Kaniadakis statistics

Kaniadakis statistics (also known as κ-statistics) is a generalization of Boltzmann–Gibbs statistical mechanics, based on a relativistic generalization of the classical Boltzmann–Gibbs–Shannon entropy (commonly referred to as Kaniadakis entropy or κ-entropy). Introduced by the Greek Italian physicist Giorgio Kaniadakis in 2001, κ-statistical mechanics preserve the main features of ordinary statistical mechanics and have attracted the interest of many researchers in recent years. The κ-distribution is currently considered one of the most viable candidates for explaining complex physical, natural or artificial systems involving power-law tailed statistical distributions. Kaniadakis statistics have been adopted successfully in the description of a variety of systems in the fields of cosmology, astrophysics, condensed matter, quantum physics, seismology, genomics, economics, epidemiology, and many others.

Mathematical formalism The mathematical formalism of κ-statistics is generated by κ-deformed functions, especially the κ-exponential function.

κ-exponential function

The Kaniadakis exponential (or κ-exponential) function is a one-parameter generalization of an exponential function, given by:

exp κ ⁡ ( x ) = { ( 1 + κ 2 x 2 + κ x ) 1 κ if 0 < κ < 1. exp ⁡ ( x ) if κ = 0 , {\displaystyle \exp _{\kappa }(x)={\begin{cases}{\Big (}{\sqrt {1+\kappa ^{2}x^{2}}}+\kappa x{\Big )}^{\frac {1}{\kappa }}&{\text{if }}0<\kappa <1.\\[6pt]\exp(x)&{\text{if }}\kappa =0,\\[8pt]\end{cases}}}

with exp − κ ⁡ ( x ) = exp κ ⁡ ( x ) {\displaystyle \exp _{-\kappa }(x)=\exp _{\kappa }(x)} . The κ-exponential for 0 < κ < 1 {\displaystyle 0<\kappa <1} can also be written in the form:

exp κ ⁡ ( x ) = exp ⁡ ( 1 κ arsinh ( κ x ) ) . {\displaystyle \exp _{\kappa }(x)=\exp {\Bigg (}{\frac {1}{\kappa }}{\text{arsinh}}(\kappa x){\Bigg )}.}

The first five terms of the Taylor expansion of exp κ ⁡ ( x ) {\displaystyle \exp _{\kappa }(x)} are given by:

exp κ ⁡ ( x ) = 1 + x + x 2 2 + ( 1 − κ 2 ) x 3 3 ! + ( 1 − 4 κ 2 ) x 4 4 ! + ⋯ {\displaystyle \exp _{\kappa }(x)=1+x+{\frac {x^{2}}{2}}+(1-\kappa ^{2}){\frac {x^{3}}{3!}}+(1-4\kappa ^{2}){\frac {x^{4}}{4!}}+\cdots }

where the first three are the same as a typical exponential function. Basic properties The κ-exponential function has the following properties of an exponential function:

exp κ ⁡ ( x ) ∈ C ∞ ( R ) {\displaystyle \exp _{\kappa }(x)\in \mathbb {C} ^{\infty }(\mathbb {R} )}

d d x exp κ ⁡ ( x ) > 0 {\displaystyle {\frac {d}{dx}}\exp _{\kappa }(x)>0}

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