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.

Keywords

  1. McKean--Vlasov PDEs
  2. nonlocal Fokker--Planck equations
  3. interacting particles
  4. Desai--Zwanzig model
  5. colored noise
  6. Hermite spectral methods
  7. phase transitions

MSC codes

  1. 35Q70
  2. 35Q83
  3. 35Q84
  4. 65N35
  5. 65M70
  6. 82B26

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Information & Authors

Information

Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 1343 - 1370
ISSN (online): 1540-3467

History

Submitted: 24 April 2019
Accepted: 28 April 2020
Published online: 2 September 2020

Keywords

  1. McKean--Vlasov PDEs
  2. nonlocal Fokker--Planck equations
  3. interacting particles
  4. Desai--Zwanzig model
  5. colored noise
  6. Hermite spectral methods
  7. phase transitions

MSC codes

  1. 35Q70
  2. 35Q83
  3. 35Q84
  4. 65N35
  5. 65M70
  6. 82B26

Authors

Affiliations

Funding Information

Leverhulme Trust https://doi.org/10.13039/501100000275 : ECF-2018-536
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/P031587/1, EP/L024926/1, EP/L020564/1, EP/K034154/1

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