Derivation of nonlinear Fokker-Planck equations by means of approximations to the master equation

Abstract

Nonlinear Fokker-Planck equations (FPEs) are derived as approximations to the master equation, in cases of transitions among both discrete and continuous sets of states. The nonlinear effects, introduced through the transition probabilities, are argued to be relevant for many real phenomena within the class of anomalous-diffusion problems. The nonlinear FPEs obtained appear to be more general than some previously proposed (on a purely phenomenological basis) ones. In spite of this, the same kind of solution applies, i.e., it is shown that the time-dependent Tsallis's probability distribution is a solution of both equations, obtained either from discrete or continuous sets of states, and that the corresponding stationary solution is, in the infinite-time limit, a stable solution.