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_ (x) = \begin
\Big(\sqrt+\kappa x \Big)^\frac & \text 0 < \kappa < 1. \\ pt\exp(x) & \text\kappa = 0, \\ pt\end

with $\exp_ (x) = \exp_ (x)$.

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

: $\exp_ (x) = \exp\Bigg(\frac \text (\kappa x)\Bigg).$
The first five terms of the Taylor expansion of $\exp_\kappa(x)$ are given by:

$\exp_ (x) = 1 + x + \frac + (1 - \kappa^2) \frac + (1 - 4 \kappa^2) \frac + \cdots$

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:
: $\exp_ (x) \in \mathbb^\infty(\mathbb)$
: $\frac\exp_ (x) > 0$
: $\frac\exp_ (x) > 0$
: $\exp_ (-\infty) = 0^+$
: $\exp_ (0) = 1$
: $\exp_ (+\infty) = +\infty$
: $\exp_ (x) \exp_ (-x) = -1$
For a real number $r$, the κ-exponential has the property:
: \Big exp_ (x)\Bigr = \exp_ (rx)
.

κ-logarithm function

The Kaniadakis logarithm (or κ-logarithm) is a relativistic one-parameter generalization of the ordinary logarithm function,
: \ln_ (x) = \begin
\frac & \text 0 < \kappa < 1, \\ pt\ln(x) & \text\kappa = 0, \\ pt\end

with $\ln_ (x) = \ln_ (x)$, which is the inverse function of the κ-exponential:

: $\ln_\Big( \exp_(x)\Big) = \exp_\Big( \ln_(x)\Big) = x.$
The κ-logarithm for $0 < \kappa < 1$ can also be written in the form:

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

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

:$\ln_ (1+x) = x - \frac + \left( 1 + \frac\right) \frac - \cdots$

following the rule

:$\ln_(1+x) = \sum_^ b_n(\kappa)\,(-1)^ \,\frac$

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

: b_(\kappa) (x) = \begin
1 & \text n = 1, \\ pt\frac\Big(1-\kappa\Big)\Big(1-\frac\Big)...
\Big(1-\frac\Big) ,\,+\,\frac\Big(1+\kappa\Big)\Big(1+\frac\Big)...
\Big(1+\frac\Big) & \text n > 1, \\ pt\end

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:
: $\ln_ (x) \in \mathbb^\infty(\mathbb^+)$
: $\frac\ln_ (x) > 0$
: $\frac\ln_ (x) < 0$
: $\ln_ (0^+) = -\infty$
: $\ln_ (1) = 0$
: $\ln_ (+\infty) = +\infty$
: $\ln_ (1/x) = -\ln_ (x)$
For a real number $r$, the κ-logarithm has the property:
: $\ln_ (x^r) = r \ln_ (x)$

κ-Algebra

κ-sum

For any $x,y \in \mathbb$ and $, \kappa, < 1$, the Kaniadakis sum (or κ-sum) is defined by the following composition law:
: $x\stackrely=x\sqrt+y\sqrt$,
that can also be written in form:
: $x\stackrely=\,\sinh \left(\,(\kappa x)\,+\,\,(\kappa y)\,\right)$,
where the ordinary sum is a particular case in the classical limit $\kappa \rightarrow 0$: $x\stackrely=x + y$.

The κ-sum, like the ordinary sum, has the following properties:
: $\text \quad (x\stackrely)\stackrelz =x \stackrel (y \stackrel z)$
: $\text \quad x \stackrel 0 = 0 \stackrelx=x$
: $\text \quad x\stackrel(-x)=(-x) \stackrelx=0$
: $\text \quad x\stackrely=y\stackrelx$
The κ-difference $\stackrel$ is given by $x\stackrely=x\stackrel(-y)$.

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

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

κ-product

For any $x,y \in \mathbb$ and $, \kappa, < 1$, the Kaniadakis product (or κ-product) is defined by the following composition law:
: $x\stackrely=\,\sinh \left(\,\,\,\,(\kappa x)\,\,\,(\kappa y)\,\right)$,
where the ordinary product is a particular case in the classical limit $\kappa \rightarrow 0$: $x\stackrely=x \times y=xy$.

The κ-product, like the ordinary product, has the following properties:
: $\text \quad (x \stackrely) \stackrelz=x \stackrel(y \stackrelz)$
: $\text \quad x \stackrelI=I \stackrelx= x \quad \text \quad I=\kappa^\sinh \kappa \stackrelx=x$
: $\text \quad x \stackrel\overline x= \overline x \stackrelx=I \quad \text \quad \overline x=\kappa^\sinh(\kappa^2/ \,(\kappa x))$
: $\text \quad x\stackrely=y\stackrelx$
The κ-division $\stackrel$ is given by $x\stackrely=x\stackrel\overline y$.

