# Derivation of a Non-autonomous Linear Boltzmann Equation from a Heterogeneous Rayleigh Gas

## TL;DR

This paper rigorously derives a non-autonomous linear Boltzmann equation for a tagged particle in a heterogeneous Rayleigh gas, valid over long times, by connecting microscopic dynamics with kinetic theory.

## Contribution

It introduces a derivation of a non-autonomous linear Boltzmann equation from deterministic particle dynamics in a heterogeneous setting, extending previous models.

## Key findings

- Validity of the Boltzmann equation for long times
- Comparison between empirical and Boltzmann dynamics
- Conditions on initial distributions for derivation

## Abstract

A linear Boltzmann equation with nonautonomous collision operator is rigorously derived in the Boltzmann-Grad limit for the deterministic dynamics of a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with heterogeneously distributed background particles, which do not interact among each other. The validity of the linear Boltzmann equation holds for arbitrary long times under moderate assumptions on spatial continuity and higher moments of the initial distributions of the tagged particle and the heterogeneous, non-equilibrium distribution of the background. The empiric particle dynamics are compared to the Boltzmann dynamics using evolution semigroups for Kolmogorov equations of associated probability measures on collision histories.