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

TL;DR

This paper introduces a generalized distance measure between solutions and Maxwellian functions for inhomogeneous Boltzmann equations, compares classical and quantum cases, and analyzes solution asymptotics and stability.

Contribution

It generalizes the Kullback-Leibler distance for inhomogeneous Boltzmann equations and solves an extremal problem for minimal distance solutions.

Findings

Generalized distance measure for inhomogeneous Boltzmann equations.

Solved extremal problem for solutions with fixed moments.

Compared classical and quantum Boltzmann equations and analyzed their asymptotics.

Abstract

The notion of distance between a global Maxwellian function and an arbitrary solution $f$ (with the same total density $\rho$ 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.