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 $S_q\equiv k\frac{1-\sum_{i=1}^W p_i^q}{q-1} \; \Bigl(q \in {\cal R}; \, \sum_{i=1}^W p_i=1 \Bigr)$, with $S_1=S_{BG}\equiv -k\sum_{i=1}^W p_i \ln p_i$. Each class of such 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 $q$-triplet, namely $(q_{sensitivity}, q_{relaxation}, q_{stationary\,state})$, defined in the body of this paper. In the BG limit we have $q_{sensitivity}=q_{relaxation}=q_{stationary\,state}=1$. 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 {\bf 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.