Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus

TL;DR

This paper analyzes the long-term behavior and phase transitions of the McKean-Vlasov equation on the torus, identifying bifurcations and conditions for phase changes in various interaction models.

Contribution

It establishes the existence of bifurcations and phase transitions for the McKean-Vlasov equation with different potentials, extending understanding of its stationary solutions.

Findings

Existence of nontrivial stationary solutions through bifurcations.

Conditions for continuous and discontinuous phase transitions.

Application to models like Kuramoto, Keller-Segel, and opinion dynamics.

Abstract

We study the McKean-Vlasov equation \[ \partial_t \varrho= \beta^{-1} \Delta \varrho + \kappa \nabla \cdot (\varrho \nabla (W \star \varrho)) \, , \] with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller--Segel model for bacterial chemotaxis, and the noisy Hegselmann--Krausse model for opinion dynamics.