Physics:Kaniadakis statistics

Short description: Statistical physics approach

}} 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

Plot of the κ-exponential function [math]\displaystyle{ \exp_\kappa(x) }[/math] for three different κ-values. The solid black curve corresponding to the ordinary exponential function [math]\displaystyle{ \exp(x) }[/math] ([math]\displaystyle{ \kappa = 0 }[/math]).

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

[math]\displaystyle{ \exp_{\kappa} (x) = \begin{cases} \Big(\sqrt{1+\kappa^2 x^2}+\kappa x \Big)^\frac{1}{\kappa} & \text{if } 0 \lt \kappa \lt 1. \\[6pt] \exp(x) & \text{if }\kappa = 0, \\[8pt] \end{cases} }[/math]

with [math]\displaystyle{ \exp_{-\kappa} (x) = \exp_{\kappa} (x) }[/math].

The κ-exponential for [math]\displaystyle{ 0 \lt \kappa \lt 1 }[/math] can also be written in the form:

[math]\displaystyle{ \exp_{\kappa} (x) = \exp\Bigg(\frac{1}{\kappa} \text{arcsinh} (\kappa x)\Bigg). }[/math]

The first five terms of the Taylor expansion of [math]\displaystyle{ \exp_\kappa(x) }[/math] are given by:

[math]\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 }[/math]

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:

[math]\displaystyle{ \exp_{\kappa} (x) \in \mathbb{C}^\infty(\mathbb{R}) }[/math]

[math]\displaystyle{ \frac{d}{dx}\exp_{\kappa} (x) \gt 0 }[/math]

[math]\displaystyle{ \frac{d^2}{dx^2}\exp_{\kappa} (x) \gt 0 }[/math]

[math]\displaystyle{ \exp_{\kappa} (-\infty) = 0^+ }[/math]

[math]\displaystyle{ \exp_{\kappa} (0) = 1 }[/math]

[math]\displaystyle{ \exp_{\kappa} (+\infty) = +\infty }[/math]

[math]\displaystyle{ \exp_{\kappa} (x) \exp_{\kappa} (-x) = -1 }[/math]

For a real number [math]\displaystyle{ r }[/math], the κ-exponential has the property:

[math]\displaystyle{ \Big[\exp_{\kappa} (x)\Big]^r = \exp_{\kappa/r} (rx) }[/math].

κ-logarithm function

Plot of the κ-logarithmic function [math]\displaystyle{ \ln_\kappa(x) }[/math] for three different κ-values. The solid black curve corresponding to the ordinary logarithmic function [math]\displaystyle{ \ln(x) }[/math] ([math]\displaystyle{ \kappa = 0 }[/math]).

The Kaniadakis logarithm (or κ-logarithm) is a relativistic one-parameter generalization of the ordinary logarithm function,

[math]\displaystyle{ \ln_{\kappa} (x) = \begin{cases} \frac{x^\kappa - x^{-\kappa}}{2\kappa} & \text{if } 0 \lt \kappa \lt 1, \\[8pt] \ln(x) & \text{if }\kappa = 0, \\[8pt] \end{cases} }[/math]

with [math]\displaystyle{ \ln_{-\kappa} (x) = \ln_{\kappa} (x) }[/math], which is the inverse function of the κ-exponential:

[math]\displaystyle{ \ln_{\kappa}\Big( \exp_{\kappa}(x)\Big) = \exp_{\kappa}\Big( \ln_{\kappa}(x)\Big) = x. }[/math]

The κ-logarithm for [math]\displaystyle{ 0 \lt \kappa \lt 1 }[/math] can also be written in the form:

[math]\displaystyle{ \ln_{\kappa}(x) = \frac{1}{\kappa}\sinh\Big(\kappa \ln(x)\Big) }[/math]

The first three terms of the Taylor expansion of [math]\displaystyle{ \ln_\kappa(x) }[/math] are given by:

[math]\displaystyle{ \ln_{\kappa} (1+x) = x - \frac{x^2}{2} + \left( 1 + \frac{\kappa^2}{2}\right) \frac{x^3}{3} - \cdots }[/math]

following the rule

[math]\displaystyle{ \ln_{\kappa}(1+x) = \sum_{n=1}^{\infty} b_n(\kappa)\,(-1)^{n-1} \,\frac{x^n}{n} }[/math]

with [math]\displaystyle{ b_1(\kappa)= 1 }[/math], and

[math]\displaystyle{ b_{n}(\kappa) (x) = \begin{cases} 1 & \text{if } n = 1, \\[8pt] \frac{1}{2}\Big(1-\kappa\Big)\Big(1-\frac{\kappa}{2}\Big)... \Big(1-\frac{\kappa}{n-1}\Big) ,\,+\,\frac{1}{2}\Big(1+\kappa\Big)\Big(1+\frac{\kappa}{2}\Big)... \Big(1+\frac{\kappa}{n-1}\Big) & \text{for } n \gt 1, \\[8pt] \end{cases} }[/math]

where [math]\displaystyle{ b_n(0)=1 }[/math] and [math]\displaystyle{ b_n(-\kappa)=b_n(\kappa) }[/math]. The two first terms of the Taylor expansion of [math]\displaystyle{ \ln_\kappa(x) }[/math] are the same as an ordinary logarithmic function.

