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_{\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_{-\kappa} (x) = \exp_{\kappa} (x) $$.

The κ-exponential for $$0 < \kappa < 1 $$ can also be written in the form:


 * $$\exp_{\kappa} (x) = \exp\Bigg(\frac{1}{\kappa} \text{arcsinh} (\kappa x)\Bigg).

$$ The first five terms of the Taylor expansion of $$\exp_\kappa(x) $$ are given by: $$\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_{\kappa} (x) \in \mathbb{C}^\infty(\mathbb{R})

$$
 * $$\frac{d}{dx}\exp_{\kappa} (x) > 0

$$
 * $$\frac{d^2}{dx^2}\exp_{\kappa} (x) > 0

$$
 * $$\exp_{\kappa} (-\infty) = 0^+

$$
 * $$\exp_{\kappa} (0) = 1

$$
 * $$\exp_{\kappa} (+\infty) = +\infty

$$
 * $$\exp_{\kappa} (x) \exp_{\kappa} (-x) = -1

$$ For a real number $$r $$, the κ-exponential has the property:
 * $$\Big[\exp_{\kappa} (x)\Big]^r = \exp_{\kappa/r} (rx)

$$.

κ-logarithm function
The Kaniadakis logarithm (or κ-logarithm) is a relativistic one-parameter generalization of the ordinary logarithm function,
 * $$\ln_{\kappa} (x) = \begin{cases}

\frac{x^\kappa - x^{-\kappa}}{2\kappa} & \text{if } 0 < \kappa < 1, \\[8pt] \ln(x) & \text{if }\kappa = 0, \\[8pt] \end{cases} $$ with $$\ln_{-\kappa} (x) = \ln_{\kappa} (x) $$, which is the inverse function of the κ-exponential:


 * $$\ln_{\kappa}\Big( \exp_{\kappa}(x)\Big) = \exp_{\kappa}\Big( \ln_{\kappa}(x)\Big) = x.$$

The κ-logarithm for $$0 < \kappa < 1 $$ can also be written in the form:

$$\ln_{\kappa}(x) = \frac{1}{\kappa}\sinh\Big(\kappa \ln(x)\Big) $$

The first three terms of the Taylor expansion of $$\ln_\kappa(x) $$ are given by:


 * $$\ln_{\kappa} (1+x) = x - \frac{x^2}{2} + \left( 1 + \frac{\kappa^2}{2}\right) \frac{x^3}{3} - \cdots

$$

following the rule


 * $$ \ln_{\kappa}(1+x) = \sum_{n=1}^{\infty} b_n(\kappa)\,(-1)^{n-1}

\,\frac{x^n}{n} $$

with $$ b_1(\kappa)= 1$$, and


 * $$ 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 > 1, \\[8pt] \end{cases} $$

where $$ b_n(0)=1 $$ and $$ b_n(-\kappa)=b_n(\kappa) $$. The two first terms of the Taylor expansion of $$\ln_\kappa(x) $$ are the same as an ordinary logarithmic function.

Basic properties

The κ-logarithm function has the following properties of a logarithmic function:
 * $$\ln_{\kappa} (x) \in \mathbb{C}^\infty(\mathbb{R}^+)

$$
 * $$\frac{d}{dx}\ln_{\kappa} (x) > 0

$$
 * $$\frac{d^2}{dx^2}\ln_{\kappa} (x) < 0

$$
 * $$\ln_{\kappa} (0^+) = -\infty

$$
 * $$\ln_{\kappa} (1) = 0

$$
 * $$\ln_{\kappa} (+\infty) = +\infty

$$
 * $$\ln_{\kappa} (1/x) = -\ln_{\kappa} (x)

$$ For a real number $$r $$, the κ-logarithm has the property:
 * $$\ln_{\kappa} (x^r) = r \ln_{r \kappa} (x)

$$

κ-sum
For any $$x,y \in \mathbb{R}$$ and $$|\kappa| < 1$$, the Kaniadakis sum (or κ-sum) is defined by the following composition law:
 * $$x\stackrel{\kappa}{\oplus}y=x\sqrt{1+\kappa^2y^2}+y\sqrt{1+\kappa^2x^2}

$$, that can also be written in form:
 * $$x\stackrel{\kappa}{\oplus}y={1\over\kappa}\,\sinh

\left({\rm arcsinh}\,(\kappa x)\,+\,{\rm arcsinh}\,(\kappa y)\,\right) $$, where the ordinary sum is a particular case in the classical limit $$\kappa \rightarrow 0 $$: $$x\stackrel{0}{\oplus}y=x + y $$.

