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We study a general class of nonlinear mean field Fokker-Planck equations inrelation with an effective generalized thermodynamical formalism. We show thatthese equations describe several physical systems such as: chemotaxis ofbacterial 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 Brownianparticles, Debye-Huckel theory of electrolytes, two-dimensional turbulence...In particular, we show that nonlinear mean field Fokker-Planck equations canprovide generalized Keller-Segel models describing the chemotaxis of biologicalpopulations. As an example, we introduce a new model of chemotaxisincorporating both effects of anomalous diffusion and exclusion principle(volume filling). Therefore, the notion of generalized thermodynamics can haveapplications for concrete physical systems. We also consider nonlinear meanfield Fokker-Planck equations in phase space and show the passage from thegeneralized Kramers equation to the generalized Smoluchowski equation in astrong friction limit. Our formalism is simple and illustrated by severalexplicit examples corresponding to Boltzmann, Tsallis and Fermi-Dirac entropiesamong others.

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