Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions


Abstract in English

In this paper, we study the mean field limit of interacting particles with memory that are governed by a system of interacting non-Markovian Langevin equations. Under the assumption of quasi-Markovianity (i.e. that the memory in the system can be described using a finite number of auxiliary processes), we pass to the mean field limit to obtain the corresponding McKean-Vlasov equation in an extended phase space. We obtain the fundamental solution (Greens function) for this equation, for the case of a quadratic confining potential and a quadratic (Curie-Weiss) interaction. Furthermore, for nonconvex confining potentials we characterize the stationary state(s) of the McKean-Vlasov equation, and we show that the bifurcation diagram of the stationary problem is independent of the memory in the system. In addition, we show that the McKean-Vlasov equation for the non-Markovian dynamics can be written in the GENERIC formalism and we study convergence to equilibrium and the Markovian asymptotic limit.

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