A magnífica mecânica estatística de Boltzmann-Gibbs, amálgama de primeiros princípios e teoria de probabilidades, constitui um dos pilares da física teórica contemporânea. Entretanto, ela não se aplica a grande número dos sistemas ditos complexos, caracterizados essencialmente por um forte emaranhamento espaço-temporal de seus elementos. Revisamos aqui a proposta de generalização chamada mecânica estatística não extensiva, que emergiu em 1988. Ela está baseada em entropias não aditivas (com índice q ≠ 1), em contraste com a entropia de Boltzmann-Gibbs-von Neumann-Shannon, que é aditiva (com índice q = 1). Sua fundamentação básica, assim como aplicações selecionadas em física e fora dela, são brevemente descritas.

The magnificent Boltzmann-Gibbs statistical mechanics, amalgam of first principles and theory of probabilities, constitutes one of the pillars of contemporary theoretical physics. However, it does not apply to a wide number of the so called complex systems, characterized essentially by a strong space-time entanglement of its elements. We tutorially review here the proposal for its generalization, referred to as nonextensive statistical mechanics, which emerged in 1988. It is based on nonadditive entropies (with index q ≠ 1), in contrast with the Boltzmann-Gibbs-von Neumann-Shannon entropy, which is additive (with index q = 1). Its basic foundations, as well as selected applications in physics and elsewhere, are briefly described.

Statistical mechanics constitutes one of the pillars of contemporary physics. Recognized as such - together with mechanics (classical, quantum, relativistic), electromagnetism and thermodynamics -, it is one of the mandatory theories studied at virtually all the intermediate-and advanced-level courses of physics around the world. As it normally happens with such basic scientific paradigms, it is placed at a crossroads of various other branches of knowledge. In the case of statistical mechanics, the standard theory - hereafter referred to as the Boltzmann-Gibbs (BG) statistical mechanics - exhibits highly relevant connections at a variety of microscopic, mesoscopic and macroscopic physical levels, as well as with the theory of probabilities (in particular, with the Central Limit Theorem, CLT ). In many circumstances, the ubiquitous efects of the CLT , with its Gaussian attractors (in the space of the distributions of probabilities), are present. Within this complex ongoing frame, a possible generalization of the BG theory was advanced in 1988 (C.T., J. Stat. Phys. 52, 479). The extension of the standard concepts is intended to be useful in those "pathological", and nevertheless very frequent, cases where the basic assumptions (molecular chaos hypothesis, ergodicity) for applicability of the BG theory would be violated. Such appears to be, for instance, the case in classical long-range-interacting many-body Hamiltonian systems (at the so-called quasi-stationary state). Indeed, in such systems, the maximal Lyapunov exponent vanishes in the thermodynamic limit N → ∞. This fact creates a quite novel situation with regard to typical BG systems, which generically have a positive value for this central nonlinear dynamical quantity. This peculiarity has sensible effects at all physical micro-, meso-and macroscopic levels. It even poses deep challenges at the level of the CLT . In the present occasion, after 20 years of the 1988 proposal, we undertake here an overview of some selected successes of the approach, and of some interesting points that still remain as open questions. Various theoretical, experimental, observational and computational aspects will be addressed.

Revisamos sumariamente o estado presente da mecânica estatística não-extensiva. Focalizamos em (i) as equacões centrais do formalismo; (ii) as aplicações mais recentes na física e em outras ciências, (iii) a determinação a priori (da dinâmica microscópica) do índice entrópico q para duas classes importantes de sistemas físicos, a saber, mapas de baixa dimensão (tanto dissipativos quanto conservativos) e sistemas clássicos hamiltonianos de muitos corpos com interações de longo alcance.

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 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.

The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cover a variety of anomalous systems. The generalization concerns nonextensive systems, where nonextensivity is understood in the thermodynamical sense. This generalization was first proposed in 1988 inspired by the probabilistic description of multifractal geometries, and has been intensively studied during this decade. In the present effort, after introducing some historical background, we briefly describe the formalism, and then exhibit the present status in what concerns theoretical, experimental and computational evidences and connections, as well as some perspectives for the future. In addition to these, here and there we point out various (possibly) relevant questions, whose answer would certainly clarify our current understanding of the foundations of statistical mechanics and its thermodynamical implications.