Mean-field approximation, Gibbs relaxation, and cross estimates

Abstract: This work focuses on the propagation of chaos and the relaxation to Gibbs equilibrium for a system of $N$ classical Brownian particles with weak mean-field interactions. While it is known that propagation of chaos holds at rate $O(N^{-1})$ uniformly in time, and Gibbs relaxation at rate $O(e^{-ct})$ uniformly in $N$, we go a step further by showing that the cross error between chaos propagation and Gibbs relaxation is $O(N^{-1}e^{-ct})$. For translation-invariant systems on the torus, this leads to an improved mean-field approximation error at the level of the one-particle density: the error decreases 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. We also derive new results on Gibbs relaxation and present partial extensions beyond the weak interaction regime.