Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive

Abstract

Phase space can be constructed for $N$ equal and distinguishable subsystems that could be (probabilistically) either {\it weakly} (or {\it "locally"}) correlated (e.g., independent, i.e., uncorrelated), or {\it strongly} (or {\it globally}) correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy $S_{BG}\equiv -k{\sum }_i{p}_i\ln p_i$ to be {\it extensive}, i.e., $S_{BG}(N)\propto N$ for $N\to \infty$. In particular, if they are independent, $S_{BG}$ is {\it strictly additive}, i.e., $S_{BG}(N)=N{S}_{BG}(1),\forall N$. However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy $S_q\equiv k[1-{\sum }_i{p}_i^q]/(q-1)$ (with $S_1=S_{BG}$) for some special value of $q\ne 1$ to be the one which extensive (i.e., $S_q(N)\propto N$ for $N\to \infty$).

15 pages, including 9 figures and 8 Tables. The new version is considerably enlarged with regard to the previous ones. New examples and new references have been included