Kaniadakis statistics

Kaniadakis statistics (also known as κ-statistics) is a generalization of Boltzmann–Gibbs statistical mechanics,^[1] based on a relativistic^[2]^[3]^[4] 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,^[5] 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,^[6]^[7] 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,^[8]^[9] condensed matter, quantum physics,^[10]^[11] seismology,^[12]^[13] genomics,^[14]^[15] economics,^[16]^[17] epidemiology,^[18] 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

\exp _{\kappa }(x)

for three different κ-values. The solid black curve corresponding to the ordinary exponential function

\exp(x)

(

\kappa =0

).

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

Plot of the κ-logarithmic function

\ln _{\kappa }(x)

for three different κ-values. The solid black curve corresponding to the ordinary logarithmic function

\ln(x)

(

\kappa =0

).

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-Algebra

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 ^{2}y^{2}}}+y{\sqrt {1+\kappa ^{2}x^{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)

\ln _{\kappa }(x\,{\stackrel {\kappa }{\otimes }}\,y)=\ln _{\kappa }(x)+\ln _{\kappa }(y)

κ-Calculus

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-Trigonometry

κ-Cyclic Trigonometry

$Plot of the κ-sine and κ-cosine functions for {\displaystyle \kappa =0} (black curve) and {\displaystyle \kappa =0.1} (blue curve).$

[click on the figure] Plot of the κ-sine and κ-cosine functions for

\kappa =0

(black curve) and

\kappa =0.1

(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:

\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^[19]^[20] 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:^[21]

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}}_{\kappa }^{-1}\{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 ^{2}t^{2}}}}\,ds}

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

Properties

Let two functions $f(t)={\cal {L}}_{\kappa }^{-1}\{F_{\kappa }(s)\}(t)$ and $g(t)={\cal {L}}_{\kappa }^{-1}\{G_{\kappa }(s)\}(t)$ , and their respective κ-Laplace transforms $F_{\kappa }(s)$ and $G_{\kappa }(s)$ , the following table presents the main properties of κ-Laplace transform:^[21]

Properties of the κ-Laplace transform
Property	$f(t)$	$F_{\kappa }(s)$
Linearity	$a\,f(t)+b\,g(t)$	$a\,F_{\kappa }(s)+b\,G_{\kappa }(s)$
Time scaling	$f(at)$	${\frac {1}{a}}\,F_{\kappa /a}({\frac {s}{a}})$
Frequency shifting	$f(t)\,[\exp _{\kappa }(-t)]^{a}$	$F_{\kappa }(s-a)$
Derivative	${\frac {d\,f(t)}{dt}}$	$s\,{\cal {L}}_{\kappa }\left\{{\frac {f(t)}{\sqrt {1+\kappa ^{2}t^{2}}}}\right\}(s)-f(0)$
Derivative	${\frac {d}{dt}}\,{\sqrt {1+\kappa ^{2}t^{2}}}\,f(t)$	$s\,F_{\kappa }(s)-f(0)$
Time-domain integration	${\frac {1}{\sqrt {1+\kappa ^{2}t^{2}}}}\,\int _{0}^{t}f(w)dw$	${\frac {1}{s}}\,F_{\kappa }(s)$
	$f(t)\,[\ln(\exp _{\kappa }(t))]^{n}$	$(-1)^{n}{\frac {d^{n}F_{\kappa }(s)}{ds^{n}}}$
	$f(t)\,[\ln(\exp _{\kappa }(t))]^{-n}$	$\int _{s}^{+\infty }dw_{n}\int _{w_{n}}^{+\infty }dw_{n-1}...\int _{w_{3}}^{+\infty }dw_{2}\int _{w_{2}}^{+\infty }dw_{1}\,F_{\kappa }(w_{1})$
Dirac delta-function	$\delta (t-\tau )$	$[\exp _{\kappa }(-\tau )]^{s}$
Heaviside unit function	$u(t-\tau )$	${\frac {s{\sqrt {1+\kappa ^{2}\tau ^{2}}}+\kappa ^{2}\tau }{s^{2}-\kappa ^{2}}}\,[\exp _{\kappa }(-\tau )]^{s}$
Power function	$t^{\nu -1}$	${\frac {s^{2}}{s^{2}-\kappa ^{2}\nu ^{2}}}\,{\frac {\Gamma _{\frac {\kappa }{s}}(\nu +1)}{\nu \,s^{\nu }}}={\frac {s}{s+\|\kappa \|\nu }}\,{\frac {\Gamma (\nu )}{\|2\kappa \|^{\nu }}}\,{\frac {\Gamma \left({\frac {s}{\|2\kappa \|}}-{\frac {\nu }{2}}\right)}{\Gamma \left({\frac {s}{\|2\kappa \|}}+{\frac {\nu }{2}}\right)}}$
Power function	$t^{2m-1},\ \ m\in Z^{+}$	${\frac {(2m-1)!}{\prod _{j=1}^{m}\left[s^{2}-(2j)^{2}\kappa ^{2}\right]}}$
Power function	$t^{2m},\ \ m\in Z^{+}$	${\frac {(2m)!\,s}{\prod _{j=1}^{m+1}\left[s^{2}-(2j-1)^{2}\kappa ^{2}\right]}}$