Basic properties

The κ-logarithm function has the following properties of a logarithmic function:

[math]\displaystyle{ \ln_{\kappa} (x) \in \mathbb{C}^\infty(\mathbb{R}^+) }[/math]

[math]\displaystyle{ \frac{d}{dx}\ln_{\kappa} (x) \gt 0 }[/math]

[math]\displaystyle{ \frac{d^2}{dx^2}\ln_{\kappa} (x) \lt 0 }[/math]

[math]\displaystyle{ \ln_{\kappa} (0^+) = -\infty }[/math]

[math]\displaystyle{ \ln_{\kappa} (1) = 0 }[/math]

[math]\displaystyle{ \ln_{\kappa} (+\infty) = +\infty }[/math]

[math]\displaystyle{ \ln_{\kappa} (1/x) = -\ln_{\kappa} (x) }[/math]

For a real number [math]\displaystyle{ r }[/math], the κ-logarithm has the property:

[math]\displaystyle{ \ln_{\kappa} (x^r) = r \ln_{r \kappa} (x) }[/math]

κ-Algebra

κ-sum

For any [math]\displaystyle{ x,y \in \mathbb{R} }[/math] and [math]\displaystyle{ |\kappa| \lt 1 }[/math], the Kaniadakis sum (or κ-sum) is defined by the following composition law:

[math]\displaystyle{ x\stackrel{\kappa}{\oplus}y=x\sqrt{1+\kappa^2y^2}+y\sqrt{1+\kappa^2x^2} }[/math],

that can also be written in form:

[math]\displaystyle{ x\stackrel{\kappa}{\oplus}y={1\over\kappa}\,\sinh \left({\rm arcsinh}\,(\kappa x)\,+\,{\rm arcsinh}\,(\kappa y)\,\right) }[/math],

where the ordinary sum is a particular case in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math]: [math]\displaystyle{ x\stackrel{0}{\oplus}y=x + y }[/math].

The κ-sum, like the ordinary sum, has the following properties:

[math]\displaystyle{ \text{1. associativity:} \quad (x\stackrel{\kappa}{\oplus}y)\stackrel{\kappa}{\oplus}z =x \stackrel{\kappa}{\oplus} (y \stackrel{\kappa}{\oplus} z) }[/math]

[math]\displaystyle{ \text{2. neutral element:} \quad x \stackrel{\kappa}{\oplus} 0 = 0 \stackrel{\kappa}{\oplus}x=x }[/math]

[math]\displaystyle{ \text{3. opposite element:} \quad x\stackrel{\kappa}{\oplus}(-x)=(-x) \stackrel{\kappa}{\oplus}x=0 }[/math]

[math]\displaystyle{ \text{4. commutativity:} \quad x\stackrel{\kappa}{\oplus}y=y\stackrel{\kappa}{\oplus}x }[/math]

The κ-difference [math]\displaystyle{ \stackrel{\kappa}{\ominus} }[/math] is given by [math]\displaystyle{ x\stackrel{\kappa}{\ominus}y=x\stackrel{\kappa}{\oplus}(-y) }[/math].

The fundamental property [math]\displaystyle{ \exp_{\kappa}(-x)\exp_{\kappa}(x)=1 }[/math] arises as a special case of the more general expression below: [math]\displaystyle{ \exp_{\kappa}(x)\exp_{\kappa}(y)=exp_\kappa(x\stackrel{\kappa}{\oplus}y) }[/math]

Furthermore, the κ-functions and the κ-sum present the following relationships:

[math]\displaystyle{ \ln_\kappa(x\,y) = \ln_\kappa(x) \stackrel{\kappa}{\oplus}\ln_\kappa(y) }[/math]

κ-product

For any [math]\displaystyle{ x,y \in \mathbb{R} }[/math] and [math]\displaystyle{ |\kappa| \lt 1 }[/math], the Kaniadakis product (or κ-product) is defined by the following composition law:

[math]\displaystyle{ x\stackrel{\kappa}{\otimes}y={1\over\kappa}\,\sinh \left(\,{1\over\kappa}\,\,{\rm arcsinh}\,(\kappa x)\,\,{\rm arcsinh}\,(\kappa y)\,\right) }[/math],

where the ordinary product is a particular case in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math]: [math]\displaystyle{ x\stackrel{0}{\otimes}y=x \times y=xy }[/math].

