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The nonadditive entropy $S_q$: A door open to the nonuniversality of the mathematical expression of the Clausius thermodynamic entropy in terms of the probabilities of the microscopic configurations
Clausius introduced, in the 1860s, a thermodynamical quantity which he named {\it entropy} $S$. This thermodynamically crucial quantity was proposed to be {\it extensive}, i.e., in contemporary terms, $S(N) \propto N$ in the thermodynamic limit $N \to\infty$. A decade later, Boltzmann proposed a functional form for this quantity which connects $S$ with the occurrence probabilities of the microscopic configurations (referred to as {\it complexions} at that time) of the system. This functional is, if written in modern words referring to a system with $W$ possible discrete states, $S_{BG}=-k_B \sum_{i=1}^W p_i \ln p_i$. The BG entropy is {\it additive}, meaning that, if A and B are two probabilistically independent systems, then $S_{BG}(A+B)=S_{BG}(A)+S_{BG}(B)$. The words, {\it extensive} and {\it additive}, were practically treated, for over more than one century, as almost synonyms, and $S_{BG}$ was considered to be the unique form that $S$ could take. In other words, the functional $S_{BG}$ was considered to be universal. It has become increasingly clear today that it is {\it not} so, and that those two words are {\it not} synonyms, but happen to coincide whenever we are dealing with paradigmatic Hamiltonians involving {\it short-range} interactions between their elements, presenting no strong frustration and other "pathologies". These facts constitute the basis of a generalization of the BG entropy and statistical mechanics, introduced in 1988, and frequently referred to as nonadditive entropy $S_q$ and nonextensive statistical mechanics, respectively. We briefly review herein these points, and exhibit recent as well as typical applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational predictions and verifications.
