Boltzmann-Grad limit for the inelastic Lorentz gas: Part I. Existence, uniqueness, and rigorous derivation via weak convergence

TL;DR

This paper rigorously derives the inelastic linear Boltzmann equation from a dissipative Lorentz gas in arbitrary dimensions, establishing existence, uniqueness, and convergence of solutions in the Boltzmann-Grad limit.

Contribution

It provides the first rigorous derivation of the inelastic Boltzmann equation from a microscopic particle system with inelastic collisions, including convergence rates.

Findings

Existence and uniqueness of weak solutions to the inelastic Boltzmann equation.

Weak-* convergence of microscopic dynamics to the Boltzmann equation.

Construction of strong solutions via series representation.

Abstract

In this paper we provide a rigorous derivation of the inelastic linear Boltzmann equation, in the Boltzmann-Grad limit, from a dissipative, random, Lorentz gas in arbitrary dimensions d $\geq$ 2. Specifically, we consider a microscopic particle system where scatterers are randomly distributed according to a Poisson process, and a tagged light particle undergoes inelastic collisions with the scatterers following a reflection law characterized by a fixed restitution coefficient. We establish the existence and uniqueness of weak solutions to the inelastic linear Boltzmann equation within the class of non-negative Radon measures, assuming that the initial data has a finite exponential moment. We first show that the forward dynamics of the dissipative particle system is globally defined almost surely and then prove the weak$-*$ convergence of the microscopic solution towards the weak solutions of the inelastic linear Boltzmann equation, providing an explicit rate of convergence. Furthermore, under the same initial data assumptions, we prove the existence of strong solutions to the inelastic linear Boltzmann equation, constructed via a series representation of the solutions.