Derivation of kinetic and diffusion equations from a hard-sphere Rayleigh gas using collision trees and semigroups

TL;DR

This paper revisits the derivation of kinetic and diffusion equations from a hard-sphere Rayleigh gas, employing collision trees and semigroup methods to connect microscopic dynamics with macroscopic behavior.

Contribution

It introduces a semigroup approach to rigorously derive Boltzmann and diffusion equations from deterministic hard-sphere dynamics in the Boltzmann-Grad limit.

Findings

Convergence of particle dynamics to Boltzmann-type equations

Derivation of diffusion equation from microscopic model

Application of collision trees and semigroup methods

Abstract

We will revisit the classical questions of understanding the statistics of various deterministic dynamics of $N$ hard spheres of diameter $\varepsilon$ with random initial data in the Boltzmann-Grad scaling as $\varepsilon$ tends to zero and $N$ tends to infinity. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. As an application we derive the diffusion equation by a further rescaling.