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:^[22]

{\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}}}}\,dx

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 ${\sqrt {1+\kappa ^{2}\,x^{2}}}$ .

Real (top panel) and imaginary (bottom panel) part of the kernel

h_{\kappa }(x,\omega )

for typical

\kappa

-values and

\omega =1

.

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:^[22]

{\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:^[22]

κ-Fourier transform of several functions
	$f(x)$	${\cal {F}}_{\kappa }[f(x)](\omega )$
Step function	$\theta (x)$	${\sqrt {2\,\pi }}\,\delta (\omega )+{1 \over {\sqrt {2\,\pi }}\,i\,\omega _{\{\kappa \}}}$
Modulation	$\cos _{\kappa }(a{\stackrel {\kappa }{\oplus }}x)$	${\sqrt {\pi \over 2}}\,u_{\kappa }(\exp _{\kappa }(a))\,\left(\delta (\omega +a)+\delta (\omega -a)\right)$
Causal $\kappa$ -exponential	$\theta (x)\,\exp _{\kappa }(-a{\stackrel {\kappa }{\otimes }}x)$	${1 \over {\sqrt {2\,\pi }}}{1 \over a_{\{\kappa \}}+i\,\omega _{\{\kappa \}}}$
Symmetric $\kappa$ -exponential	$\exp _{\kappa }(-a{\stackrel {\kappa }{\otimes }}\|x\|)$	${\sqrt {2 \over \pi }}\,{a_{\{\kappa \}} \over a_{\{\kappa \}}^{2}+\omega _{\{\kappa \}}^{2}}$
Constant	$1$	${\sqrt {2\,\pi }}\,\delta (\omega )$
$\kappa$ -Phasor	$\exp _{\kappa }\,(a{\stackrel {\kappa }{\otimes }}x)^{i}$	${\sqrt {2\,\pi }}\,u_{\kappa }(\exp _{\kappa }(a))\,\delta (\omega -a)$
Impuslse	$\delta (x-a)$	${1 \over {\sqrt {2\,\pi }}}{\exp _{\kappa }\,(\omega {\stackrel {\kappa }{\otimes }}a)^{i} \over u_{\kappa }\left(\exp _{\kappa }\,(a)\right)}$
Signum	Sgn $(x)$	${\sqrt {2 \over \pi }}\,\,{1 \over i\,\omega _{\{\kappa \}}}$
Rectangular	$\Pi \left({x \over a}\right)$	${\sqrt {2 \over \pi }}\,\,a_{\{\kappa \}}\,{\rm {sinc}}_{\kappa }(\omega {\stackrel {\kappa }{\otimes }}a)$

