Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations
arXiv:2008.10403
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.
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