The Nonadditive Entropy Sq: A Door Open to the Nonuniversality of the Mathematical Expression of the Clausius Thermodynamic Entropy in Terms of the Probabilities of the Microscopic Configurations

Abstract

Clausius introduced, in the 1860s, a thermodynamical quantity which he named entropy S. This thermodynamically crucial quantity was proposed to be extensive, i.e., in contemporary terms, S(N) ∝ N in the thermodynamic limit N →∞. 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 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<sub>BG</sub> = −k<sub>B</sub> ∑<sup>w</sup><sub>i</sub>=1 pi ln pi, with ∑<sup>W</sup><sub>i</sub>=1 pi=1, k<sub>B</sub> being nowadays called the Boltzmann constant (BG stands for Boltzmann-Gibbs, to also acknowledge the fact that Gibbs provided a wider sense for W). The BG entropy is additive, meaning that, if A and B are two probabilistically independent systems, then S<sub>BG</sub>(A+B) = S<sub>BG</sub>(A)+S<sub>BG</sub>(B). These two words, extensive and additive, were practically treated by physicists, for over more than one century, as almost synonyms, and S<sub>BG</sub> was considered to be the unique form that S could take. In other words, the functional SBG was considered to be universal. It has become increasingly clear today that it is not so, and that those two words are not synonyms, but happen to coincide whenever we are dealing with paradigmatic Hamiltonians involving short-range interactions between their elements, presenting no strong frustration and other “pathologies”. Consistently, it is today allowed to think that the entropic functional connecting S with the microscopic world transparently appears to be nonuniversal, but is rather dictated by the nature of possible strong correlations between the elements of the system. 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 Sq 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.