Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions

TL;DR

This paper investigates the connection between phase transitions, logarithmic Sobolev inequalities, and the long-term behavior of weakly interacting diffusions, revealing how the LSI constant influences propagation of chaos and equilibrium fluctuations.

Contribution

It establishes a link between the LSI constant's behavior and phase transitions in mean field limits, extending previous results with new coupling and gradient flow techniques.

Findings

Non-degeneracy of LSI constant prevents phase transitions.

Uniform-in-time propagation of chaos is achieved under certain conditions.

New proofs of Talagrand's inequality and propagation of chaos are provided.

Abstract

In this article, we study the mean field limit of weakly interacting diffusions for confining and interaction potentials that are not necessarily convex. We explore the relationship between the large $N$ limit of the constant in the logarithmic Sobolev inequality (LSI) for the $N$-particle system and the presence or absence of phase transitions for the mean field limit. The non-degeneracy of the LSI constant is shown to have far reaching consequences, especially in the context of uniform-in-time propagation of chaos and the behaviour of equilibrium fluctuations. Our results extend previous results related to unbounded spin systems and recent results on propagation of chaos using novel coupling methods. As incidentals, we provide concise and, to our knowledge, new proofs of a generalised form of Talagrand's inequality and of quantitative propagation of chaos by employing techniques from the theory of gradient flows, specifically the Riemannian calculus on the space of probability measures.