Senses along Which the Entropy Sq Is Unique

Abstract

The Boltzmann–Gibbs–von Neumann–Shannon additive entropy SBG=−k∑ipilnpi 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=k1−∑ipiqq−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.

Non-Additive Entropic Forms and Evolution Equations for Continuous and Discrete Probabilities

“…The solution of this NLFPE is the so-called q-Gaussian distribution (cf. Equation ( 4)), which was shown to cover a large number of experimental verifications within the context of anomalous diffusion phenomena, for which the value of q may vary for different systems [4][5][6][7][8]. An NLFPE with a variable index q, its solution, as well as a possible H-theorem, require a particular nontrivial analysis, which, to our knowledge, has not been addressed at present.…”

“…Recently, there has been a growing interest in generalized entropic forms for an appropriate description of complex systems [4][5][6][7][8][9][10][11]. In most cases, these entropic forms may be written as…”

“…These systems are usually characterized by a large number of components immersed in random or disordered media that interact through long-range forces and/or possess long time memories; as a consequence, they may present a collective behavior very different from those of their individual components. Many of the above-mentioned phenomena have been understood appropriately by means of proposals of generalized entropies [4][5][6][7][8][9][10][11][12], which have found grounds on diverse applications within the realm of complex systems (see, e.g., Ref. [9] for a comprehensive list of entropic forms available in the literature up to 2011).…”

The Chromatic Entropy of Linear Supertrees and Its Application

“…Shannon entropy is of great importance in the field of graph structure information theory. Based on Shannon entropy and some graph variables, many graph entropies were proposed; we refer to the reader to [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. For graph entropy, there are lots of applications in chemistry, network, biology and so on; we refer to the reader to [16][17][18][19][20][21][22][23].…”