Exponential Convergence to the Maxwell Distribution For Some Class of Boltzmann Equations

TL;DR

This paper proves that solutions to a class of nonlinear Boltzmann equations describing particle gases in a thermal medium converge exponentially fast to Maxwellian distributions, demonstrating rapid return to thermal equilibrium.

Contribution

It establishes exponential convergence rates to Maxwellian equilibrium for a class of Boltzmann equations in a specific function space, extending understanding of thermalization.

Findings

Solutions converge exponentially fast to Maxwellian distributions.

Convergence holds for smooth initial conditions in weighted L^1 space.

Results apply to dilute gases in thermal media.

Abstract

We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3).$ Special solutions of these equations, called "Maxwellians," are spatially homogenous static Maxwell velocity distributions at the temperature of the medium. We prove that, for dilute gases, the solutions corresponding to smooth initial conditions in a weighted $L^{1}$-space converge to a Maxwellian in $L^{1},$ exponentially fast in time.