Towards the thermodynamics of localization processes

We study the entropy time evolution of a quantum mechanical model, which is frequently used as a prototype for Anderson's localization Recently Latora and Baranger [Phys. Rev. Lett. 82, 520 (1999)] found that there exist thre e entropy regimes, a transient regime of passage from dynamics to thermodyn amics, a linear-in-time regime of entropy increase, that is, a thermodynami c regime of Kolmogorov kind, and a saturation regime. We use the nonextensi ve entropic indicator advocated by Tsallis [J. Stat. Phys. 52, 479 (1988)] with a mobile entropic index q, and we find that the adoption of the "magic " value q = Q = 1/2, compared to the traditional entropic index q = 1, redu ces the length of the transient regime and makes earlier the emergence of t he Kolmogorov regime. We adopt a two-site model to explain these properties by means of an analytical treatment and we argue that Q=1/2 might be a typ ical signature of the occurrence of Anderson localization.