Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem

Increasing the number N of elements of a system typically makes the entropy to increase. The question arises on what particular entropic form we have in mind and how it increases with N. Thermodynamically speaking it makes sense to choose an entropy which increases linearly with N for large N, i.e., which is extensive. If the N elements are probabilistically independent (no interactions) or quasi-independent (e.g., short-range interacting), it is known that the entropy which is extensive is that of Boltzmann–Gibbs–Shannon, SBG≡-k∑i=1Wpilnpi. If they are, however, globally correlated (e.g., through long-range interactions), the answer depends on the particular nature of the correlations. There is a large class of correlations (in one way or another related to scale-invariance) for which an appropriate entropy is that on which nonextensive statistical mechanics is based, i.e., Sq≡k(1-∑i=1Wpiq)/q-1 (S1=SBG), where q is determined by the specific correlations. We briefly review and illustrate these ideas through simple examples of occupation of phase space. A very similar scenario emerges with regard to the central limit theorem (CLT). If the variables that are being summed are independent (or quasi-independent, in the sense that they gradually become independent if N→∞), two basic possibilities exist: if the variance of the random variables that are being composed is finite, the N→∞ attractor in the space of distributions is a Gaussian, whereas if it diverges, it is a Lévy distribution. If the variables that are being summed are however globally correlated, there is no reason to expect the usual CLTs to hold. The N→∞ attractor is expected to depend on the nature of the correlations. That class of correlations (or part of it) that makes Sq to be extensive for q≠1 is expected to have a qe-Gaussian as its N→∞ attractor, where qe depends on q [qe(q) such that qe(1)=1], and where qe-Gaussians are proportional to [1-(1-qe)βx2]1/(1-qe) (β>0; qe<3). We present some numerical indications along these lines. The full clarification of such a possible connection would have considerable interest: it would help qualifying the class of systems for which the nonextensive statistical concepts are applicable, and, concomitantly, it would enlighten the reason for which q-exponentials are ubiquitous in many natural and artificial systems.