Mean field limits for non-Markovian 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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2018
- DOI:
- 10.48550/arXiv.1805.04959
- arXiv:
- arXiv:1805.04959
- Bibcode:
- 2018arXiv180504959D
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematical Physics;
- Mathematics - Probability
- E-Print:
- 28 pages. Comments are welcome