The κ-sum, like the ordinary sum, has the following properties:
 * $$\text{1. associativity:} \quad (x\stackrel{\kappa}{\oplus}y)\stackrel{\kappa}{\oplus}z

=x \stackrel{\kappa}{\oplus} (y \stackrel{\kappa}{\oplus} z) $$
 * $$\text{2. neutral element:} \quad x \stackrel{\kappa}{\oplus} 0 = 0

\stackrel{\kappa}{\oplus}x=x $$
 * $$\text{3. opposite element:} \quad x\stackrel{\kappa}{\oplus}(-x)=(-x) \stackrel{\kappa}{\oplus}x=0

$$
 * $$\text{4. commutativity:} \quad x\stackrel{\kappa}{\oplus}y=y\stackrel{\kappa}{\oplus}x

$$ The κ-difference $$\stackrel{\kappa}{\ominus}$$ is given by $$x\stackrel{\kappa}{\ominus}y=x\stackrel{\kappa}{\oplus}(-y)$$.

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

Furthermore, the κ-functions and the κ-sum present the following relationships:
 * $$\ln_\kappa(x\,y) = \ln_\kappa(x) \stackrel{\kappa}{\oplus}\ln_\kappa(y)

$$

κ-product
For any $$x,y \in \mathbb{R}$$ and $$|\kappa| < 1$$, the Kaniadakis product (or κ-product) is defined by the following composition law:
 * $$x\stackrel{\kappa}{\otimes}y={1\over\kappa}\,\sinh

\left(\,{1\over\kappa}\,\,{\rm arcsinh}\,(\kappa x)\,\,{\rm arcsinh}\,(\kappa y)\,\right) $$, where the ordinary product is a particular case in the classical limit $$\kappa \rightarrow 0 $$: $$x\stackrel{0}{\otimes}y=x \times y=xy $$.

The κ-product, like the ordinary product, has the following properties:
 * $$\text{1. associativity:} \quad (x \stackrel{\kappa}{\otimes}y)

\stackrel{\kappa}{\otimes}z=x \stackrel{\kappa}{\otimes}(y \stackrel{\kappa}{\otimes}z) $$
 * $$\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 $$
 * $$\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)) $$
 * $$\text{4. commutativity:} \quad x\stackrel{\kappa}{\otimes}y=y\stackrel{\kappa}{\otimes}x

$$ The κ-division $$\stackrel{\kappa}{\oslash}$$ is given by $$x\stackrel{\kappa}{\oslash}y=x\stackrel{\kappa}{\otimes}\overline y$$.

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

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


 * $$\ln_\kappa(x\,y) = \ln_\kappa(x)\stackrel{\kappa}{\oplus} \ln_\kappa(y)

$$
 * Furthermore, the κ-functions and the κ-product present the following relationships:
 * $$\exp_\kappa(x) \stackrel{\kappa}{\otimes} \exp_\kappa(y) = \exp_\kappa(x\,+\,y)
 * $$\exp_\kappa(x) \stackrel{\kappa}{\otimes} \exp_\kappa(y) = \exp_\kappa(x\,+\,y)

$$
 * $$\ln_\kappa(x\,\stackrel{\kappa}{\otimes}\,y) = \ln_\kappa(x) + \ln_\kappa(y)

$$

κ-Differential
The Kaniadakis differential (or κ-differential) of $$x$$ is defined by:
 * $$\mathrm{d}_{\kappa}x= \frac{\mathrm{d}\,x}{\displaystyle{\sqrt{1+\kappa^2\,x^2} }}

$$.

So, the κ-derivative of a function $$f(x)$$ is related to the Leibniz derivative through:
 * $$ \frac{\mathrm{d} f(x)}{\mathrm{d}_{\kappa}x} = \gamma_\kappa (x) \frac{\mathrm{d} f(x)}{\mathrm{d} x} $$,

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

κ-Integral
The Kaniadakis integral (or κ-integral) is the inverse operator of the κ-derivative defined through
 * $$\int \mathrm{d}_{\kappa}x \,\, f(x)= \int \frac{\mathrm{d}\, x}{\sqrt{1+\kappa^2\,x^2}}\,\,f(x) $$,

which recovers the ordinary integral in the classical limit $$\kappa \rightarrow 0$$.

