Exponential or power law? How to select a stable distribution of probability in a physical system

A mapping of non-extensive statistical mechanics into standard, Gibbs' statistical mechanics exists [S. Abe, Physica A 300 417-423 (2001), E. Vives, A. Planes, PRL 88 2, 020601 (2002)] which preserves the concavity of entropy and generalizes Gibbs-Duhem equation. This mapping allows generalization of the 'general evolution criterion' of [P. Glansdorff, I. Prigogine, Physica 30 351 (1964)] to non-extensive statistical mechanics with non-additivity parameter q. GEC leads to a necessary criterion for the stability of relaxed states of a wide class of physical systems [A. Di Vita, Phys. Rev. E 81 041137 (2010)]. In its generalized version, thiscriterion holds for arbitrary q and implies a suitably constrained minimization of the amount Pi-q of non-extensive entropy Sq produced per unit time in the bulk of the system. Moreover, if we assume q to be uniform throughout a non-isolated, relaxed system, then the distribution of probabilities of microstates in each small part of the system is a power law (a Boltzmann exponential) for q different from (equal to) 1. Accordingly, constrained minimization of Pi-q selects the value of q of a stable distribution of probabilities: if Pi-q = min for q equal to (different from) 1, then the probability distribution of a stable, relaxed state is a Boltzmann exponential (a power law). This value of q depends on both the detailed dynamics and on the boundary conditions of the system. As an example, we apply our result to a simple, one-dimensional system.