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

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

2005-02-11

arxiv.org/abs/cond-mat/0502274v3

Proc.Nat.Acad.Sci. 102 (2005) 15377-15382

Physics

Condensed Matter

Statistical Mechanics

15 pages, including 9 figures and 8 Tables. The new version is considerably enlarged with regard to the previous ones. New exa

Scientific paper

10.1073/pnas.0503807102

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 $q\ne1$ to be the one which extensive (i.e., $S_q(N)\propto N $ for $N \to\infty$).

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