Is the entropy Sq extensive or nonextensive?

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).Comment: 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 figur

On the extensivity of the entropy Sq, the q-generalized central limit theorem and the q-triplet

First, we briefly review the conditions under which the entropy $S_q$ can be extensive (Tsallis, Gell-Mann and Sato, Proc. Natl. Acad. Sc. USA (2005), in press; cond-mat/0502274), as well as the possible $q$-generalization of the central limit theorem (Moyano, Tsallis and Gell-Mann, cond-mat/0509229). Then, we address the $q$-triplet recently determined in the solar wind (Burlaga and Vinas, Physica A {\bf 356}, 375 (2005)) and its possible relation with the space-dimension $d$ and with the range of the interactions (characterized by $\alpha$, the attractive potential energy being assumed to decay as $r^{-\alpha}$ at long distances $r$).Comment: 9 pages including 3 figures. Invited lecture at the International Conference on Complexity and Nonextensivity - New Trends in Statistical Mechanics (Yukawa Institute for Theoretical Physics, Kyoto, 14-18 March 2005). To appear in Prog. Theor. Phys. Supplement, eds. M. Sakagami, N. Suzuki and S. Ab

Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems

It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional ($S_{BG}$) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely mandate the BG form $S_{BG}=-k\sum_i p_i \ln p_i$; the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached with nonadditive entropies such as the $S_q=k \frac{1-\sum_i p_i^q}{q-1}$ one ($q \in {\cal R}; \,S_1=S_{BG}$), basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate $S_q$. We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. We arrive to this conclusion by analyzing specific classes of strongly correlated complex systems that await such generalization.Comment: This new version has been sensibly modified and updated. The title and abstract have been modifie

Nonextensive statistical mechanics: A brief review of its present status

We briefly review the present status of nonextensive statistical mechanics. We focus on (i) the central equations of the formalism, (ii) the most recent applications in physics and other sciences, (iii) the {\it a priori} determination (from microscopic dynamics) of the entropic index $q$ for two important classes of physical systems, namely low-dimensional maps (both dissipative and conservative) and long-range interacting many-body hamiltonian classical systems.Comment: Brief review to appear in Annals of the Brazilian Academy of Sciences [http://www.scielo.br/scielo.php] Latex, 7 fig