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Entropy 2013, 15(10), 3983-4010; doi:10.3390/e15103983
Review

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

Received: 29 July 2013; in revised form: 17 September 2013 / Accepted: 17 September 2013 / Published: 25 September 2013
(This article belongs to the collection Advances in Applied Statistical Mechanics)
Download PDF [303 KB, uploaded 25 September 2013]
Abstract: We present the main features of the mathematical theory generated by the κ-deformed exponential function exp k (x) = ( 1 + k 2 x 2 + kx) 1 k , 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.
Keywords: κ-statistical mechanics; κ-mathematics; κ-exponential; κ-logarithm; power-law tailed statistical distributions κ-statistical mechanics; κ-mathematics; κ-exponential; κ-logarithm; power-law tailed statistical distributions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Kaniadakis, G. Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions. Entropy 2013, 15, 3983-4010.

AMA Style

Kaniadakis G. Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions. Entropy. 2013; 15(10):3983-4010.

Chicago/Turabian Style

Kaniadakis, Giorgio. 2013. "Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions." Entropy 15, no. 10: 3983-4010.


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