The European Physical Journal B

, Volume 62, Issue 2, pp 179–208

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

Statistical and Nonlinear Physics

DOI: 10.1140/epjb/e2008-00142-9

Cite this article as:
Chavanis, P. Eur. Phys. J. B (2008) 62: 179. doi:10.1140/epjb/e2008-00142-9


We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) 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-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for 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, Fermi-Dirac and Bose-Einstein entropies among others.


05.20.-y Classical statistical mechanics 05.45.-a Nonlinear dynamics and nonlinear dynamical systems 89.75.-k Complex systems 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2008

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique, Université Paul SabatierToulouseFrance