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

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 empiral 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.

Authors

Thierry Bodineau

CMAP, CNRS, Ecole Polytechnique, I.P. Paris, Route de Saclay, 91128 Palaiseau Cedex, France

Isabelle Gallagher

École Normale Supérieure, CNRS, PSL Research University 45 rue d'Ulm, 75005 Paris, France, and Université de Paris,

Laure Saint-Raymond

UMPA UMR 5669 du CNRS, ENS de Lyon, Université de Lyon, 46 allée d'Italie, 69007 Lyon, France

Sergio Simonella

UMPA UMR 5669 du CNRS, ENS de Lyon, Université de Lyon, 46 allée d'Italie, 69007 Lyon, France