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

TL;DR

This paper investigates the conditions under which the nonextensive entropy $S_q$ remains extensive for small systems of up to three correlated binary subsystems, challenging the traditional association of $S_q$ with nonextensivity.

Contribution

It provides a detailed analysis of the extensivity of $S_q$ for systems with $N  2$ and $3$ correlated binary subsystems, highlighting special correlations that preserve additivity.

Findings

$S_q$ can be extensive for small correlated systems with specific correlations.

For $N=2,3$, certain correlations make $S_q$ additive even when $q eq 1$.

The growth of effective states $W^{eff}$ can be polynomial, not exponential, for nonextensive systems.

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 $S_1=-k\Sigma_{i=1}^{W} p_i \ln p_i$ (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.