Gradient Flow Structure of the Spontaneous Aggregation Model

Abstract

We identify a previously unnoticed gradient-flow structure of the Fokker--Planck equation arising in the spontaneous particle aggregation model. For an exponential response function, the equation is a generalized Wasserstein gradient flow of the attractive McKean--Vlasov free energy with a nonlocal mobility. We show that, within a natural class of local entropies and symmetric interaction energies, this structure essentially singles out the exponential response. We discuss consequences for stationary states, global minimizers, the single-cluster structure in one dimension, and formally outline convergence to equilibrium using a Wasserstein--\mbox{\L}ojasiewicz inequality.