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@article{Duong:2018:10.4310/CMS.2018.v16.n8.a7,
author = {Duong, MH and Pavliotis, GA},
doi = {10.4310/CMS.2018.v16.n8.a7},
journal = {Communications in Mathematical Sciences},
pages = {2199--2230},
title = {Mean field limits for non-Markovian interacting particles: Convergence to equilibrium, generic formalism, asymptotic limits and phase transitions},
url = {http://dx.doi.org/10.4310/CMS.2018.v16.n8.a7},
volume = {16},
year = {2018}
}

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TY  - JOUR
AB  - © 2018 International Press. 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.
AU  - Duong,MH
AU  - Pavliotis,GA
DO  - 10.4310/CMS.2018.v16.n8.a7
EP  - 2230
PY  - 2018///
SN  - 1539-6746
SP  - 2199
TI  - Mean field limits for non-Markovian interacting particles: Convergence to equilibrium, generic formalism, asymptotic limits and phase transitions
T2  - Communications in Mathematical Sciences
UR  - http://dx.doi.org/10.4310/CMS.2018.v16.n8.a7
VL  - 16
ER  -

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Professor Grigorios A. Pavliotis

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