  This paper develops a non-asymptotic, local approach to quantitative
propagation of chaos for a wide class of mean field diffusive dynamics. For a
system of $n$ interacting particles, the relative entropy between the marginal
law of $k$ particles and its limiting product measure is shown to be
$O((k/n)^2)$ at each time, as long as the same is true at time zero. A simple
Gaussian example shows that this rate is optimal. The main assumption is that
the limiting measure obeys a certain functional inequality, which is shown to
encompass many potentially irregular but not too singular finite-range
interactions, as well as some infinite-range interactions. This unifies the
previously disparate cases of Lipschitz versus bounded measurable interactions,
improving the best prior bounds of $O(k/n)$ which were deduced from global
estimates involving all $n$ particles. We also cover a class of models for
which qualitative propagation of chaos and even well-posedness of the
McKean-Vlasov equation were previously unknown. At the center of a new approach
is a differential inequality, derived from a form of the BBGKY hierarchy, which
bounds the $k$-particle entropy in terms of the $(k+1)$-particle entropy.