The κ-product, like the ordinary product, has the following properties:

[math]\displaystyle{ \text{1. associativity:} \quad (x \stackrel{\kappa}{\otimes}y) \stackrel{\kappa}{\otimes}z=x \stackrel{\kappa}{\otimes}(y \stackrel{\kappa}{\otimes}z) }[/math]

[math]\displaystyle{ \text{2. neutral element:} \quad x \stackrel{\kappa}{\otimes}I=I \stackrel{\kappa}{\otimes}x= x \quad \text{for} \quad I=\kappa^{-1}\sinh \kappa \stackrel{\kappa}{\oplus}x=x }[/math]

[math]\displaystyle{ \text{3. inverse element:} \quad x \stackrel{\kappa}{\otimes}\overline x= \overline x \stackrel{\kappa}{\otimes}x=I \quad \text{for} \quad \overline x=\kappa^{-1}\sinh(\kappa^2/{\rm arcsinh} \,(\kappa x)) }[/math]

[math]\displaystyle{ \text{4. commutativity:} \quad x\stackrel{\kappa}{\otimes}y=y\stackrel{\kappa}{\otimes}x }[/math]

The κ-division [math]\displaystyle{ \stackrel{\kappa}{\oslash} }[/math] is given by [math]\displaystyle{ x\stackrel{\kappa}{\oslash}y=x\stackrel{\kappa}{\otimes}\overline y }[/math].

The κ-sum [math]\displaystyle{ \stackrel{\kappa}{\oplus} }[/math] and the κ-product [math]\displaystyle{ \stackrel{\kappa}{\otimes} }[/math] obey the distributive law: [math]\displaystyle{ z\stackrel{\kappa}{\otimes}(x \stackrel{\kappa}{\oplus}y) = (z \stackrel{\kappa}{\otimes}x) \stackrel{\kappa}{\oplus}(z \stackrel{\kappa}{\otimes}y) }[/math].

The fundamental property [math]\displaystyle{ \ln_{\kappa}(1/x)=-\ln_{\kappa}(x) }[/math] arises as a special case of the more general expression below:

[math]\displaystyle{ \ln_\kappa(x\,y) = \ln_\kappa(x)\stackrel{\kappa}{\oplus} \ln_\kappa(y) }[/math]

Furthermore, the κ-functions and the κ-product present the following relationships:

[math]\displaystyle{ \exp_\kappa(x) \stackrel{\kappa}{\otimes} \exp_\kappa(y) = \exp_\kappa(x\,+\,y) }[/math]

[math]\displaystyle{ \ln_\kappa(x\,\stackrel{\kappa}{\otimes}\,y) = \ln_\kappa(x) + \ln_\kappa(y) }[/math]

κ-Calculus

κ-Differential

The Kaniadakis differential (or κ-differential) of [math]\displaystyle{ x }[/math] is defined by:

[math]\displaystyle{ \mathrm{d}_{\kappa}x= \frac{\mathrm{d}\,x}{\displaystyle{\sqrt{1+\kappa^2\,x^2} }} }[/math].

So, the κ-derivative of a function [math]\displaystyle{ f(x) }[/math] is related to the Leibniz derivative through:

[math]\displaystyle{ \frac{\mathrm{d} f(x)}{\mathrm{d}_{\kappa}x} = \gamma_\kappa (x) \frac{\mathrm{d} f(x)}{\mathrm{d} x} }[/math],

where [math]\displaystyle{ \gamma_\kappa(x) = \sqrt{1+\kappa^2 x^2} }[/math] is the Lorentz factor. The ordinary derivative [math]\displaystyle{ \frac{\mathrm{d} f(x)}{\mathrm{d} x} }[/math] is a particular case of κ-derivative [math]\displaystyle{ \frac{\mathrm{d} f(x)}{\mathrm{d}_{\kappa}x} }[/math] in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

κ-Integral

The Kaniadakis integral (or κ-integral) is the inverse operator of the κ-derivative defined through

[math]\displaystyle{ \int \mathrm{d}_{\kappa}x \,\, f(x)= \int \frac{\mathrm{d}\, x}{\sqrt{1+\kappa^2\,x^2}}\,\,f(x) }[/math],

which recovers the ordinary integral in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

κ-Trigonometry

κ-Cyclic Trigonometry

[click on the figure] Plot of the κ-sine and κ-cosine functions for [math]\displaystyle{ \kappa = 0 }[/math] (black curve) and [math]\displaystyle{ \kappa = 0.1 }[/math] (blue curve).

