The Mean Field Fokker-Planck Equation with Nonlinear No-flux Boundary Conditions

TL;DR

This paper analyzes the mean field Fokker-Planck equation with nonlinear no-flux boundary conditions, revealing how boundary effects influence spectral properties and applying numerical methods to various physical models.

Contribution

It introduces a spectral analysis of the Fokker-Planck equation with nonlinear boundary conditions and demonstrates numerical solutions across diverse physical systems.

Findings

Spectral properties vary significantly with boundary conditions.

Numerical methods effectively solve complex mean field models.

Applications include models of liquids, oscillators, and particle interactions.

Abstract

We consider the mean field Fokker-Planck equation subject to nonlinear no-flux boundary conditions, which necessarily arise when subjecting a system of Brownian particles interacting via a pair potential in a bounded domain. With the additional presence of an external potential $V_1$, we show, by analysing the linearised Fokker-Planck operator, that the spectral properties of the equilibrium densities can differ considerably when compared with previous studies, e.g., with periodic boundary conditions. Amongst other mean field models of complex many-body particle systems, we present numerical experiments encompassing in a wide range of physical applications, including: generalised exponential models (Gaussian, Morse); a Kuramoto model, for noisy coupled oscillators; and an Onsager model for liquid crystals. We showcase our results by using the numerical methods developed in the pseudospectral collocation scheme 2DChebClass.