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

Nonadditive entropy: the concept and its use

The entropic form $S_q$ is, for any $q \neq 1$, {\it nonadditive}. Indeed, for two probabilistically independent subsystems, it satisfies $S_q(A+B)/k=[S_q(A)/k]+[S_q(B)/k]+(1-q)[S_q(A)/k][S_q(B)/k] \ne S_q(A)/k+S_q(B)/k$. This form will turn out to be {\it extensive} for an important class of nonlocal correlations, if $q$ is set equal to a special value different from unity, noted $q_{ent}$ (where $ent$ stands for $entropy$). In other words, for such systems, we verify that $S_{q_{ent}}(N) \propto N (N>>1)$, thus legitimating the use of the classical thermodynamical relations. Standard systems, for which $S_{BG}$ is extensive, obviously correspond to $q_{ent}=1$. Quite complex systems exist in the sense that, for them, no value of $q$ exists such that $S_q$ is extensive. Such systems are out of the present scope: they might need forms of entropy different from $S_q$, or perhaps -- more plainly -- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with $S_q$, the $q$-generalizations of the Central Limit Theorem and of its extended L\'evy-Gnedenko form have been achieved. These recent theorems could of course be the cause of the ubiquity of $q$-exponentials, $q$-Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as well as presently available experimental, observational and computational confirmations -- in high energy physics and elsewhere --, are briefly reviewed. Finally, we address a confusion which is quite common in the literature, namely referring to distinct physical mechanisms {\it versus} distinct regimes of a single physical mechanism.Comment: Brief review to appear in "Statistical Power-Law Tails in High Energy Phenomena", ed. T.S. Biro, Eur. Phys. J. A (2009);10 pages including 3 figure

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

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