The nonlinear Fokker-Planck equation with state-dependent diffusion - a nonextensive maximum entropy approach

Nonlinear Fokker-Planck equations (e.g., the diffusion equation for porous medium) are important candidates for describing anomalous diffusion in a va riety of systems. In this paper we introduce such nonlinear Fokker-Planck e quations with general state-dependent diffusion, thus significantly general izing the case of constant diffusion which has been discussed previously. A n approximate maximum entropy (MaxEnt) approach based on the Tsallis nonext ensive entropy is developed for the study of these equations. The MaxEnt so lutions are shown to preserve the functional relation between the time deri vative of the entropy and the time dependent solution. In some particular i mportant cases of diffusion with pou er-law multiplicative noise, our MaxEn t scheme provides exact time dependent solutions. We also prove that the st ationary solutions of the nonlinear Fokker-Planck equation with diffusion o f the (generalized) Stratonovich type exhibit the Tsallis MaxEnt form.