Communications in Mathematical Sciences

Volume 16 (2018)

Number 8

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

Pages: 2199 – 2230



Manh Hong Duong (School of Mathematics, University of Birmingham, United Kingdom)

Grigorios A. Pavliotis (Department of Mathematics, Imperial College London,United Kingdom)


In this paper, we study the mean field limit of weakly interacting particles with memory that are governed by a system of non-Markovian Langevin equations. Under the assumption of quasi-Markovianity (i.e. 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. For the case of a quadratic confining potential and a quadratic (Curie–Weiss) interaction, we obtain the fundamental solution (Green’s function). 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.


mean field limits, non-Markovian interacting particles, convergence to equilibrium, GENERIC, asymptotic limit, phase transitions

2010 Mathematics Subject Classification

35K10, 60F17, 60H10, 60J60, 82C31

G.P. is partially supported by the EPSRC under Grants No. EP/P031587/1, EP/L024926/1, and EP/L020564/1.

Received 25 May 2018

Accepted 22 August 2018

Published 18 April 2019