The Kaniadakis cyclic trigonometry (or κ-cyclic trigonometry) is based on the κ-cyclic sine (or κ-sine) and κ-cyclic cosine (or κ-cosine) functions defined by:

[math]\displaystyle{ \sin_{\kappa}(x) =\frac{\exp_{\kappa}(ix) -\exp_{\kappa}(-ix)}{2i} }[/math],

[math]\displaystyle{ \cos_{\kappa}(x) =\frac{\exp_{\kappa}(ix) +\exp_{\kappa}(-ix)}{2} }[/math],

where the κ-generalized Euler formula is

[math]\displaystyle{ \exp_{\kappa}(\pm ix)=\cos_{\kappa}(x)\pm i\sin_{\kappa}(x) }[/math].:

The κ-cyclic trigonometry preserves fundamental expressions of the ordinary cyclic trigonometry, which is a special case in the limit κ → 0, such as:

[math]\displaystyle{ \cos_{\kappa}^2(x) + \sin_{\kappa}^2(x)=1 }[/math]

[math]\displaystyle{ \sin_{\kappa}(x \stackrel{\kappa}{\oplus} y) = \sin_{\kappa}(x)\cos_{\kappa}(y) + \cos_{\kappa}(x)\sin_{\kappa}(y) }[/math].

The κ-cyclic tangent and κ-cyclic cotangent functions are given by:

[math]\displaystyle{ \tan_{\kappa}(x)=\frac{\sin_{\kappa}(x)}{\cos_{\kappa}(x)} }[/math]

[math]\displaystyle{ \cot_{\kappa}(x)=\frac{\cos_{\kappa}(x)}{\sin_{\kappa}(x)} }[/math].

The κ-cyclic trigonometric functions become the ordinary trigonometric function in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

κ-Inverse cyclic function

The Kaniadakis inverse cyclic functions (or κ-inverse cyclic functions) are associated to the κ-logarithm:

[math]\displaystyle{ {\rm arcsin}_{\kappa}(x)=-i\ln_{\kappa}\left(\sqrt{1-x^2}+ix\right) }[/math],

[math]\displaystyle{ {\rm arccos}_{\kappa}(x)=-i\ln_{\kappa}\left(\sqrt{x^2-1}+x\right) }[/math],

[math]\displaystyle{ {\rm arctan}_{\kappa}(x)=i\ln_{\kappa}\left(\sqrt{\frac{1-ix}{1+ix}}\right) }[/math],

[math]\displaystyle{ {\rm arccot}_{\kappa}(x)=i\ln_{\kappa}\left(\sqrt{\frac{ix+1}{ix-1}}\right) }[/math].

κ-Hyperbolic Trigonometry

The Kaniadakis hyperbolic trigonometry (or κ-hyperbolic trigonometry) is based on the κ-hyperbolic sine and κ-hyperbolic cosine given by:

[math]\displaystyle{ \sinh_{\kappa}(x) =\frac{\exp_{\kappa}(x) -\exp_{\kappa}(-x)}{2} }[/math],

[math]\displaystyle{ \cosh_{\kappa}(x) =\frac{\exp_{\kappa}(x) +\exp_{\kappa}(-x)}{2} }[/math],

where the κ-Euler formula is

[math]\displaystyle{ \exp_{\kappa}(\pm x)=\cosh_{\kappa}(x)\pm \sinh_{\kappa}(x) }[/math].

The κ-hyperbolic tangent and κ-hyperbolic cotangent functions are given by:

[math]\displaystyle{ \tanh_{\kappa}(x)=\frac{\sinh_{\kappa}(x)}{\cosh_{\kappa}(x)} }[/math]

[math]\displaystyle{ \coth_{\kappa}(x)=\frac{\cosh_{\kappa}(x)}{\sinh_{\kappa}(x)} }[/math].

The κ-hyperbolic trigonometric functions become the ordinary hyperbolic trigonometric functions in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

From the κ-Euler formula and the property [math]\displaystyle{ \exp_{\kappa}(-x)\exp_{\kappa}(x)=1 }[/math] the fundamental expression of κ-hyperbolic trigonometry is given as follows:

[math]\displaystyle{ \cosh_{\kappa}^2(x)- \sinh_{\kappa}^2(x)=1 }[/math]

κ-Inverse hyperbolic function

The Kaniadakis inverse hyperbolic functions (or κ-inverse hyperbolic functions) are associated to the κ-logarithm:

[math]\displaystyle{ {\rm arcsinh}_{\kappa}(x)=\ln_{\kappa}\left(\sqrt{1+x^2}+x\right) }[/math],

[math]\displaystyle{ {\rm arccosh}_{\kappa}(x)=\ln_{\kappa}\left(\sqrt{x^2-1}+x\right) }[/math],

[math]\displaystyle{ {\rm arctanh}_{\kappa}(x)=\ln_{\kappa}\left(\sqrt{\frac{1+x}{1-x}}\right) }[/math],

[math]\displaystyle{ {\rm arccoth}_{\kappa}(x)=\ln_{\kappa}\left(\sqrt{\frac{1-x}{1+x}}\right) }[/math],

in which are valid the following relations:

[math]\displaystyle{ {\rm arcsinh}_{\kappa}(x) = {\rm sign}(x){\rm arccosh}_{\kappa}\left(\sqrt{1+x^2}\right) }[/math],

[math]\displaystyle{ {\rm arcsinh}_{\kappa}(x) = {\rm arctanh}_{\kappa}\left(\frac{x}{\sqrt{1+x^2}}\right) }[/math],

[math]\displaystyle{ {\rm arcsinh}_{\kappa}(x) = {\rm arccoth}_{\kappa}\left(\frac{\sqrt{1+x^2}}{x}\right) }[/math].

