Mean field limits for nonMarkovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions
Abstract
In this paper, we study the mean field limit of interacting particles with memory that are governed by a system of interacting nonMarkovian Langevin equations. Under the assumption of quasiMarkovianity (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 McKeanVlasov 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 (CurieWeiss) interaction. Furthermore, for nonconvex confining potentials we characterize the stationary state(s) of the McKeanVlasov 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 McKeanVlasov equation for the nonMarkovian dynamics can be written in the GENERIC formalism and we study convergence to equilibrium and the Markovian asymptotic limit.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.04959
 Bibcode:
 2018arXiv180504959D
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 28 pages. Comments are welcome