Non-extensive thermodynamics and stationary processes of localization

TL;DR

This paper explores how non-extensive thermodynamics with an entropic index Q<1 leads to localization in stationary processes, supported by numerical analysis of the logistic map at criticality.

Contribution

It demonstrates that stationary subdiffusive processes with Q<1 inherently undergo localization within finite time scales.

Findings

Subdiffusive processes with Q<1 are associated with localization.

Stationary conditions induce localization in non-extensive systems.

Numerical simulations confirm localization in the logistic map at criticality.

Abstract

We focus our attention on dynamical processes characterized by an entropic index Q<1. According to the probabilistic arguments of Tsallis and Bukman [C.Tsallis, D.J. Bukman, Phys. Rev. E 54,R2197 (1996)] these processes are subdiffusional in nature. The non-extensive generalization of the Kolmogorov-Sinai entropy yielding the same entropic index implies the stationary condition. We note, on the other hand, that enforcing the stationary property on subdiffusion has the effect of producing a localization process occurring within a finite time scale. We thus conclude that the stationary dynamic processes with Q<1 must undergo a localization process occurring at finite time. We check the validity of this conclusion by means of a numerical treatment of the dynamics of the logistic map at the critical point.