  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.
