Thierry Bodineau (Ecole Polytechnique)

Fluctuating Boltzmann equation and large deviations for a hard sphere gas

Abstract: A gas dynamics can be modelled by a billiard made of hard spheres, moving according to the laws of classical mechanics. Initially the spheres are randomly distributed according to a probability measure which is then transported by the flow of the deterministic dynamics. Since the seminal work of Lanford, it is known in the kinetic limit that the gas density converges towards the Boltzmann equation (at least for a short time). In this talk, we are going to discuss the fluctuations of the microscopic dynamics around the Boltzmann equation and the convergence of the fluctuation field to the fluctuating Boltzmann equation. We will also show that the occurence of atypical evolutions can be quantified by a large deviation principle. One key feature of these results is that the statistical behaviour arising from the deterministic dynamics has a very similar structure to the one arising from a stochastic collision process of Kac’s type. Our analysis sheds also some light on the emergence of the irreversibility in the kinetic limit.