Derivation of the Higher Order Boltzmann Equation for Hard Spheres

TL;DR

This paper rigorously derives a higher order Boltzmann equation for hard spheres, accounting for complex multi-particle collisions and generalizing geometric analysis techniques for correlation studies.

Contribution

It introduces a new scaling regime and systematically extends geometric methods to analyze higher order recollisions in gas particle systems.

Findings

Derived a Boltzmann-type equation with higher order collision terms

Identified a novel scaling regime for complex collisions

Generalized geometric techniques for recollision analysis

Abstract

In this paper we complete the program initiated by the first and second authors and rigorously derive a Boltzmann-type equation that incorporates higher order collisions among gas particles. More precisely, starting from a finite $N$-particle system where the particles can perform symmetric hard sphere type interactions up to arbitrarily high order $M$, we derive a kinetic equation which consists of a linear combination of higher order collisional terms. We identify the new scaling regime that facilitates such collisions and systematically generalize geometric techniques allowing us to analyze the correlation of potentially distinct order recollisions as time evolves.