Separation of time scales in weakly interacting diffusions

TL;DR

This paper rigorously analyzes the separation of time scales in weakly interacting Brownian particles, demonstrating metastability and quantifying convergence rates and localization properties of the droplet state.

Contribution

It provides a quantitative, rigorous characterization of metastable behavior and time scale separation in weakly interacting diffusions at the empirical measure level.

Findings

Convergence to the droplet state occurs at rate O(1) as temperature increases.

Leakage from the droplet state is exponentially small in inverse temperature.

The droplet's localization scale is proportional to the inverse square root of temperature.

Abstract

We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a "droplet state" which is $metastable$, i.e. persists on a much longer time scale than the time scale of convergence, before eventually diffusing to $0$. In this article, we provide rigorous evidence and a quantitative characterisation of this separation of time scales. Working at the level of the empirical measure, we show that (after quotienting out the motion of the centre of mass) the rate of convergence to the quasi-stationary distribution, which corresponds with the droplet state, is $O(1)$ as the inverse temperature $\beta \to \infty$. Meanwhile the rate of leakage away from its centre of mass is $O(e^{-\beta})$. Futhermore, the quasi-stationary distribution is localised on a length scale of order $O(\beta^{-\frac12})$. We thus provide a partial answer to a question posed by Carrillo, Craig, and Yao (Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits. In: Active Particles, Volume 2, 2019) in the microscopic setting.