Stability of stationary states for mean field models with multichromatic interaction potentials

TL;DR

This paper investigates the stability and structure of stationary states in mean field models with multichromatic interaction potentials, revealing phase transitions and multipeak states influenced by Fourier modes and confining potentials.

Contribution

It demonstrates the existence of multipeak stationary states in mean field models with non-H-stable interactions, linking peaks to Fourier modes and analyzing effects of confining potentials.

Findings

Multipeak stationary states correspond to nonzero Fourier modes.

Phase transitions occur due to non-H-stable interactions.

Confining potentials influence the structure of steady states.

Abstract

We consider weakly interacting diffusions on the torus, for multichromatic interaction potentials. We consider interaction potentials that are not H-stable, leading to phase transitions in the mean field limit. We show that the mean field dynamics can exhibit multipeak stationary states, where the number of peaks is related to the number of nonzero Fourier modes of the interaction. We also consider the effect of a confining potential on the structure of non-uniform steady states. We approach the problem by means of analysis, perturbation theory and numerical simulations for the interacting particle systems and the PDEs.