On the extensivity of the entropy $S_q$ for $N \le 3$ specially correlated binary subsystems

Abstract

Many natural and artificial systems whose range of interaction is long enough are known to exhibit (quasi)stationary states that defy the standard, Boltzmann-Gibbs statistical mechanical prescriptions. For handling such anomalous systems (or at least some classes of them), {\it nonextensive} statistical mechanics has been proposed based on the entropy $S_{q}\equiv k (1-\sum_{i=1}^Wp_i^{q})/(q-1)$, with $S_1=-kΣ_{i=1}^{W} p_i \ln p_i$ (Boltzmann-Gibbs entropy). Special collective correlations can be mathematically constructed such that the strictly {\it additive} entropy is now $S_q$ for an adequate value of $q \ne 1$, whereas Boltzmann-Gibbs entropy is {\it nonadditive}. Since important classes of systems exist for which the strict additivity of Boltzmann-Gibbs entropy is replaced by asymptotic additivity (i.e., extensivity), a variety of classes are expected to exist for which the strict additivity of $S_q (q\ne 1)$ is similarly replaced by asymptotic additivity (i.e., extensivity). All probabilistically well defined systems whose adequate entropy is $S_{1}$ are called {\it extensive} (or {\it normal}). They correspond to a number $W^{\it eff}$ of {\it effectively} occupied states which grows {\it exponentially} with the number $N$ of elements (or subsystems). Those whose adequate entropy is $S_q (q \ne 1)$ are currently called {\it nonextensive} (or {\it anomalous}). They correspond to $W^{\it eff}$ growing like a {\it power} of $N$. To illustrate this scenario, recently addressed, we provide in this paper details about systems composed by $N=2,3$ two-state subsystems.

10 pages including 7 figures. Invited paper to appear in a special issue of the International Journal of Bifurcation and Chaos: Proceedings of the Summer School and Conference on Complexity at Patras and Olympia (July 2004), Ed. T. Bountis