The κ-cyclic and κ-hyperbolic trigonometric functions are connected by the following relationships:

[math]\displaystyle{ {\rm sin}_{\kappa}(x) = -i{\rm sinh}_{\kappa}(ix) }[/math],

[math]\displaystyle{ {\rm cos}_{\kappa}(x) = {\rm cosh}_{\kappa}(ix) }[/math],

[math]\displaystyle{ {\rm tan}_{\kappa}(x) = -i{\rm tanh}_{\kappa}(ix) }[/math],

[math]\displaystyle{ {\rm cot}_{\kappa}(x) = i{\rm coth}_{\kappa}(ix) }[/math],

[math]\displaystyle{ {\rm arcsin}_{\kappa}(x)=-i\,{\rm arcsinh}_{\kappa}(ix) }[/math],

[math]\displaystyle{ {\rm arccos}_{\kappa}(x)\neq -i\,{\rm arccosh}_{\kappa}(ix) }[/math],

[math]\displaystyle{ {\rm arctan}_{\kappa}(x)=-i\,{\rm arctanh}_{\kappa}(ix) }[/math],

[math]\displaystyle{ {\rm arccot}_{\kappa}(x)=i\,{\rm arccoth}_{\kappa}(ix) }[/math].

Kaniadakis entropy

The Kaniadakis statistics is based on the Kaniadakis κ-entropy, which is defined through:

[math]\displaystyle{ S_\kappa \big(p\big) = -\sum_i p_i \ln_{\kappa}\big(p_i\big) = \sum_i p_i \ln_{\kappa}\bigg(\frac{1}{p_i} \bigg) }[/math]

where [math]\displaystyle{ p = \{p_i = p(x_i); x \in \mathbb{R}; i = 1, 2, ..., N; \sum_i p_i = 1\} }[/math] is a probability distribution function defined for a random variable [math]\displaystyle{ X }[/math], and [math]\displaystyle{ 0 \leq |\kappa| \lt 1 }[/math] is the entropic index.

The Kaniadakis κ-entropy is thermodynamically and Lesche stable and obeys the Shannon-Khinchin axioms of continuity, maximality, generalized additivity and expandability.

Kaniadakis distributions

A Kaniadakis distribution (or κ-distribution) is a probability distribution derived from the maximization of Kaniadakis entropy under appropriate constraints. In this regard, several probability distributions emerge for analyzing a wide variety of phenomenology associated with experimental power-law tailed statistical distributions.

Kaniadakis integral transform

κ-Laplace Transform

The Kaniadakis Laplace transform (or κ-Laplace transform) is a κ-deformed integral transform of the ordinary Laplace transform. The κ-Laplace transform converts a function [math]\displaystyle{ f }[/math] of a real variable [math]\displaystyle{ t }[/math] to a new function [math]\displaystyle{ F_\kappa(s) }[/math] in the complex frequency domain, represented by the complex variable [math]\displaystyle{ s }[/math]. This κ-integral transform is defined as:

[math]\displaystyle{ F_{\kappa}(s)={\cal L}_{\kappa}\{f(t)\}(s)=\int_{\, 0}^{\infty}\!f(t) \,[\exp_{\kappa}(-t)]^s\,dt }[/math]

The inverse κ-Laplace transform is given by:

[math]\displaystyle{ f(t)={\cal L}^{-1}_{\kappa}\{F_{\kappa}(s)\}(t)={\frac{1}{2\pi i}\int_{c-i \infty}^{c+i \infty}\!F_{\kappa}(s) \,\frac{[\exp_{\kappa}(t)]^s}{\sqrt{1+\kappa^2t^2}}\,ds} }[/math]

The ordinary Laplace transform and its inverse transform are recovered as [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Properties

Let two functions [math]\displaystyle{ f(t) = {\cal L}^{-1}_{\kappa}\{F_{\kappa}(s)\}(t) }[/math] and [math]\displaystyle{ g(t) = {\cal L}^{-1}_{\kappa}\{G_{\kappa}(s)\}(t) }[/math], and their respective κ-Laplace transforms [math]\displaystyle{ F_\kappa(s) }[/math] and [math]\displaystyle{ G_\kappa(s) }[/math], the following table presents the main properties of κ-Laplace transform:

