Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations

TL;DR

This paper develops a mathematical framework for understanding fluctuations and large deviations in a hard sphere gas, extending the Boltzmann equation to include stochastic effects and analyzing their behavior in the Boltzmann-Grad limit.

Contribution

It introduces a rigorous theory of dynamical fluctuations and large deviations for the hard sphere gas, connecting empirical measure fluctuations to a fluctuating Boltzmann equation.

Findings

Fluctuations converge to a Gaussian process driven by the fluctuating Boltzmann equation.

Large deviations are exponentially small in the number of particles.

Results hold for short times away from thermal equilibrium.

Abstract

We present a mathematical theory of dynamical fluctuations for the hard sphere gas in the Boltzmann-Grad limit. We prove that: (1) fluctuations of the empirical measure from the solution of the Boltzmann equation, scaled with the square root of the average number of particles, converge to a Gaussian process driven by the fluctuating Boltzmann equation, as predicted in [67]; (2) large deviations are exponentially small in the average number of particles and are characterized, under regularity assumptions, by a large deviation functional as previously obtained in [61] for dynamics with stochastic collisions. The results are valid away from thermal equilibrium, but only for short times. Our strategy is based on uniform a priori bounds on the cumulant generating function, characterizing the fine structure of the small correlations.