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

• Main Author: Giorgio Kaniadakis
• Format: Article
• Language: English
• Published: MDPI AG 2013-09-01
• Series: Entropy
• Subjects: κ-statistical mechanics; κ-mathematics; κ-exponential; κ-logarithm; power-law tailed statistical distributions
• Online Access: http://www.mdpi.com/1099-4300/15/10/3983

We present the main features of the mathematical theory generated by the √ κ-deformed exponential function expκ(x) = ( 1 + κ2x2 + κx)1/κ, with 0 ≤ κ < 1, 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.