of dynamic conductivity of a normal metal at low temperatures. The method is applied to the ordinary Boltzmann equation. The following three cases are considered: i) when only electron-impurity interactions are dominant, ii) when the metal may be considered as pure plasma, and iii) when both of the interactions are present. The expressions for conductivity are presented in the binary collision approximation. The connected diagram expansion method is extended for the direct calculation of the conductivity. Finally the method is extended for the calculation of conductivity by the correlation function formula.     

Exact solutions of the Boltzmann equation in the VHP model with removal interaction

The linear and nonlinear Boltzmann equation for very hard particles (VHP) is considered in the case when the collision between two particles may lead not only to elastic scattering, but also to a removal event with the disappearance of the molecules. The extended transport equation is solved for arbitrary initial distributions. The computations are carried out explicitly for a special class of initial distributions and for various removal rates. The results are demonstrated graphically. Finally, source terms fulfilling physically reasonable conditions are introduced into the VHP model, and the time-dependent particle number is calculated.     

Exact solution to the one-dimensional Dirac equation of linear potential

In this paper the one-dimensional Dirac equation with linear potential has been solved by the method of canonical transformation. The bound-state wavefunctions and the corresponding energy spectrum have been obtained for all bound states.     

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.     

A study on the inclusion of body forces in the lattice Boltzmann BGK equation to recover steady-state hydrodynamics

When the lattice Boltzmann (LB) method is used to solve hydrodynamic problems containing a body force term varying in space and/or time, its modelling at the mesoscopic scale must be verified in terms of consistency in order to avoid the appearance of non-hydrodynamic error terms at the macroscopic scale. In the present work it is shown that the modelling of spatially varying steady body force terms in the LB equation must be different from the time-dependent case, when a steady-state flow solution is sought. For that, the Chapman-Enskog analysis is used to derive the LB body force model for the LB BGK equations in a steady-state flow problem. The theoretical findings are supported by numerical tests performed on two different 2D steady-state laminar flows driven by spatially varying body forces with known analytical solutions.     

The Boltzmann equation

An abstract form of the spatially non-homogeneous Boltzmann equation is derived which includes the usual, more concrete form for any kind of potential, hard or soft, with finite cutoff. It is assumed that the corresponding gas is confined to a bounded domain by some sort of reflection law. The problem then considered is the corresponding initial-boundary value problem, locally in time.Two proofs of existence are given. Both are constructive, and the first, at least, provides two sequences, one converging to the solution from above, the other from below, thus producing, at the same time as existence, approximations to the solution and error bounds for the approximation.The solution is found within a space of functions bounded by a multiple of a Maxwellian, and, in this space, uniqueness is also proved.Research supported, in part, by the National Research Council of Canada (NRC A8560)     

Stationary Boltzmann equation for a degenerate gas in a slab: Boundary value problem and hydrodynamics

The boundary value problem for the stationary Boltzmann equation for a model gas in a plane slab is solved in full generality. The asymptotic behavior as the size of the slab goes to infinity is studied via a Chapman-Enskog expansion.