Graduate Seminar Analysis

 

Tuesday, January 16, 2018, 10:30am

Phase transitions for the McKean-Vlasov equation on the torus

Rishabh Gvalani, Imperial College London

Abstract: We study the McKean-Vlasov equation on the flat torus which 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, stellar dynamics etc.

After commenting on the well-posedness of the equation, we study its long time behaviour and convergence to equilibrium. We then focus our attention on the stationary problem - under certain assumptions on the interaction potential, we show that the system exhibits multiple equilibria which arise from the uniform state through continuous bifurcations. This relates closely with previous work on phase transitions for the Mckean-Vlasov equation (cf. Chayes and Panferov, J. Stat. Phys., 2010). Finally, we attempt to classify continuous and discontinuous transitions for this system and show how this work, in conjunction with previous studies of the system, can be used to recover classical results on phase transitions for the noisy Kuramoto model. This is joint work with José Carrillo, Greg Pavliotis, and André Schlichting.

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