A procedure for obtaining general nonlinear Fokker–Planck equations

Abstract

A procedure for deriving general nonlinear Fokker–Planck equations (FPEs) directly from the master equation is presented. The nonlinear effects are introduced in the transition probabilities, which present a dependence on the probabilities for finding the system in a given state. It is shown that the FPEs, obtained from master equations describing transitions among discrete and continuous sets of states, are identical. Within such a procedure, we construct nonlinear FPEs that appear to be very general. Our general FPEs recover, as particular cases, nonlinear FPEs investigated previously by many authors, introduced on a purely phenomenological basis, and they lead to the possibility of more complete and complex diffusive equations.