RESUMO
Ehrenfest's urn model is modified by introducing nonlinear terms in the associated transition probabilities. It is shown that these modifications lead, in the continuous limit, to a Fokker-Planck equation characterized by two competing diffusion terms, namely, the usual linear one and a nonlinear diffusion term typical of anomalous diffusion. By considering a generalized H theorem, the associated entropy is calculated, resulting in a sum of Boltzmann-Gibbs and Tsallis entropic forms. It is shown that the stationary state of the associated Fokker-Planck equation satisfies precisely the same equation obtained by extremization of the entropy. Moreover, the effects of the nonlinear contributions on the entropy production phenomenon are also analyzed.
RESUMO
The time rate of generalized entropic forms, defined in terms of discrete probabilities following a master equation, is investigated. Both contributions, namely entropy production and flux, are obtained, extending works carried previously for the Boltzmann-Gibbs entropy to a wide class of entropic forms. Particularly, it is shown that the entropy-production contribution is always non-negative for such entropies. Some illustrative examples for known generalized entropic forms in the literature are also worked out. Since generalized entropies have been lately associated with several complex systems in nature, the present analysis should be applicable to irreversible processes in these systems.
Assuntos
Algoritmos , Transferência de Energia , Entropia , Modelos Estatísticos , Simulação por ComputadorRESUMO
The entropy time rate of systems described by nonlinear Fokker-Planck equations--which are directly related to generalized entropic forms--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.