Generalized Fokker-Planck equations derived from nonextensive entropies asymptotically equivalent to Boltzmann-Gibbs.

Abstract

We derive generalized Fokker-Planck equations (FPEs) based on two nonextensive entropy measures S_{±} that depend exclusively on the probability. These entropies have been originally obtained from the superstatistics framework, therefore they regard nonequilibrium systems outlined by a long-term stationary state in view of a spatiotemporally fluctuating intensive quantity. Moreover, entropies S_{±} as well as Boltzmann-Gibbs (BG) entropy S_{B} both pertain to the same asymptotical equivalence class, thus suggesting that S_{±} could depict a consistent thermodynamic generalization of BG. For these reasons, we assert that transport phenomena to be accounted for by our models shall coincide with the portrait given by the conventional FPEs for systems comprehending short-range interactions or a high number of accessible microstates, whereas, for systems composed of a small number of microstates, or those with long-range interactions, the governing equations of motion are to be the FPEs here derived, as long as the system fulfills the attributes mentioned above. We discuss the anomalous diffusion exhibited by the two generalized FPEs and also present some numerical applications. In particular, we find that there are models regarding biological sciences, for the study of congregation and aggregation behavior, the structure of which coincides with the one of our models.