Formation of clusters and coarsening in weakly interacting diffusions
4 affiliations
- 1Ulm University
- 2University of Edinburgh
- 3EPFL
- 4Imperial College London
Abstract
This paper studies the clustering behaviour of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behaviour 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 analyse 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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BibTeX
@article{gerber2026formation,
title = {{Formation of clusters and coarsening in weakly interacting diffusions}},
author = {Nicolai Jurek Gerber and Rishabh S. Gvalani and Martin Hairer and G A Pavliotis and André Schlichting},
journal = {Nonlinearity},
year = {2026},
month = jul,
doi = {10.1088/1361-6544/ae7ee7},
url = {https://doi.org/10.1088/1361-6544/ae7ee7},
note = {CC BY},
}
RIS
TY - JOUR
TI - Formation of clusters and coarsening in weakly interacting diffusions
AU - Nicolai Jurek Gerber
AU - Rishabh S. Gvalani
AU - Martin Hairer
AU - G A Pavliotis
AU - André Schlichting
PY - 2026
DA - 2026/07/23
JO - Nonlinearity
DO - 10.1088/1361-6544/ae7ee7
UR - https://doi.org/10.1088/1361-6544/ae7ee7
AB - This paper studies the clustering behaviour of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behaviour 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 analyse 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.
SN - 0951-7715
SN - 1361-6544
ER -