Approach of complexity in nature: Entropic nonuniqueness

Boltzmann introduced in the 1870's a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His entropic functional for classical systems was extended by Gibbs to the entire phase space of a many-body system, and by von Neumann in order to cover quantum systems as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of $W$ microscopic possibilities, and is given by $S_{BG}= -k\sum_{i=1}^W p_i \ln p_i$ ($k$ is a positive universal constant; {\it BG} stands for {\it Boltzmann-Gibbs}). This relation enables the construction of BG statistical mechanics. The BG theory has provided uncountable important applications. Its application in physical systems is legitimate whenever the hypothesis of {\it ergodicity} is satisfied. However, {\it what can we do when ergodicity and similar simple hypotheses are violated?}, which indeed happens in very many natural, artificial and social complex systems. It was advanced in 1988 the possibility of generalizing BG statistical mechanics through a family of nonadditive entropies, namely $S_q=k\frac{1-\sum_{i=1}^W p_i^q}{q-1}$, which recovers the additive $S_{BG}$ entropy in the $q \to1$ limit. The index $q$ is to be determined from mechanical first principles. Along three decades, this idea intensively evolved world-wide (see Bibliography in \url{http://tsallis.cat.cbpf.br/biblio.htm}), and led to a plethora of predictions, verifications, and applications in physical systems and elsewhere. As expected whenever a {\it paradigm shift} is explored, some controversy naturally emerged as well in the community. The present status of the general picture is here described, starting from its dynamical and thermodynamical foundations, and ending with its most recent physical applications.

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