Nonextensive Statistical Mechanics

8.3 Open Questions 327This is a frequently asked question whose full answer is still unclear. We havepresented in a previous section a variety of mathematical reasons pointing the relevanceof the escort distributions within the present theory. However, a clear-cutphysical interpretation in terms of the dynamics and occupancy geometry within thefull phase-space is still lacking. Some important hints can be found in [55–57].(d) What is the logical connection between the class of systems whose extensivityrequires the adoption of the entropy S q with q ≠ 1, and the class of systems whoseprobabilities distributions of occupancy of phase-space leads, in the limit N →∞,to anomalous central limit theorems?The present scenario is that asymptotic scale-invariance is necessary but not sufficient.Hints can be found in [245], in Section 4.6.4, and in the nonlinear Fokker–Planck equation.(e) Under what generic conditions nonlinear dynamics such as those emerging at theedge of chaos as well as in long-range-interacting many-body classical Hamiltoniansat their quasi-stationary state tend to create, in the full phase-space, structuresgeometrically similar to scale-free networks?The scenario is that probabilistic correlations of the q-independent class tendto create a (multi)fractal occupation of phase-space. The clarification of this pointwould most probably also provide an answer to the above point (c).(f) What are the precise physical quantitities associated with the infinite set of interrelatedvalues of q emerging in relations such as Eq. (4.90)? What is their preciseconnection to sets such as the q-triplet?This is a most important open question. The scenario is that somehow the q-triplet essentially corresponds to central elements (such as n = 0, ±1, ±2, etc) ofthe relation (4.90).The solution, or at least crucial hints pointing along that direction, of these andother similar questions would be more than welcome!