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

Abstract

Increasing the number $N$ of elements of a system typically makes the entropy to increase. The question arises on {\it what particular entropic form} we have in mind and {\it how it increases} with $N$. Thermodynamically speaking it makes sense to choose an entropy which increases {\it linearly} with $N$ for large $N$, i.e., which is {\it extensive}. If the $N$ elements are probabilistically {\it independent} (no interactions) or quasi-independent (e.g., {\it short}-range interacting), it is known that the entropy which is extensive is that of Boltzmann-Gibbs-Shannon, $S_{BG} \equiv -k \sum_{i=1}^W p_i \ln p_i$. If they are however {\it globally correlated} (e.g., through {\it 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., $S_q \equiv k \frac{1-\sum_{i=1}^W p_i^q}{q-1}$ ($S_1=S_{BG}$), 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. We present some numerical indications along these lines. The full clarification of such a possible connection 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.