The κ-sum $\stackrel$ and the κ-product $\stackrel$ obey the distributive law: $z\stackrel(x \stackrely) = (z \stackrelx) \stackrel(z \stackrely)$.

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

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

κ-Calculus

κ-Differential

The Kaniadakis differential (or κ-differential) of $x$ is defined by:
: $d _x= \frac$.

So, the κ-derivative of a function $f(x)$ is related to the Leibniz derivative through:
: $\frac = \gamma_\kappa (x) \frac$,
where $\gamma_\kappa(x) = \sqrt$ is the Lorentz factor. The ordinary derivative $\frac$ is a particular case of κ-derivative $\frac$ 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
: $\int d_x \,\, f(x)= \int \frac\,\,f(x)$,
which recovers the ordinary integral in the classical limit $\kappa \rightarrow 0$.

κ-Trigonometry

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:
: $\sin_(x) =\frac$,
: $\cos_(x) =\frac$,
where the κ-generalized Euler formula is
: $\exp_(\pm ix)=\cos_(x)\pm i\sin_(x)$.
:

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

The κ-cyclic tangent and κ-cyclic cotangent functions are given by:
: $\tan_(x)=\frac$
: $\cot_(x)=\frac$.

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:
: $_(x)=-i\ln_\left(\sqrt+ix\right)$,
: $_(x)=-i\ln_\left(\sqrt+x\right)$,
: $_(x)=i\ln_\sqrt$,
: $_(x)=i\ln_\sqrt$.

κ-Hyperbolic Trigonometry

The Kaniadakis hyperbolic trigonometry (or κ-hyperbolic trigonometry) is based on the κ-hyperbolic sine and κ-hyperbolic cosine given by:
: $\sinh_(x) =\frac$,
: $\cosh_(x) =\frac$,
where the κ-Euler formula is
: $\exp_(\pm x)=\cosh_(x)\pm \sinh_(x)$.

The κ-hyperbolic tangent and κ-hyperbolic cotangent functions are given by:
: $\tanh_(x)=\frac$
: $\coth_(x)=\frac$.

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 $\exp_(-x)\exp_(x)=1$ the fundamental expression of κ-hyperbolic trigonometry is given as follows:
: $\cosh_^2(x)- \sinh_^2(x)=1$

κ-Inverse hyperbolic function

The Kaniadakis inverse hyperbolic functions (or κ-inverse hyperbolic functions) are associated to the κ-logarithm:
: $_(x)=\ln_\left(\sqrt+x\right)$,
: $_(x)=\ln_\left(\sqrt+x\right)$,
: $_(x)=\ln_\sqrt$,
: $_(x)=\ln_\sqrt$,
in which are valid the following relations:
: $_(x) = _\sqrt$,
: $_(x) = _\frac$,
: $_(x) = _\frac$.
The κ-cyclic and κ-hyperbolic trigonometric functions are connected by the following relationships:
: $_(x) = -i_(ix)$,
: $_(x) = _(ix)$,
: $_(x) = -i_(ix)$,
: $_(x) = i_(ix)$,
: $_(x)=-i\,_(ix)$,
: $_(x)= -i\,_(x)$,
: $_(x)=-i\,_(ix)$,
: $_(x)=i\,_(ix)$.

Kaniadakis entropy

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

κ-Gaussian distribution

κ-Gamma distribution

κ-Weibull distribution

κ-Logistic distribution

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 $f$ 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_(s)=_\(s)=\int_^\!f(t) \, exp_(-t)s\,dt

The inverse κ-Laplace transform is given by:
:$f(t)=^_\(t)=$
The ordinary Laplace transform and its inverse transform are recovered as $\kappa \rightarrow 0$.

Properties

Let two functions $f(t) = ^_\(t)$ and $g(t) = ^_\(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:

:$_\kappa$(x)\omega)=\int\limits_\limits^f(x)\,
\exp_\kappa(-x\otimes_\kappa\omega)^i\,d_\kappa x

which can be rewritten as

:$_\kappa$(x)\omega)=\int\limits_\limits^f(x)\,
\,d x

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

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

$h_\kappa(x,\omega) = \frac\sqrt$

The inverse κ-Fourier transform is defined as:

:
_\kappa hat f(\omega)x)=\int\limits_\limits^\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 $\kappa \rightarrow 0$.

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

References

*
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External links

Giorgio Kaniadakis Google Scholar pageKaniadakis Statistics on arXiv.org

Statistical mechanics