Cluster expansion for a dilute hard sphere gas dynamics

In a previous paper by the authors, a cluster expansion method had been developed to study the fluctuations of the hard sphere dynamics around the Boltzmann equation. This method provides a precise control on the exponential moments of the empirical measure, from which the fluctuating Boltzmann equation and large deviation estimates have been deduced. The cluster expansion originally devised by the authors was implemented at the level of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, which is a standard tool to investigate the deterministic dynamics. In this paper, we introduce an alternative approach in which the cluster expansion is applied directly on real trajectories of the particle system. This offers a fresh perspective on the study of the hard sphere dynamics in the low density limit, allowing us to recover the results previously obtained by the authors and also to describe the actual clustering of particle trajectories.