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

## TL;DR

This paper investigates how colored noise influences phase transitions in interacting particle systems, employing spectral methods to solve McKean-Vlasov equations and validating results with simulations.

## Contribution

It introduces a spectral method for solving mean-field Fokker-Planck equations with colored noise, analyzing bifurcations and phase transitions.

## Key findings

- Colored noise affects bifurcation diagrams of particle systems.
- Spectral method accurately solves complex Fokker-Planck equations.
- Small correlation time regime analyzed via perturbation theory.

## 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.