Properties of the κ-Laplace transform

Property	[math]\displaystyle{ f(t) }[/math]	[math]\displaystyle{ F_\kappa(s) }[/math]
Linearity	[math]\displaystyle{ a\, f (t)+ b\, g (t) }[/math]	[math]\displaystyle{ a\, F_{\kappa} (s)+ b\, G_{\kappa} (s) }[/math]
Time scaling	[math]\displaystyle{ f(at) }[/math]	[math]\displaystyle{ \frac{1}{a}\, F_{\kappa / a} (\frac{s}{a}) }[/math]
Frequency shifting	[math]\displaystyle{ f(t)\, [\exp_{\kappa}(-t)]^{a} }[/math]	[math]\displaystyle{ F_{\kappa}(s-a) }[/math]
Derivative	[math]\displaystyle{ \frac{d\, f(t)}{dt} }[/math]	[math]\displaystyle{ s\, {\cal L}_{\kappa}\left \{\frac{f(t)}{\sqrt{1+\kappa^2 t^2}}\right \}(s)-f(0) }[/math]
Derivative	[math]\displaystyle{ \frac{d}{dt} \, \sqrt{1+\kappa^2 t^2} \, f(t) }[/math]	[math]\displaystyle{ s \, F_{\kappa} (s) -f(0) }[/math]
Time-domain integration	[math]\displaystyle{ \frac{1}{\sqrt{1+\kappa^2 t^2}}\, \int_0^t f(w)dw }[/math]	[math]\displaystyle{ \frac{1}{s} \, F_{\kappa} (s) }[/math]
	[math]\displaystyle{ f(t)\, [\ln (\exp_{\kappa}(t))]^n }[/math]	[math]\displaystyle{ (-1)^n \frac{d^{n} F_{\kappa}(s)}{ds^n} }[/math]
	[math]\displaystyle{ f(t) \,[\ln (\exp_{\kappa}(t))]^{-n} }[/math]	[math]\displaystyle{ \int_s^{+\infty}dw_{n} \int_{w_n}^{+\infty}dw_{n-1}...\int_{w_3}^{+\infty}dw_{2}\int_{w_2}^{+\infty}dw_{1} \,F_{\kappa}(w_1) }[/math]
Dirac delta-function	[math]\displaystyle{ \delta (t-\tau) }[/math]	[math]\displaystyle{ [\exp_{\kappa}(-\tau)]^s }[/math]
Heaviside unit function	[math]\displaystyle{ u(t-\tau) }[/math]	[math]\displaystyle{ \frac{s\sqrt{1+\kappa^2 \tau^2}+\kappa^2 \tau}{s^2-\kappa^2}\, [\exp_{\kappa}(-\tau)]^{s} }[/math]
Power function	[math]\displaystyle{ t^{\nu-1} }[/math]	[math]\displaystyle{ \frac{s^2}{s^2-\kappa^2\nu^2}\,\frac{\Gamma_{\frac{\kappa}{s}}(\nu+1)}{\nu\, s^{\nu}}=\frac{s}{s+|\kappa|\nu}\, \frac{\Gamma (\nu)}{|2\kappa|^{\nu}}\, \frac{\Gamma\left( \frac{s}{|2\kappa|} - \frac{\nu}{2} \right )} {\Gamma\left( \frac{s}{|2\kappa|} + \frac{\nu}{2} \right )} }[/math]
Power function	[math]\displaystyle{ t^{2m-1}, \ \ m \in Z^+ }[/math]	[math]\displaystyle{ \frac{(2m-1)!}{\prod_{j=1}^{m}\left[s^2-(2j)^2\kappa^2\right] } }[/math]
Power function	[math]\displaystyle{ t^{2m}, \ \ m \in Z^+ }[/math]	[math]\displaystyle{ \frac{(2m)!\, s}{\prod_{j=1}^{m+1}\left[s^2-(2j-1)^2\kappa^2\right] } }[/math]

The κ-Laplace transforms presented in the latter table reduce to the corresponding ordinary Laplace transforms in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

κ-Fourier Transform

The Kaniadakis Fourier transform (or κ-Fourier transform) is a κ-deformed integral transform of the ordinary Fourier transform, which is consistent with the κ-algebra and the κ-calculus. The κ-Fourier transform is defined as:

[math]\displaystyle{ {\cal F}_\kappa[f(x)](\omega)={1\over\sqrt{2\,\pi}}\int\limits_{-\infty}\limits^{+\infty}f(x)\, \exp_\kappa(-x\otimes_\kappa\omega)^i\,d_\kappa x }[/math]

which can be rewritten as

[math]\displaystyle{ {\cal F}_\kappa[f(x)](\omega)={1\over\sqrt{2\,\pi}}\int\limits_{-\infty}\limits^{+\infty}f(x)\, {\exp(-i\,x_{\{\kappa\}}\,\omega_{\{\kappa\}})\over\sqrt{1+\kappa^2\,x^2}} \,d x }[/math]

where [math]\displaystyle{ x_{\{\kappa\}}=\frac{1}{\kappa}\, {\rm arcsinh} \,(\kappa\,x) }[/math] and [math]\displaystyle{ \omega_{\{\kappa\}}=\frac{1}{\kappa}\, {\rm arcsinh} \,(\kappa\,\omega) }[/math]. The κ-Fourier transform imposes an asymptotically log-periodic behavior by deforming the parameters [math]\displaystyle{ x }[/math] and [math]\displaystyle{ \omega }[/math] in addition to a damping factor, namely [math]\displaystyle{ \sqrt{1+\kappa^2\,x^2} }[/math].

