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, SBG = −kB ∑wi=1 pi ln pi, with ∑Wi=1 pi=1,
kB 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 SBG(A+B) = SBG(A)+SBG(B). These two words, extensive and additive, were
practically treated by physicists, for over more than one century, as almost synonyms,
and SBG 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.
Keywords: Nonadditive entropy, Nonextensive statistical mechanics, Complex systems,
Central Limit theorems.
