Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

TL;DR

This paper investigates how mean field and homogenization limits interact for diffusions in a two-scale potential, revealing non-commuting behavior over long times and effects on stationary states.

Contribution

It demonstrates that mean field and homogenization limits do not commute in the long term and analyzes their impact on stationary solutions in two-scale potentials.

Findings

Mean field and homogenization limits commute at finite times.

Long-term limits do not commute, affecting bifurcation diagrams.

Multiple local minima influence the number and stability of stationary states.

Abstract

In this paper we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in~\cite{DuncanPavliotis2016}. We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean-Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.