Formation of clusters and coarsening in weakly interacting diffusions

TL;DR

This paper investigates clustering and coarsening phenomena in weakly interacting diffusions on a one-dimensional torus, revealing phase transitions, analyzing timescales, and introducing a new mass exchange model supported by numerical experiments.

Contribution

It introduces a novel analysis connecting clustering behavior to phase transitions and proposes a new mass exchange model for the PDE, supported by numerical validation.

Findings

Global minimizers are either uniform or single-cluster states.

Particle systems can coarsen via coalescence and diffusive mass exchange.

The PDE exhibits dynamical metastability due to mass exchange processes.

Abstract

This paper studies the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.