Is the entropy Sq extensive or nonextensive?

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 $S_q\equiv k(1-{\sum }_{i=1}^W{p}_i^q)/(q-1)(q\in \mathbb{R};S_1=S_{BG})$. 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).

To appear in the Proceedings of the 31st Workshop of the International School of Solid State Physics ``Complexity, Metastability and Nonextensivity", held at the Ettore Majorana Foundation and Centre for Scientific Culture, Erice (Sicily) in 20-26 July 2004, eds. C. Beck, A. Rapisarda and C. Tsallis (World Scientific, Singapore, 2005). 10 pages including 1 figure

Cited by in corpus (2)

• Algebra and calculus for Tsallis thermostatistics
• From Boltzmann-Gibbs ensemble to generalized ensembles