Nonadditive entropy: the concept and its use
Abstract
The entropic form Sq is, for any q ≠ 1, nonadditive. Indeed, for two probabilistically independent subsystems, it satisfies Sq(A+B)/k=[Sq(A)/k]+[Sq(B)/k]+(1-q)[Sq(A)/k][Sq(B)/k] Sq(A)/k+Sq(B)/k. This form will turn out to be extensive for an important class of nonlocal correlations, if q is set equal to a special value different from unity, noted qent (where ent stands for entropy). In other words, for such systems, we verify that Sqent(N) N (N>>1), thus legitimating the use of the classical thermodynamical relations. Standard systems, for which SBG is extensive, obviously correspond to qent=1. Quite complex systems exist in the sense that, for them, no value of q exists such that Sq is extensive. Such systems are out of the present scope: they might need forms of entropy different from Sq, or perhaps -- more plainly -- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with Sq, the q-generalizations of the Central Limit Theorem and of its extended L\'evy-Gnedenko form have been achieved. These recent theorems could of course be the cause of the ubiquity of q-exponentials, q-Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as well as presently available experimental, observational and computational confirmations -- in high energy physics and elsewhere --, are briefly reviewed. Finally, we address a confusion which is quite common in the literature, namely referring to distinct physical mechanisms versus distinct regimes of a single physical mechanism.