Mean-field approximation, Gibbs relaxation, and cross estimates

Abstract

We study the propagation of chaos and relaxation to Gibbs equilibrium for a system of $N$ classical Brownian particles with weak mean-field interactions. It is well known that propagation of chaos holds uniformly in time with rate $O(N^{-1})$ and that Gibbs relaxation holds uniformly in $N$ with exponential rate $O(e^{-ct})$. We go one step further by establishing a cross estimate that simultaneously captures both effects: the joint deviation between chaos propagation and Gibbs relaxation is of order $O(N^{-1}e^{-ct})$. In particular, for translation-invariant systems, this yields an accelerated propagation of chaos, with the mean-field approximation error at the level of the one-particle density improving from $O(N^{-1})$ to $O(N^{-1}e^{-ct})$. Our approach relies on a detailed analysis of the BBGKY hierarchy for correlation functions, and applies to both underdamped and overdamped Langevin dynamics with merely bounded interaction forces. In addition, we obtain new quantitative results on Gibbs relaxation and provide partial extensions beyond the weak interaction regime.