Cluster Formation and coarsening in locally attractive interacting particle systems

We investigate the behavior of weakly interacting diffusions on the one-dimensional torus subject to finite-range attractive interactions. In particular, we study the emergence of clusters and the subsequent coarsening of multi-cluster configurations into a single-cluster state. Depending on the parameter regime, this coarsening is driven by two competing mechanisms:

(1) coalescence, whereby whole clusters move and merge in a manner analogous to coalescing Brownian motions; and

(2) mass exchange, in which individual particles detach from one cluster and attach to another, leading to an effective transfer of mass between clusters.

Based on an Eyring–Kramers-type asymptotic analysis, we propose a unified effective model capturing the interplay between these mechanisms and argue that the associated deterministic mean-field PDE exhibits dynamical metastability induced by mass exchange. In addition, we introduce a new variant of the strict Riesz rearrangement inequality to characterize the global minimizers of the free energy, showing that they are either spatially uniform or single-cluster states, i.e., symmetrically decreasing.

Joint work with N. Gerber, R. Gvalani, M. Hairer, and G. Pavliotis.