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

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

References

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↑ Kaniadakis, G. (2009). "Relativistic entropy and related Boltzmann kinetics". The European Physical Journal A. 40 (3): 275–287. arXiv:0901.1058. Bibcode:2009EPJA...40..275K. doi:10.1140/epja/i2009-10793-6. ISSN 1434-6001. S2CID 119190011.
↑ Kaniadakis, G. (2002). "Statistical mechanics in the context of special relativity". Physical Review E. 66 (5): 056125. arXiv:cond-mat/0210467. Bibcode:2002PhRvE..66e6125K. doi:10.1103/PhysRevE.66.056125. ISSN 1063-651X. PMID 12513574. S2CID 45635888.
↑ Kaniadakis, G. (2005). "Statistical mechanics in the context of special relativity. II". Physical Review E. 72 (3): 036108. arXiv:cond-mat/0507311. Bibcode:2005PhRvE..72c6108K. doi:10.1103/PhysRevE.72.036108. ISSN 1539-3755. PMID 16241516. S2CID 18115408.
↑ Kaniadakis, G. (2011). "Power-law tailed statistical distributions and Lorentz transformations". Physics Letters A. 375 (3): 356–359. arXiv:1110.3944. Bibcode:2011PhLA..375..356K. doi:10.1016/j.physleta.2010.11.057. ISSN 0375-9601. S2CID 118435479.
↑ Kaniadakis, G. (2001). "Non-linear kinetics underlying generalized statistics". Physica A: Statistical Mechanics and Its Applications. 296 (3): 405–425. arXiv:cond-mat/0103467. Bibcode:2001PhyA..296..405K. doi:10.1016/S0378-4371(01)00184-4. ISSN 0378-4371. S2CID 44275064.
↑ Kaniadakis, G. (2009). "Maximum entropy principle and power-law tailed distributions". The European Physical Journal B. 70 (1): 3–13. arXiv:0904.4180. Bibcode:2009EPJB...70....3K. doi:10.1140/epjb/e2009-00161-0. ISSN 1434-6028. S2CID 55421804.
↑ Kaniadakis, G. (2021). "New power-law tailed distributions emerging in κ-statistics (a)". Europhysics Letters. 133 (1): 10002. arXiv:2203.01743. Bibcode:2021EL....13310002K. doi:10.1209/0295-5075/133/10002. ISSN 0295-5075. S2CID 234144356.
↑ Carvalho, J. C.; Silva, R.; do Nascimento Jr., J. D.; De Medeiros, J. R. (2008). "Power law statistics and stellar rotational velocities in the Pleiades". EPL (Europhysics Letters). 84 (5): 59001. arXiv:0903.0836. Bibcode:2008EL.....8459001C. doi:10.1209/0295-5075/84/59001. ISSN 0295-5075. S2CID 7123391.
↑ Curé, Michel; Rial, Diego F.; Christen, Alejandra; Cassetti, Julia (2014). "A method to deconvolve stellar rotational velocities". Astronomy & Astrophysics. 565: A85. arXiv:1401.1054. Bibcode:2014A&A...565A..85C. doi:10.1051/0004-6361/201323344. ISSN 0004-6361. S2CID 59375612.
↑ Ourabah, Kamel; Hamici-Bendimerad, Amel Hiba; Tribeche, Mouloud (2015). "Quantum entanglement and Kaniadakis entropy". Physica Scripta. 90 (4): 045101. Bibcode:2015PhyS...90d5101O. doi:10.1088/0031-8949/90/4/045101. ISSN 0031-8949. S2CID 123776127.
↑ Abreu, Everton M. C.; Ananias Neto, Jorge; Mendes, Albert C. R.; de Paula, Rodrigo M. (2019). "Loop quantum gravity Immirzi parameter and the Kaniadakis statistics". Chaos, Solitons & Fractals. 118: 307–310. arXiv:1808.01891. Bibcode:2019CSF...118..307A. doi:10.1016/j.chaos.2018.11.033. ISSN 0960-0779. S2CID 119207713.
