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.

We study the dynamics of a system of N classical spins with infinite-range interaction. We show that, if the thermodynamic limit is taken before the infinite-time limit, the system does not relax to the Boltzmann-Gibbs equilibrium, but exhibits different equilibrium properties, characterized by stable non-Gaussian velocity distributions, Lévy walks, and dynamical correlation in phase space.

Through the use of a recently introduced, nonextensive, entropy, we generalize that of Kullback and Leibler [Ann. Math. Stat. 22, 79 (1961)] and study its properties. This in turn enables the proposal of a consistent criterion for testing relevant hypotheses such as the independence of random variables. Straightforward applications are shown to be possible for (physical, geophysical, economic, and biological) time series.

The nonadditive entropy S_q has been introduced in 1988 focusing on a generalization of Boltzmann–Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which the BG theory is expected to be valid. It is now known that S_q has a large applicability; more specifically speaking, even outside Hamiltonian systems and their thermodynamical approach. In the present paper we review and comment some relevant aspects of this entropy, namely (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) The analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (iv) The analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. Finally, we exhibit recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results.

Phase space can be constructed for N equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy S _BG ≡ - k Σ _i p _i ln p _i to be extensive , i.e., S _BG ( N ) ∝ N for N → ∞. In particular, if they are independent, S _BG is strictly additive , i.e., S _BG ( N ) = NS _BG (1), ∀ N . However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy S _q ≡ k [1 - Σ _i p ^q _i ]/( q - 1) (with S ₁ = S _BG ) for some special value of q ≠ 1 to be the one which is extensive [i.e., S _q ( N ) ∝ N for N → ∞]. Another concept which is relevant is strict or asymptotic scale-freedom (or scale-invariance ), defined as the situation for which all marginal probabilities of the N -system coincide or asymptotically approach (for N → ∞) the joint probabilities of the ( N - 1)-system. If each subsystem is a binary one, scale-freedom is guaranteed by what we hereafter refer to as the Leibnitz rule , i.e., the sum of two successive joint probabilities of the N -system coincides or asymptotically approaches the corresponding joint probability of the ( N - 1)-system. The kinds of interplay of these various concepts are illustrated in several examples. One of them justifies the title of this paper. We conjecture that these mechanisms are deeply related to the very frequent emergence, in natural and artificial complex systems, of scale-free structures and to their connections with nonextensive statistical mechanics. Summarizing, we have shown that, for asymptotically scale-invariant systems, it is S _q with q ≠ 1, and not S _BG , the entropy which matches standard, clausius-like, prescriptions of classical thermodynamics.