Corrections to mean field for classical interacting particles

Abstract: We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we optimally analyze the propagation of chaos in form of sharp estimates on the many-particle correlation functions. While approaches based on the BBGKY hierarchy are doomed by uncontrolled losses of derivatives, we propose a novel non-hierarchical approach that focusses on the empirical measure of the system and exploits a Glauber type calculus with respect to initial data. This main result allows to rigorously truncate the BBGKY hierarchy to an arbitrary precision on the mean-field timescale, thus justifying the Bogolyubov corrections to mean field. As a corollary, we discuss the justification of the Lenard-Balescu limit for a spatially homogeneous system.