Real (top panel) and imaginary (bottom panel) part of the kernel [math]\displaystyle{ h_\kappa(x,\omega) }[/math] for typical [math]\displaystyle{ \kappa }[/math]-values and [math]\displaystyle{ \omega = 1 }[/math].

The kernel of the κ-Fourier transform is given by:

[math]\displaystyle{ h_\kappa(x,\omega) = \frac{\exp(-i\,x_{\{\kappa\}}\,\omega_{\{\kappa\}})}\sqrt{1+\kappa^2\,x^2} }[/math]

The inverse κ-Fourier transform is defined as:

[math]\displaystyle{ {\cal F}_\kappa[\hat f(\omega)](x)={1\over\sqrt{2\,\pi}}\int\limits_{-\infty}\limits^{+\infty}\hat f(\omega)\, \exp_\kappa(\omega \otimes_\kappa x)^i\,d_\kappa \omega }[/math]

Let [math]\displaystyle{ u_\kappa(x) = \frac 1 \kappa \cosh\Big(\kappa\ln(x) \Big) }[/math], the following table shows the κ-Fourier transforms of several notable functions:

κ-Fourier transform of several functions

	[math]\displaystyle{ f(x) }[/math]	[math]\displaystyle{ {\cal F}_\kappa[f(x)](\omega) }[/math]
Step function	[math]\displaystyle{ \theta(x) }[/math]	[math]\displaystyle{ \sqrt{2\,\pi}\,\delta(\omega)+{1\over\sqrt{2\,\pi}\,i\,\omega_{\{\kappa\}}} }[/math]
Modulation	[math]\displaystyle{ \cos_\kappa(a \stackrel{\kappa}{\oplus} x) }[/math]	[math]\displaystyle{ \sqrt{\pi\over2}\,u_\kappa(\exp_\kappa(a))\,\left(\delta(\omega+a)+\delta(\omega-a)\right) }[/math]
Causal [math]\displaystyle{ \kappa }[/math]-exponential	[math]\displaystyle{ \theta(x)\,\exp_\kappa(-a \stackrel{\kappa}{\otimes} x) }[/math]	[math]\displaystyle{ {1\over\sqrt{2\,\pi}}{1\over a_{\{\kappa\}}+i\,\omega_{\{\kappa\}}} }[/math]
Symmetric [math]\displaystyle{ \kappa }[/math]-exponential	[math]\displaystyle{ \exp_\kappa(-a \stackrel{\kappa}{\otimes} |x|) }[/math]	[math]\displaystyle{ \sqrt{2\over\pi}\,{a_{\{\kappa\}}\over a_{\{\kappa\}}^2+\omega_{\{\kappa\}}^2} }[/math]
Constant	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ \sqrt{2\,\pi}\,\delta(\omega) }[/math]
[math]\displaystyle{ \kappa }[/math]-Phasor	[math]\displaystyle{ \exp_\kappa\,(a \stackrel{\kappa}{\otimes} x)^i }[/math]	[math]\displaystyle{ \sqrt{2\,\pi}\,u_\kappa(\exp_\kappa(a))\,\delta(\omega-a) }[/math]
Impuslse	[math]\displaystyle{ \delta(x-a) }[/math]	[math]\displaystyle{ {1\over\sqrt{2\,\pi}}{\exp_\kappa\,(\omega \stackrel{\kappa}{\otimes} a)^i\over u_\kappa\left(\exp_\kappa\,(a)\right)} }[/math]
Signum	Sgn[math]\displaystyle{ (x) }[/math]	[math]\displaystyle{ \sqrt{2\over\pi}\,\,{1\over i\,\omega_{\{\kappa\}}} }[/math]
Rectangular	[math]\displaystyle{ \Pi\left({x\over a}\right) }[/math]	[math]\displaystyle{ \sqrt{2\over\pi}\,\,a_{\{\kappa\}}\,{\rm sinc}_\kappa(\omega \stackrel{\kappa}{\otimes} a) }[/math]

The κ-deformed version of the Fourier transform preserves the main properties of the ordinary Fourier transform, as summarized in the following table.

