Senses along Which the Entropy Sq Is Unique

The Boltzmann-Gibbs-von Neumann-Shannon additive entropy SBG = k a i pi ln pi as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988 into the nonextensive statistical mechanics-as currently referred to-grounded on the nonadditive entropy Sq = k(1) Sigma p(i)(q) / q- 1 as well as its corresponding continuous and quantum counterparts. In the literature, there exist nowadays over fifty mathematically well defined entropic functionals. Sq plays a special role among them. Indeed, it constitutes the pillar of a great variety of theoretical, experimental, observational and computational validations in the area of complexity-plectics, as Murray Gell-Mann used to call it. Then, a question emerges naturally, namely In what senses is entropy Sq unique? The present effort is dedicated to a-surely non exhaustive-mathematical answer to this basic question.

Automated summary

The Boltzmann-Gibbs-von Neumann-Shannon additive entropy and its nonextensive counterpart have provided a foundation for statistical mechanics in both classical and quantum systems. However, the increasing complexity of natural, artificial, and social systems has made it necessary to develop nonadditive entropic functionals. Among them, the nonextensive entropy Sq has played a special role in the study of complex systems.