Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

TL;DR

This paper explores nonlinear mean field Fokker-Planck equations, linking them to various physical systems and introducing a new chemotaxis model that accounts for anomalous diffusion and volume exclusion effects.

Contribution

It demonstrates the broad applicability of nonlinear mean field Fokker-Planck equations and introduces a novel chemotaxis model with anomalous diffusion and exclusion principles.

Findings

Equations describe systems like chemotaxis, Bose-Einstein condensation, and turbulence.

Generalized Keller-Segel models for biological chemotaxis.

Transition from Kramers to Smoluchowski equations in strong friction limit.

Abstract

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Huckel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models describing the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis and Fermi-Dirac entropies among others.