Identification and existence of Boltzmann processes

TL;DR

This paper establishes a stochastic differential equation framework for Boltzmann processes, linking McKean-Vlasov SDEs with the Boltzmann equation to model particle dynamics in vacuum.

Contribution

It identifies a specific McKean-Vlasov SDE whose solutions correspond to Boltzmann processes, providing a new probabilistic representation of the Boltzmann equation.

Findings

Existence of solutions to the McKean-Vlasov SDE for non-cutoff hard sphere case

Connection between Boltzmann equation and stochastic processes

Framework for modeling particle dynamics in vacuum

Abstract

The stochastic differential equation of McKean-Vlasov type is identified such that the Fokker-Planck equation associated to it is the Boltzmann equation. Hence, we call its solutions as Boltzmann processes. They describe the dynamics (in position and velocity) of particles expanding in vacuum in accordance with the Boltzmann equation. Given a solution $f:=$ $\{f(t,x,v\}_{0 \leq t \leq T} $ of the Boltzmann equation, the existence of solutions to the McKean-Vlasov SDE is established for the non-cutoff hard sphere case.