Eur. Phys. J. B 12, 285-297 (1999)
https://doi.org/10.1007/s100510051006

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

¹ Centro Brasileiro de Pesquisas Fisicas (CBPF), Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil
² Department of Physics, National University La Plata, C.C. 727, 1900 La Plata, Argentina
³ Departament de Física, Universitat de les Illes Balears, 07071 Palma de Mallorca, Spain
⁴ Faculty of Astronomy and Geophysics, National University La Plata, C.C. 727, 1900 La Plata, Argentina

Corresponding author: ^a lisa@cat.cbpf.br

Received: 26 February 1999
Published online: 15 November 1999

Abstract

Nonlinear Fokker-Planck equations (e.g., the diffusion equation for porous medium) are important candidates for describing anomalous diffusion in a variety of systems. In this paper we introduce such nonlinear Fokker-Planck equations with general state-dependent diffusion, thus significantly generalizing the case of constant diffusion which has been discussed previously. An approximate maximum entropy (MaxEnt) approach based on the Tsallis nonextensive entropy is developed for the study of these equations. The MaxEnt solutions are shown to preserve the functional relation between the time derivative of the entropy and the time dependent solution. In some particular important cases of diffusion with power-law multiplicative noise, our MaxEnt scheme provides exact time dependent solutions. We also prove that the stationary solutions of the nonlinear Fokker-Planck equation with diffusion of the (generalized) Stratonovich type exhibit the Tsallis MaxEnt form.