Nonextensive Statistical Mechanics

316 8 Final Comments and Perspectivesrespective central limit theorems. Such question has been preliminarily addressedlong ago in [826], and once again in [585] as a rebuttal to [825]. The answer basicallyreminds that the stability observed in the usual central limit theorems isintimately related to the hypothesis of independence (or quasi-independence) ofthe random variables that are being composed. If important global correlations arepresent even in the N →∞limit, different central limit theorems are applicable,as proved in [247–249, 251–253]. Under these circumstances, stable distributionsdiffering from Gaussians and Levy ones are to be expected in nature.(h) Is entropy S q “physical”?Another question (or line of critique) that might emerge concerns the “physicality”of S q (see [812]). Or whether it could exist a “physical” entropy differentfrom S BG . Since such issues appear to be of a rather discursive/philosophical nature,we prefer to put these critiques on slightly different, more objective, grounds. Weprefer to ask, for instance, (i) whether S q is useful in theoretical physics in a sensesimilar to that in which S BG undoubtedly is useful; (ii) whether q necessarily isa fitting parameter, or whether it can be determined a priori, as it should if wewish the present theory to be a complete one; (iii) whether there is no other wayof addressing the thermal physics of the anomalous systems addressed here, veryspecifically whether one could not do so by just using S BG ; (iv) whether S q is specialin some physical sense, or whether it is to be put on the same grounds as the thirtyor forty entropic functionals popular in cybernetics, control theory, and informationtheory—generally speaking.Such questions have received answers in [150, 813–818] and elsewhere. (i) Theusefulness of this theory seems to be answered by the large amount of applicationsit has already received, and by the ubiquity of the q-exponential form in nature.(ii) The a priori calculation of q from microscopic dynamics has been specificallyillustrated in Chapter 5 (see also point (m) here below). (iii) The optimization ofS q , as well as of almost any other entropic form, with a few constraints has beenshown in [814] to be equivalent to the optimization of S BG with an infinite numberof appropriately chosen constraints. Therefore, we could in principle restrainto the exclusive use of S BG . If we followed that line, we would be doing like ahypothetical classical astronomer who, instead of using the extremely convenientKeplerian elliptic form for the planetary orbits, would (equivalently) use an infinitenumber of Ptolemaic epicycles. Obviously, it is appreciably much simpler tocharacterize, whenever possible, a complex structure of constraints with a singleindex q ≠ 1 (in analogy with the fact that the ellipticity of a Keplerian orbit canbe simply specified by a single parameter, namely the eccentricity of the ellipse).(iv) The entropy S q shares with S BG an impressive set of important properties (seealso point (j) here below), which includes, among others, concavity, extensivity,Lesche-stability, and finiteness of the entropy production per unit time, ∀q > 0. Thedifficulty of simultaneously satisfying all these four properties can be illustrated bythe fact that the (additive) Renyi entropy (usefully used in the geometric characterizationof multifractals) satisfies, under the hypothesis of probabilistic independenceor quasi-independence (and only then), extensivity ∀q, and none of the other three