↑ Hristopulos, Dionissios T.; Petrakis, Manolis P.; Kaniadakis, Giorgio (2014). "Finite-size effects on return interval distributions for weakest-link-scaling systems". Physical Review E. 89 (5): 052142. arXiv:1308.1881. Bibcode:2014PhRvE..89e2142H. doi:10.1103/PhysRevE.89.052142. ISSN 1539-3755. PMID 25353774. S2CID 22310350.
↑ da Silva, Sérgio Luiz E. F. (2021). "κ-generalised Gutenberg–Richter law and the self-similarity of earthquakes". Chaos, Solitons & Fractals. 143: 110622. Bibcode:2021CSF...14310622D. doi:10.1016/j.chaos.2020.110622. ISSN 0960-0779. S2CID 234063959.
↑ Souza, N. T. C. M.; Anselmo, D. H. A. L.; Silva, R.; Vasconcelos, M. S.; Mello, V. D. (2014). "A κ -statistical analysis of the Y-chromosome". EPL (Europhysics Letters). 108 (3): 38004. doi:10.1209/0295-5075/108/38004. ISSN 0295-5075. S2CID 122456729.
↑ Costa, M. O.; Silva, R.; Anselmo, D. H. A. L.; Silva, J. R. P. (2019). "Analysis of human DNA through power-law statistics". Physical Review E. 99 (2): 022112. Bibcode:2019PhRvE..99b2112C. doi:10.1103/PhysRevE.99.022112. ISSN 2470-0045. PMID 30934358. S2CID 91186653.
↑ Clementi, Fabio; Gallegati, Mauro; Kaniadakis, Giorgio (2012). "A new model of income distribution: the κ-generalized distribution". Journal of Economics. 105 (1): 63–91. doi:10.1007/s00712-011-0221-0. hdl:11393/73598. ISSN 0931-8658. S2CID 155080665.
↑ Trivellato, Barbara (2013). "Deformed Exponentials and Applications to Finance". Entropy. 15 (12): 3471–3489. Bibcode:2013Entrp..15.3471T. doi:10.3390/e15093471. ISSN 1099-4300.
↑ Kaniadakis, Giorgio; Baldi, Mauro M.; Deisboeck, Thomas S.; Grisolia, Giulia; Hristopulos, Dionissios T.; Scarfone, Antonio M.; Sparavigna, Amelia; Wada, Tatsuaki; Lucia, Umberto (2020). "The κ-statistics approach to epidemiology". Scientific Reports. 10 (1): 19949. arXiv:2012.00629. Bibcode:2020NatSR..1019949K. doi:10.1038/s41598-020-76673-3. ISSN 2045-2322. PMC 7673996. PMID 33203913.
↑ Abe, S.; Kaniadakis, G.; Scarfone, A. M. (2004) [2004]. "Stabilities of generalized entropies". Journal of Physics A: Mathematical and General. 37 (44): 10513–10519. arXiv:cond-mat/0401290. Bibcode:2004JPhA...3710513A. doi:10.1088/0305-4470/37/44/004. S2CID 16080176.
↑ Kaniadakis, G. (2001). "H-theorem and generalized entropies within the framework of nonlinear kinetics". Physics Letters A. 288 (5–6): 283–291. arXiv:cond-mat/0109192. Bibcode:2001PhLA..288..283K. doi:10.1016/S0375-9601(01)00543-6. S2CID 119445915.
1 2 Kaniadakis, Giorgio (2013-09-25). "Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions". Entropy. 15 (12): 3983–4010. arXiv:1309.6536. Bibcode:2013Entrp..15.3983K. doi:10.3390/e15103983. ISSN 1099-4300.
1 2 3 Scarfone, A.M. (2017). "κ -deformed Fourier transform". Physica A: Statistical Mechanics and Its Applications. 480: 63–78. arXiv:2206.06869. Bibcode:2017PhyA..480...63S. doi:10.1016/j.physa.2017.03.036. S2CID 126079408.

