On the size of chaos via Glauber calculus in the classical mean-field dynamics

TL;DR

This paper introduces a novel non-hierarchical method using Glauber calculus to analyze chaos propagation in classical mean-field particle systems, providing sharp estimates and justifying Bogolyubov corrections.

Contribution

It develops a new approach based on Glauber calculus and Poincaré inequalities to rigorously analyze chaos and hierarchy truncation in mean-field dynamics.

Findings

Sharp estimates on many-particle correlation functions

Rigorous truncation of the BBGKY hierarchy

Quantitative central limit theorem for fluctuations

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 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.