A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The $H$ theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard—linear Fokker-Planck equation—may be also related to a family of nonlinear Fokker-Planck equations.

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.

A system of interacting vortices under overdamped motion, which has been commonly used in the literature to model flux-front penetration in disordered type-II superconductors, was recently related to a nonlinear Fokker-Planck equation, characteristic of nonextensive statistical mechanics, through an analysis of its stationary state. Herein, this connection is extended by means of a thorough analysis of the time evolution of this system. Numerical data from molecular-dynamics simulations are presented for both position and velocity probability distributions P(x,t) and P(vx,t), respectively; both distributions are well fitted by similar q-Gaussian distributions, with the same index q=0, for all times considered. Particularly, the evolution of the system occurs in such a way that P(x,t) presents a time behavior for its width, normalization, and second moment, in full agreement with the analytic solution of the nonlinear Fokker-Planck equation. The present results provide further evidence that this system is deeply associated with nonextensive statistical mechanics.

The entropy time rate of systems described by nonlinear Fokker-Planck equations\char22{}which are directly related to generalized entropic forms\char22{}is analyzed. Both entropy production, associated with irreversible processes, and entropy flux from the system to its surroundings are studied. Some examples of known generalized entropic forms are considered, and particularly, the flux and production of the Boltzmann-Gibbs entropy, obtained from the linear Fokker-Planck equation, are recovered as particular cases. Since nonlinear Fokker-Planck equations are appropriate for the dynamical behavior of several physical phenomena in nature, like many within the realm of complex systems, the present analysis should be applicable to irreversible processes in a large class of nonlinear systems, such as those described by Tsallis and Kaniadakis entropies.

Several previous results valid for one-dimensional nonlinear Fokker-Planck equations are generalized to N-dimensions. A general nonlinear N-dimensional Fokker-Planck equation is derived directly from a master equation, by considering nonlinearitiesin the transition rates. Using nonlinear Fokker-Planck equations, the H-theorem is proved;for that, an important relation involving these equations and general entropic forms is introduced. It is shown that due to this relation, classes of nonlinear N-dimensional Fokker-Planck equations are connected to a single entropic form. A particular emphasis is given to the class of equations associated to Tsallis entropy, in both cases of the standard, and generalized definitions for the internal energy.