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 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.
Cite
@article{arxiv.2008.10403,
title = {Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations},
author = {Thierry Bodineau and Isabelle Gallagher and Laure Saint-Raymond and Sergio Simonella},
journal= {arXiv preprint arXiv:2008.10403},
year = {2022}
}
Comments
This version is reviewed following the remarks and suggestions of anonymous referees. The main modifications concern Chapters 5, 6 and 7, where the fluctuation theory is now presented in more canonical variables. An appendix has been added to collect local well-posedness results