Non-extensive thermodynamics and stationary processes of localization

We focus our attention on dynamical processes characterized by an entropic index Q < 1. According to the probabilistic arguments of Tsallis, C and Bukman, DJ [Phys Rev E 1996;54:R2197] these processes are subdiffusional in nature. The non-extensive generalization of the Kolmogorov-Sinai (KS) 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 a 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. (C) 2000 Elsevier Science Ltd. All rights reserved.