A Rigorous Derivation of a Boltzmann System for a Mixture of Hard-Sphere Gases

TL;DR

This paper rigorously derives a Boltzmann equation for a mixture of two different hard-sphere gases from many-body dynamics, establishing well-defined microscopic behavior and propagation of chaos for such mixtures.

Contribution

It introduces a two-parameter BBGKY hierarchy to handle non-symmetric interactions and provides a rigorous derivation of the Boltzmann system for gas mixtures.

Findings

Proves well-defined microscopic dynamics for gas mixtures.

Establishes propagation of chaos for mixtures of gases.

Derives a Boltzmann equation specific to gas mixtures.

Abstract

In this paper, we rigorously derive a Boltzmann equation for mixtures from the many body dynamics of two types of hard sphere gases. We prove that the microscopic dynamics of two gases with different masses and diameters is well defined, and introduce the concept of a two parameter BBGKY hierarchy to handle the non-symmetric interaction of these gases. As a corollary of the derivation, we prove Boltzmann's propagation of chaos assumption for the case of a mixtures of gases.