κ-Cyclic Trigonometry
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:
 * $$\sin_{\kappa}(x) =\frac{\exp_{\kappa}(ix) -\exp_{\kappa}(-ix)}{2i} $$,
 * $$\cos_{\kappa}(x) =\frac{\exp_{\kappa}(ix) +\exp_{\kappa}(-ix)}{2} $$,

where the κ-generalized Euler formula is
 * $$ \exp_{\kappa}(\pm ix)=\cos_{\kappa}(x)\pm i\sin_{\kappa}(x) $$.:

The κ-cyclic trigonometry preserves fundamental expressions of the ordinary cyclic trigonometry, which is a special case in the limit κ → 0, such as:
 * $$\cos_{\kappa}^2(x) + \sin_{\kappa}^2(x)=1 $$
 * $$\sin_{\kappa}(x \stackrel{\kappa}{\oplus} y) = \sin_{\kappa}(x)\cos_{\kappa}(y) + \cos_{\kappa}(x)\sin_{\kappa}(y) $$.

The κ-cyclic tangent and κ-cyclic cotangent functions are given by:
 * $$ \tan_{\kappa}(x)=\frac{\sin_{\kappa}(x)}{\cos_{\kappa}(x)} $$
 * $$ \cot_{\kappa}(x)=\frac{\cos_{\kappa}(x)}{\sin_{\kappa}(x)} $$.

The κ-cyclic trigonometric functions become the ordinary trigonometric function in the classical limit $$\kappa \rightarrow 0$$.

κ-Inverse cyclic function

The Kaniadakis inverse cyclic functions (or κ-inverse cyclic functions) are associated to the κ-logarithm:
 * $$ {\rm arcsin}_{\kappa}(x)=-i\ln_{\kappa}\left(\sqrt{1-x^2}+ix\right) $$,
 * $$ {\rm arccos}_{\kappa}(x)=-i\ln_{\kappa}\left(\sqrt{x^2-1}+x\right) $$,
 * $$ {\rm arctan}_{\kappa}(x)=i\ln_{\kappa}\left(\sqrt{\frac{1-ix}{1+ix}}\right) $$,
 * $$ {\rm arccot}_{\kappa}(x)=i\ln_{\kappa}\left(\sqrt{\frac{ix+1}{ix-1}}\right) $$.

κ-Hyperbolic Trigonometry
The Kaniadakis hyperbolic trigonometry (or κ-hyperbolic trigonometry) is based on the κ-hyperbolic sine and κ-hyperbolic cosine given by:
 * $$\sinh_{\kappa}(x) =\frac{\exp_{\kappa}(x) -\exp_{\kappa}(-x)}{2} $$,
 * $$\cosh_{\kappa}(x) =\frac{\exp_{\kappa}(x) +\exp_{\kappa}(-x)}{2} $$,

where the κ-Euler formula is
 * $$ \exp_{\kappa}(\pm x)=\cosh_{\kappa}(x)\pm \sinh_{\kappa}(x) $$.

The κ-hyperbolic tangent and κ-hyperbolic cotangent functions are given by:
 * $$ \tanh_{\kappa}(x)=\frac{\sinh_{\kappa}(x)}{\cosh_{\kappa}(x)} $$
 * $$ \coth_{\kappa}(x)=\frac{\cosh_{\kappa}(x)}{\sinh_{\kappa}(x)} $$.

The κ-hyperbolic trigonometric functions become the ordinary hyperbolic trigonometric functions in the classical limit $$\kappa \rightarrow 0$$.

