{
  "id": "1805.04959",
  "version": "v1",
  "published": "2018-05-13T21:42:58.000Z",
  "updated": "2018-05-13T21:42:58.000Z",
  "title": "Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions",
  "authors": [
    "M. H. Duong",
    "G. A. Pavliotis"
  ],
  "comment": "27 pages. Comments are welcome",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-13T21:42:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mean field limit",
      "non-markovian interacting particles",
      "generic formalism",
      "phase transitions",
      "mckean-vlasov equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}