Long time validity of the linearized Boltzmann equation for hard spheres: a proof without billiard theory

TL;DR

This paper proves that the covariance of a hard sphere system at thermal equilibrium converges to a linearized Boltzmann equation solution over time, using a new proof method that avoids billiard theory, advancing the fluctuation theory of rarefied gases.

Contribution

It provides a self-contained proof of the long-time validity of the linearized Boltzmann equation for hard spheres without relying on billiard geometric bounds.

Findings

Covariance converges to linearized Boltzmann solution over time.

Proof method does not depend on recollision bounds or billiard theory.

Advances the understanding of fluctuation theory for rarefied gases.

Abstract

We study space-time fluctuations of a hard sphere system at thermal equilibrium, and prove that the covariance converges to the solution of a linearized Boltzmann equation in the low density limit, globally in time. This result has been obtained previously in [7], by using uniform bounds on the number of recollisions of dispersing systems of hard spheres (as provided for instance in [9]). We present a self-contained proof with substantial differences, which does not use this geometric result. This can be regarded as the first step of a program aiming to derive the fluctuation theory of the rarefied gas, for interaction potentials different from hard spheres.