Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

Abstract

We present the main features of the mathematical theory generated by the κ-deformed exponential function exp κ (x) = (developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ-algebra, we present the associated κ-differential and κ-integral calculus. Then, we obtain the corresponding κ-exponential and κ-logarithm functions and give the κ-version of the main functions of the ordinary mathematics.

Shannon, Tsallis And Kaniadakis Entropies In Bi-level Image Thresholding

“…In fact, in a generalization of statistical mechanics, a deformed entropy had been proposed, the Kaniadakis entropy, also known as κ-entropy [6,7]:…”

“…In this paper, we will discuss the use of Shannon and Tsallis entropies for image thresholding. Among the other formulations of entropy [5], here, we propose the thresholding using Kaniadakis entropy, which is a quite attractive entropy based on the relativistic formulation of the statistical mechanics [6,7]. Both Tsallis and Kaniadakis entropies have an entropic index.…”

Convexity and the Euclidean Metric of Space-Time

“…This tacitly assumes that the underlying quantities characterizing geometric characteristics of the system are proportional to the effective measure(s) used in the calculations of the pertinent statistical quantities. This is clearly true for ergodic systems, but it can also be true for non-ergodic systems such as the ones whose thermodynamic behavior is conjecturally described by any of the many recently proposed entropic functionals, such as the "Tsallis entropy" [15], or the "κ-entropy" [16], for instance. So, from a geometric viewpoint, equations like (6) express this insensitivity to small-scale details.…”

Statistics of Binary Exchange of Energy or Money

“…In particular, we shall take as F the energy ratio R defined in (4). Note that L and F do not depend on the choice of the parameter t. The factor which multiplies F in the integral (9) is the parameter-invariant infinitesimal length ds of the path.…”

“…A unifying element in this analysis is the generalized κ-distribution [3][4][5][6], which has proven to be helpful for the description of both kinds of phenomena. We start from the familiar assumption of statistical mechanics according to which the macroscopic distributions (of energy or income) are determined by the microscopic interactions (between molecules or, respectively, individuals).…”