The Derivation of the Linear Boltzmann Equation from a Rayleigh Gas Particle Model

TL;DR

This paper rigorously derives a linear Boltzmann equation from a deterministic particle model of a Rayleigh gas, demonstrating its validity over long times under certain initial conditions.

Contribution

It provides a rigorous derivation of the linear Boltzmann equation from particle dynamics in a Rayleigh gas under Boltzmann-Grad scaling, including long-time validity.

Findings

Derivation of the linear Boltzmann equation from particle dynamics.

Validation of the equation's long-time applicability.

Use of Kolmogorov equations for convergence proof.

Abstract

A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using Kolmogorov equations for associated probability measures on collision histories.