κ-Fourier properties

[math]\displaystyle{ f(x) }[/math]	[math]\displaystyle{ {\cal F}_\kappa[f(x)](\omega) }[/math]
Linearity	[math]\displaystyle{ {\cal F}_\kappa[\alpha\,f(x)+\beta\,g(x)](\omega)=\alpha\,{\cal F}_\kappa[f(x)](\omega)+\beta\,{\cal F}_\kappa[g(x)](\omega) }[/math]
Scaling	[math]\displaystyle{ {\cal F}_\kappa\left[f(\alpha\,x)\right](\omega)={1\over\alpha}\,{\cal F}_{\kappa^\prime}\left[f(x)\right](\omega^\prime) }[/math] where [math]\displaystyle{ \kappa^\prime=\kappa/\alpha }[/math] and [math]\displaystyle{ \omega^\prime=(a/\kappa)\,\sinh\left({\rm arcsinh}(\kappa\,\omega)/a^2\right) }[/math]
[math]\displaystyle{ \kappa }[/math]-Scaling	[math]\displaystyle{ {\cal F}_\kappa\left[f(\alpha \stackrel{\kappa}{\otimes} x)\right](\omega)={1\over\alpha_{\{\kappa\}}}\,{\cal F}_\kappa[f(x)]\left(\frac{1}{\alpha}\stackrel{\kappa}{\otimes}\omega\right) }[/math]
Complex conjugation	[math]\displaystyle{ {\cal F}_\kappa\big[f(x)\big]^{\ast}(\omega)={\cal F}_\kappa\big[f(x)\big](-\omega) }[/math]
Duality	[math]\displaystyle{ {\cal F}_\kappa\Big[{\cal F}_\kappa\big[f(x)\big](\nu)\Big](\omega)=f(-\omega) }[/math]
Reverse	[math]\displaystyle{ {\cal F}_\kappa\left[f(-x)\right](\omega)={\cal F}_\kappa[f(x)](-\omega) }[/math]
[math]\displaystyle{ \kappa }[/math]-Frequency shift	[math]\displaystyle{ {\cal F}_\kappa\left[\exp_\kappa (\omega_0 \stackrel{\kappa}{\otimes} x)^if(x)\right](\omega)={\cal F}_\kappa[f(x)](\omega\stackrel{\kappa}{\ominus}\omega_0) }[/math]
[math]\displaystyle{ \kappa }[/math]-Time shift	[math]\displaystyle{ {\cal F}_\kappa\left[f(x \,\stackrel{\kappa}{\oplus}\,x_0)\right](\omega)=\exp_\kappa (\omega\,\stackrel{\kappa}{\otimes}\, x_0)^i\, {\cal F}_\kappa[f(x)](\omega) }[/math]
Transform of [math]\displaystyle{ \kappa }[/math]-derivative	[math]\displaystyle{ {\cal F}_\kappa\left[\frac{d\,f(x)}{d_\kappa x}\right](\omega)=i\,\omega_{\{\kappa\}}\,{\cal F}_\kappa[f(x)](\omega) }[/math]
[math]\displaystyle{ \kappa }[/math]-Derivative of transform	[math]\displaystyle{ \frac{d}{d_\kappa\omega}\,{\cal F}_\kappa[f(x)](\omega)=-i\,\omega_{\{\kappa\}}\,{\cal F}_\kappa\left[x_{\{\kappa\}}\,f(x)\right](\omega) }[/math]
Transform of integral	[math]\displaystyle{ {\cal F}_\kappa\left[\int\limits_{-\infty}\limits^x f(y)\,dy\right](\omega)={1\over i\,\omega_{\{\kappa\}}}{\cal F}_\kappa[f(x)](\omega)+2\,\pi\,{\cal F}_\kappa[f(x)](0)\,\delta(\omega) }[/math]
[math]\displaystyle{ \kappa }[/math]-Convolution	[math]\displaystyle{ {\cal F}_\kappa\left[(f \,\stackrel{\kappa}{\circledast}\, g)(x)\right](\omega)=\sqrt{2\,\pi}\,{\cal F}_\kappa[f(x)](\omega)\,{\cal F}_\kappa[g(x)](\omega) }[/math] where [math]\displaystyle{ (f \,\stackrel{\kappa}{\circledast}\, g)(x)=\int\limits_{-\infty}\limits^{+\infty} f(y)\,g(x\,\stackrel{\kappa}{\ominus}\, y)\,d_\kappa y }[/math]
Modulation	[math]\displaystyle{ {\cal F}_\kappa\left[f(x)\,g(x)\right](\omega)={1\over\sqrt{2\,\pi}}\left({\cal F}_\kappa\left[f(x)\right] \,\stackrel{\kappa}{\circledast}\, {\cal F}_\kappa\left[g(x)\right]\right)(\omega) }[/math]

The properties of the κ-Fourier transform presented in the latter table reduce to the corresponding ordinary Fourier transforms in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

See also

• Giorgio Kaniadakis
• Kaniadakis distribution
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Gamma distribution
• Kaniadakis κ-Weibull distribution
• Kaniadakis κ-Logistic distribution
• Kaniadakis κ-Erlang distribution

External links

• Giorgio Kaniadakis Google Scholar page
• Kaniadakis Statistics on arXiv.org