The propagation of chaos for a rarefied gas of hard spheres in the whole space

TL;DR

This paper revisits and strengthens the mathematical understanding of Boltzmann's equation for rarefied gases, introducing new proofs, a refined notion of chaos propagation, and partial factorization results.

Contribution

It provides new proofs of classical bounds, introduces a novel concept of strong chaos with uniform error estimates, and proves partial factorization at specific phase points.

Findings

New proofs of uniform bounds for Lanford's theorem

Introduction of a refined notion of propagation of chaos

Proof of partial factorization at certain phase points

Abstract

We discuss old and new results on the mathematical justification of Boltzmann's equation. The classical result along these lines is a theorem which was proven by Lanford in the 1970s. This paper is naturally divided into three parts. I. Classical. We give new proofs of both the uniform bounds required for Lanford's theorem, as well as the related bounds due to Illner & Pulvirenti for a perturbation of vacuum. The proofs use a duality argument and differential inequalities, instead of a fixed point iteration. II. Strong chaos. We introduce a new notion of propagation of chaos. Our notion of chaos provides for uniform error estimates on a very precise set of points; this set is closely related to the notion of strong (one-sided) chaos and the emergence of irreversibility. III. Supplemental. We announce and provide a proof (in Appendix A) of propagation of partial factorization at some phase-points where complete factorization is impossible.