From the κ-Euler formula and the property $$\exp_{\kappa}(-x)\exp_{\kappa}(x)=1$$ the fundamental expression of κ-hyperbolic trigonometry is given as follows:
 * $$\cosh_{\kappa}^2(x)- \sinh_{\kappa}^2(x)=1

$$

κ-Inverse hyperbolic function

The Kaniadakis inverse hyperbolic functions (or κ-inverse hyperbolic functions) are associated to the κ-logarithm:
 * $$ {\rm arcsinh}_{\kappa}(x)=\ln_{\kappa}\left(\sqrt{1+x^2}+x\right) $$,
 * $$ {\rm arccosh}_{\kappa}(x)=\ln_{\kappa}\left(\sqrt{x^2-1}+x\right) $$,
 * $$ {\rm arctanh}_{\kappa}(x)=\ln_{\kappa}\left(\sqrt{\frac{1+x}{1-x}}\right) $$,
 * $$ {\rm arccoth}_{\kappa}(x)=\ln_{\kappa}\left(\sqrt{\frac{1-x}{1+x}}\right) $$,

in which are valid the following relations:
 * $$ {\rm arcsinh}_{\kappa}(x) = {\rm sign}(x){\rm arccosh}_{\kappa}\left(\sqrt{1+x^2}\right) $$,
 * $$ {\rm arcsinh}_{\kappa}(x) = {\rm arctanh}_{\kappa}\left(\frac{x}{\sqrt{1+x^2}}\right) $$,
 * $$ {\rm arcsinh}_{\kappa}(x) = {\rm arccoth}_{\kappa}\left(\frac{\sqrt{1+x^2}}{x}\right) $$.

The κ-cyclic and κ-hyperbolic trigonometric functions are connected by the following relationships:
 * $$ {\rm sin}_{\kappa}(x) = -i{\rm sinh}_{\kappa}(ix) $$,
 * $$ {\rm cos}_{\kappa}(x) = {\rm cosh}_{\kappa}(ix) $$,
 * $$ {\rm tan}_{\kappa}(x) = -i{\rm tanh}_{\kappa}(ix) $$,
 * $$ {\rm cot}_{\kappa}(x) = i{\rm coth}_{\kappa}(ix) $$,
 * $$ {\rm arcsin}_{\kappa}(x)=-i\,{\rm arcsinh}_{\kappa}(ix) $$,
 * $$ {\rm arccos}_{\kappa}(x)\neq -i\,{\rm arccosh}_{\kappa}(ix) $$,
 * $$ {\rm arctan}_{\kappa}(x)=-i\,{\rm arctanh}_{\kappa}(ix) $$,
 * $$ {\rm arccot}_{\kappa}(x)=i\,{\rm arccoth}_{\kappa}(ix) $$.

Kaniadakis entropy
The Kaniadakis statistics is based on the Kaniadakis κ-entropy, which is defined through:
 * $$S_\kappa \big(p\big) = -\sum_i p_i \ln_{\kappa}\big(p_i\big) = \sum_i p_i \ln_{\kappa}\bigg(\frac{1}{p_i} \bigg)$$

where $$p = \{p_i = p(x_i); x \in \mathbb{R}; i = 1, 2, ..., N; \sum_i p_i = 1\}$$ is a probability distribution function defined for a random variable $$X$$, and $$0 \leq |\kappa| < 1$$ 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.

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 $$f$$ of a real variable  $$t$$ to a new function $$F_\kappa(s)$$ in the complex frequency domain, represented by the complex variable $$s$$. This κ-integral transform is defined as:

F_{\kappa}(s)={\cal L}_{\kappa}\{f(t)\}(s)=\int_{\, 0}^{\infty}\!f(t) \,[\exp_{\kappa}(-t)]^s\,dt $$ The inverse κ-Laplace transform is given by:

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} $$ The ordinary Laplace transform and its inverse transform are recovered as $$\kappa \rightarrow 0$$.

Properties

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

The κ-Laplace transforms presented in the latter table reduce to the corresponding ordinary Laplace transforms in the classical limit $$\kappa \rightarrow 0$$.

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:



{\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 $$

which can be rewritten as



{\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 $$

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



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

$$ h_\kappa(x,\omega) = \frac{\exp(-i\,x_{\{\kappa\}}\,\omega_{\{\kappa\}})}\sqrt{1+\kappa^2\,x^2} $$

The inverse κ-Fourier transform is defined as:



{\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 $$

Let $$u_\kappa(x) = \frac 1 \kappa \cosh\Big(\kappa\ln(x) \Big)$$, the following table shows the κ-Fourier transforms of several notable functions:

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

The properties of the κ-Fourier transform presented in the latter table reduce to the corresponding ordinary Fourier transforms in the classical limit $$\kappa \rightarrow 0$$.