Statistical mechanics for complex systems: On the structure of q-triplets*

A plethora of natural, artificial and social complex systems exists which violate the basic hypothesis (e.g., ergodicity) of Boltzmann-Gibbs (BG) statistical mechanics. Many of such cases can be satisfactorily handled by introducing nonadditive entropic functionals, such as systems can be characterized by a set of values {q}, directly corresponding to its various physical/dynamical/geometrical properties. A most important subset is usually referred to as the qtriplet, namely (qsensitivity , qrelaxation, qstationary state), defined in the body of this paper. In the BG limit we have qsensitivity = qrelaxation = qstationary state = 1. For a given class of complex systems, the set {q} contains only a few independent values of q, all the others being functions of those few. An illustration of this structure was given in 2005 [Tsallis, Gell-Mann and Sato, Proc. Natl. Acad. Sc. USA 102, 15377; TGS]. This illustration enabled a satisfactory analysis of the Voyager 1 data on the solar wind. But the general form of these structures still is an open question. This is so, for instance, for the challenging q-triplet associated with the edge of chaos of the logistic map. We introduce here a transformation which sensibly generalizes the TGS one, and which might constitute an important step towards the general solution.

2016