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).
Categories: cond-mat.stat-mech