On the evolution of the empirical measure for the Hard-Sphere dynamics

TL;DR

This paper proves that the evolution of empirical measures for hard-sphere systems follows the BBGKY hierarchy, clarifying the connection to the Boltzmann-Enskog equation and addressing solution ambiguities.

Contribution

It establishes a rigorous link between empirical measure evolution and the BBGKY hierarchy for hard spheres, including implications for the Boltzmann-Enskog equation.

Findings

Empirical measure evolution aligns with the BBGKY hierarchy.

Derived the hierarchy for $L^1$ measures as a corollary.

Discussed ambiguities in microscopic solutions to the Boltzmann-Enskog equation.

Abstract

We prove that the evolution of marginals associated to the empirical measure of a finite system of hard spheres is driven by the BBGKY hierarchical expansion. The usual hierarchy of equations for $L^1$ measures is obtained as a corollary. We discuss the ambiguities arising in the corresponding notion of microscopic series solution to the Boltzmann-Enskog equation.