Spectral asymptotics and metastability for the linear relaxation Boltzmann equation

TL;DR

This paper analyzes the spectral properties and metastability of the linear relaxation Boltzmann equation in a semiclassical setting, providing sharp asymptotics and long-term behavior estimates.

Contribution

It introduces a novel semiclassical analysis of the collision operator as a pseudo-differential operator in class S^{1/2}, leading to precise spectral and metastability results.

Findings

Sharp spectral asymptotics for the small spectrum at low temperature

Quantitative estimates on return to equilibrium

Metastability characterized by superpositions of exponentials

Abstract

We consider the linear relaxation Boltzmann equation in a semiclassical framework. We construct a family of sharp quasimodes for the associated operator which yields sharp spectral asymptotics for its small spectrum in the low temperature regime. We deduce some information on the long time behavior of the solutions with a sharp estimate on the return to equilibrium as well as a quantitative metastability result. The main novelty is that the collision operator is a pseudo-differential operator in the critical class S^1/2 and that its action on the gaussian quasimodes yields a superposition of exponentials.