On uniform in time propagation of chaos in metastable cases: the Curie-Weiss model

TL;DR

This paper investigates the metastable behavior of mean-field particle systems, specifically the Curie-Weiss model, demonstrating uniform in time propagation of chaos when conditioned on positive magnetization, despite multiple steady states.

Contribution

It provides the first analysis of uniform in time propagation of chaos in metastable regimes for the Curie-Weiss model, accounting for multiple steady states.

Findings

Uniform in time propagation of chaos established for the conditioned spin system.

Metastable behavior characterized by exponential exit times from basins of attraction.

Quasi-stationary distributions approximate nonlinear steady states before exit.

Abstract

Many low temperature particle systems in mean-field interaction are ergodic with respect to a unique invariant measure, while their (non-linear) mean-field limit may possess several steady states. In particular, in such cases, propagation of chaos (i.e. the convergence of the particle system to its mean-field limit as n, the number of particles, goes to infinity) cannot hold uniformly in time since the long-time behaviors of the two processes are a priori incompatible. However, the particle system may be metastable, and the time needed to exit the basin of attraction of one of the steady states of its limit, and go to another, is exponentially (in n) long. Before this exit time, the particle system reaches a (quasi-)stationary distribution, which we expect to be a good approximation of the corresponding non-linear steady state. Our goal is to study the typical metastable behavior of the empirical measure of such mean-field systems, starting in this work with the Curie-Weiss model. We thus show uniform in time propagation of chaos of the spin system conditioned to keeping a positive magnetization.