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### Mean field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods

File | Description | Size | Format | |
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1904.05973v5.pdf | Accepted version | 4.25 MB | Adobe PDF | View/Open |

Title: | Mean field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods |

Authors: | Gomes, SN Pavliotis, GA Vaes, U |

Item Type: | Journal Article |

Abstract: | In this paper we consider systems of weakly interacting particles driven by colored noise in a bistable potential, and we study the effect of the correlation time of the noise on the bifurcation diagram for the equilibrium states. We accomplish this by solving the corresponding McKean--Vlasov equation using a Hermite spectral method, and we verify our findings using Monte Carlo simulations of the particle system. We consider both Gaussian and non-Gaussian noise processes, and for each model of the noise we also study the behavior of the system in the small correlation time regime using perturbation theory. The spectral method that we develop in this paper can be used for solving linear and nonlinear, local and nonlocal (mean field) Fokker--Planck equations, without requiring that they have a gradient structure. |

Issue Date: | 2-Sep-2020 |

Date of Acceptance: | 28-Apr-2020 |

URI: | http://hdl.handle.net/10044/1/88275 |

DOI: | 10.1137/19M1258116 |

ISSN: | 1540-3459 |

Publisher: | Society for Industrial and Applied Mathematics |

Start Page: | 1343 |

End Page: | 1370 |

Journal / Book Title: | SIAM: Multiscale Modeling and Simulation |

Volume: | 18 |

Issue: | 3 |

Copyright Statement: | © 2020, Society for Industrial and Applied Mathematics |

Keywords: | Science & Technology Physical Sciences Mathematics, Interdisciplinary Applications Physics, Mathematical Mathematics Physics McKean-Vlasov PDEs nonlocal Fokker-Planck equations interacting particles Desai-Zwanzig model colored noise Hermite spectral methods phase transitions MARKOVIAN BROWNIAN-MOTION STOCHASTIC-SYSTEMS MODEL CONVERGENCE FLUCTUATIONS EQUILIBRIUM EQUATIONS DYNAMICS BEHAVIOR Science & Technology Physical Sciences Mathematics, Interdisciplinary Applications Physics, Mathematical Mathematics Physics McKean-Vlasov PDEs nonlocal Fokker-Planck equations interacting particles Desai-Zwanzig model colored noise Hermite spectral methods phase transitions MARKOVIAN BROWNIAN-MOTION STOCHASTIC-SYSTEMS MODEL CONVERGENCE FLUCTUATIONS EQUILIBRIUM EQUATIONS DYNAMICS BEHAVIOR math.NA math.NA cond-mat.stat-mech cs.NA 35Q70, 35Q83, 35Q84, 65N35, 65M70, 82B26 0102 Applied Mathematics Applied Mathematics |

Publication Status: | Published |

Online Publication Date: | 2020-09-02 |

Appears in Collections: | Mathematics Applied Mathematics and Mathematical Physics |