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arXiv · 2510.17629

Formation of clusters and coarsening in weakly interacting diffusions

Nicolai Gerber·Rishabh S. Gvalani·Martin Hairer·Grigorios A. Pavliotis·André Schlichting
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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.

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Nicolai Gerber, Rishabh S. Gvalani, Martin Hairer, Grigorios A. Pavliotis, André Schlichting. 2025-10-20. Formation of clusters and coarsening in weakly interacting diffusions. https://arxiv.org/abs/2510.17629

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