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Phase transitions for the McKean-Vlasov equation on the torus

Banff International Research Station for Mathematical Innovation and Discovery

In the talk, the McKean-Vlasov equation on the flat torus is studied. The model is obtained as the mean field limit of a system of interacting diffusion processes enclosed in a periodic box. The system acts as a model for several real-world phenomena from statistical physics, opinion dynamics, collective behaviour, and stellar dynamics. This work provides a systematic approach to the qualitative and quantitative analysis of the McKean-Vlasov equation. We comment on the longtime behaviour and convergence to equilibrium, for which we introduce a notion of H-stability. The main part of the talk considers the stationary problem. We show that the system exhibits multiple equilibria which arise from the uniform state through continuous bifurcations, under certain assumptions on the interaction potential. Finally, criteria for the classification of continuous and discontinuous transitions of this system are provided. This classification is based on a fine analysis of the free energy. The results are illustrated by proving and extending results for a wide range of models, including the noisy Kuramoto model, Hegselmann-Krause model, and Keller-Segel model. (joint work with José Carrillo, Rishabh Gvalani, and Greg Pavliotis)

Mathematics; Partial differential equations; Statistical mechanics, structure of matter

video/mp4

Author affiliation: University of Bonn / RWTH Aachen University

2018-10-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/67550

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/