  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 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 in form of higher-order
Poincar\'e inequalities for cumulants. 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
corollaries, we also deduce a quantitative central limit theorem for
fluctuations of the empirical measure, and we partially justify the
Lenard-Balescu limit for a spatially homogeneous system away from thermal
equilibrium.
