Is the entropy Sq extensive or nonextensive?

TL;DR

This paper examines the conditions under which the entropies S_{BG} and S_q are extensive or nonextensive, emphasizing the importance of the composition law and correlations between subsystems.

Contribution

It demonstrates how specific correlations between subsystems can alter the extensivity properties of both Boltzmann-Gibbs and nonextensive entropies.

Findings

S_q can be extensive with the right correlations

S_{BG} remains nonextensive under certain correlations

Extensivity depends on the composition law and correlations

Abstract

The cornerstones of Boltzmann-Gibbs and nonextensive statistical mechanics respectively are the entropies $S_{BG} \equiv -k \sum_{i=1}^W p_i \ln p_i $ and $S_{q}\equiv k (1-\sum_{i=1}^Wp_i^{q})/(q-1) (q\in{\mathbb R} ; S_1=S_{BG})$. Through them we revisit the concept of additivity, and illustrate the (not always clearly perceived) fact that (thermodynamical) extensivity has a well defined sense {\it only} if we specify the composition law that is being assumed for the subsystems (say $A$ and $B$). If the composition law is {\it not} explicitly indicated, it is {\it tacitly} assumed that $A$ and $B$ are {\it statistically independent}. In this case, it immediately follows that $S_{BG}(A+B)= S_{BG}(A)+S_{BG}(B)$, hence extensive, whereas $S_q(A+B)/k=[S_q(A)/k]+[S_q(B)/k]+(1-q)[S_q(A)/k][S_q(B)/k]$, hence nonextensive for $q \ne 1$. In the present paper we illustrate the remarkable changes that occur when $A$ and $B$ are {\it specially correlated}. Indeed, we show that, in such case, $S_q(A+B)=S_q(A)+S_q(B)$ for the appropriate value of $q$ (hence extensive), whereas $S_{BG}(A+B) \ne S_{BG}(A)+S_{BG}(B)$ (hence nonextensive).