Group entropies, correlation laws, and zeta functions

Abstract

The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.

Groups, information theory, and Einstein's likelihood principle

“…(5). Then Einsten's likelihood principle (1) follows immediately from the additivity property of the information functional (12). In the case of the multiplicative group (8), we recover the likelihood function recently introduced in [23].…”

“…In other words, the BG entropy and the Rényi entropy are stable with respect to definition (12): the associated group-theoretical information measure coincides with the corresponding entropy. If we consider instead the S a,q entropy, we see that…”

“…DOI: 10.1103/PhysRevE.93.040101 The study of the relations among statistical mechanics, information theory and the notion of entropy is at the heart of the science of complexity, and in the last decades has been widely explored. After the seminal works of Shannon and Tsallis [7], has been the prototype of the nonadditive entropies studied in the last decades [8][9][10][11][12][13]. These functionals are generalizations of the BG entropy; they depend on one or more parameters, in such a way that the BG entropy is recovered as a particular limit.…”

Nonextensive statistical mechanics and high energy physics

“…Summarizing the situation, remarkably enough, for the class W(N) ∝ N ρ , in order to have an extensive thermodynamic entropy we must use a nonadditive entropic functional such as S q and not the (additive) BG one! At this point let us mention another (quite striking) fact, namely that S q is directly related to the Riemann zeta function [4]. Another strongly correlated class corresponds to W(N) ∝ ν N γ (ν > 1; 0 < γ < 1).…”

Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle

“…Generalized statistical mechanics, based on κ-entropy [1,3,4], preserves the main features of ordinary Boltzmann-Gibbs statistical mechanics. For this reason, it has attracted the interest of many researchers over the last 16 years, who have studied its foundations and mathematical aspects [5][6][7][8][9][10][11][12], the underlying thermodynamics [13][14][15][16][17], and specific applications of the theory in various scientific and engineering fields. A non-exhaustive list of application areas includes quantum statistics [18][19][20], quantum entanglement [21,22], plasma physics [23][24][25][26][27], nuclear fission [28], astrophysics [29][30][31][32][33][34][35], geomechanics [36], genomics [37], complex networks [38,39], economy [40][41][42][43] and finance [44][45][46]…”