Structure of correlations for the Boltzmann-Grad limit of hard spheres

TL;DR

This paper analyzes how correlations develop among hard spheres under the Boltzmann-Grad limit, providing a structure theorem that describes the nature of these correlations over time.

Contribution

It generalizes previous results by establishing a detailed structure theorem for correlations in a gas of hard spheres under Boltzmann-Grad scaling.

Findings

Correlations develop over time even from initially independent particles.

The structure theorem describes the factorization properties of the correlation functions.

Results hold under uniform bounds, either locally or globally for large mean free path.

Abstract

We consider a gas of $N$ identical hard spheres in the whole space, and we enforce the Boltzmann-Grad scaling. We may suppose that the particles are essentially independent of each other at some initial time; even so, correlations will be created by the dynamics. We will prove a structure theorem for the correlations which develop at positive time. Our result generalizes a previous result which states that there are phase points where the three-particle marginal density factorizes into two-particle and one-particle parts, while further factorization is impossible. The result depends on uniform bounds which are known to hold on a small time interval, or globally in time when the mean free path is large.