Equivalence among different formalisms in the Tsallis entropy framework

TL;DR

This paper demonstrates the equivalence of different Tsallis entropy formalisms for systems evolving to stationary states, using nonlinear Fokker-Planck equations and time-scale conversions related to Lyapunov functions.

Contribution

It extends previous equilibrium results to out-of-equilibrium systems, establishing a correspondence among solutions of nonlinear Fokker-Planck equations in the Tsallis framework.

Findings

Existence of a correspondence among self-similar solutions

Time-scale conversion relates to Lyapunov functions

Equivalence applies to out-of-equilibrium stationary states

Abstract

In a recent paper [Phys. Lett. A {\bf335}, 351 (2005)] the authors discussed the equivalence among the various probability distribution functions of a system in equilibrium in the Tsallis entropy framework. In the present letter we extend these results to a system which is out of equilibrium and evolves to a stationary state according to a nonlinear Fokker-Planck equation. By means of time-scale conversion, it is shown that there exists a ``correspondence'' among the self-similar solutions of the nonlinear Fokker-Planck equations associated with the different Tsallis formalisms. The time-scale conversion is related to the corresponding Lyapunov functions of the respective nonlinear Fokker-Planck equations.