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

Abstract

We study the McKean-Vlasov equation \[ \partial_t \varrho= β^{-1} Δ\varrho + κ\nabla \cdot (\varrho \nabla (W \star \varrho)) \, , \] 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.

50 pages, 3 figures, Version 3

References in corpus (13)

• Uniform convergence to equilibrium for granular media
• Existence of Compactly Supported Global Minimisers for the Interaction Energy
• Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics
• Dynamical aspects of mean field plane rotators and the Kuramoto model
• Phase Transitions in a kinetic flocking model of Cucker-Smale type
• The McKean-Vlasov Equation in Finite Volume
• Mean Field Limits for Interacting Diffusions in a Two-Scale Potential
• Mean-Field Treatment of the Many-Body Fokker-Planck Equation
• Gradient flow structure for McKean-Vlasov equations on discrete spaces
• Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions
• Structure preserving schemes for the continuum Kuramoto model: phase transitions
• A non-local problem for the Fokker-Planck equation related to the Becker-Döring model
• Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions

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