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arXiv · 1805.04959

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

M. H. Duong·G. A. Pavliotis
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Abstract

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 (Green's 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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M. H. Duong, G. A. Pavliotis. 2018-05-25. Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions. https://arxiv.org/abs/1805.04959

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