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[1] Kaniadakis, G. (2009). "Relativistic entropy and related Boltzmann kinetics". The European Physical Journal A. 40 (3): 275–287. arXiv:0901.1058. Bibcode:2009EPJA...40..275K. doi:10.1140/epja/i2009-10793-6. ISSN 1434-6001. S2CID 119190011.

[2] Kaniadakis, G. (2002). "Statistical mechanics in the context of special relativity". Physical Review E. 66 (5): 056125. arXiv:cond-mat/0210467. Bibcode:2002PhRvE..66e6125K. doi:10.1103/PhysRevE.66.056125. ISSN 1063-651X. PMID 12513574. S2CID 45635888.

[3] Kaniadakis, G. (2005). "Statistical mechanics in the context of special relativity. II". Physical Review E. 72 (3): 036108. arXiv:cond-mat/0507311. Bibcode:2005PhRvE..72c6108K. doi:10.1103/PhysRevE.72.036108. ISSN 1539-3755. PMID 16241516. S2CID 18115408.

[:0-4] Kaniadakis, G. (2011). "Power-law tailed statistical distributions and Lorentz transformations". Physics Letters A. 375 (3): 356–359. arXiv:1110.3944. Bibcode:2011PhLA..375..356K. doi:10.1016/j.physleta.2010.11.057. ISSN 0375-9601. S2CID 118435479.

[5] Kaniadakis, G. (2001). "Non-linear kinetics underlying generalized statistics". Physica A: Statistical Mechanics and Its Applications. 296 (3): 405–425. arXiv:cond-mat/0103467. Bibcode:2001PhyA..296..405K. doi:10.1016/S0378-4371(01)00184-4. ISSN 0378-4371. S2CID 44275064.

[6] Kaniadakis, G. (2009). "Maximum entropy principle and power-law tailed distributions". The European Physical Journal B. 70 (1): 3–13. arXiv:0904.4180. Bibcode:2009EPJB...70....3K. doi:10.1140/epjb/e2009-00161-0. ISSN 1434-6028. S2CID 55421804.

[7] Kaniadakis, G. (2021). "New power-law tailed distributions emerging in κ-statistics (a)". Europhysics Letters. 133 (1): 10002. arXiv:2203.01743. Bibcode:2021EL....13310002K. doi:10.1209/0295-5075/133/10002. ISSN 0295-5075. S2CID 234144356.

[8] Carvalho, J. C.; Silva, R.; do Nascimento Jr., J. D.; De Medeiros, J. R. (2008). "Power law statistics and stellar rotational velocities in the Pleiades". EPL (Europhysics Letters). 84 (5): 59001. arXiv:0903.0836. Bibcode:2008EL.....8459001C. doi:10.1209/0295-5075/84/59001. ISSN 0295-5075. S2CID 7123391.

[9] Curé, Michel; Rial, Diego F.; Christen, Alejandra; Cassetti, Julia (2014). "A method to deconvolve stellar rotational velocities". Astronomy & Astrophysics. 565: A85. arXiv:1401.1054. Bibcode:2014A&A...565A..85C. doi:10.1051/0004-6361/201323344. ISSN 0004-6361. S2CID 59375612.

[10] Ourabah, Kamel; Hamici-Bendimerad, Amel Hiba; Tribeche, Mouloud (2015). "Quantum entanglement and Kaniadakis entropy". Physica Scripta. 90 (4): 045101. Bibcode:2015PhyS...90d5101O. doi:10.1088/0031-8949/90/4/045101. ISSN 0031-8949. S2CID 123776127.

[11] Abreu, Everton M. C.; Ananias Neto, Jorge; Mendes, Albert C. R.; de Paula, Rodrigo M. (2019). "Loop quantum gravity Immirzi parameter and the Kaniadakis statistics". Chaos, Solitons & Fractals. 118: 307–310. arXiv:1808.01891. Bibcode:2019CSF...118..307A. doi:10.1016/j.chaos.2018.11.033. ISSN 0960-0779. S2CID 119207713.

[12] Hristopulos, Dionissios T.; Petrakis, Manolis P.; Kaniadakis, Giorgio (2014). "Finite-size effects on return interval distributions for weakest-link-scaling systems". Physical Review E. 89 (5): 052142. arXiv:1308.1881. Bibcode:2014PhRvE..89e2142H. doi:10.1103/PhysRevE.89.052142. ISSN 1539-3755. PMID 25353774. S2CID 22310350.

[13] Silva, Sérgio Luiz E. F. (2021). "κ-generalised Gutenberg–Richter law and the self-similarity of earthquakes". Chaos, Solitons & Fractals. 143: 110622. Bibcode:2021CSF...14310622D. doi:10.1016/j.chaos.2020.110622. ISSN 0960-0779. S2CID 234063959.

[14] Souza, N. T. C. M.; Anselmo, D. H. A. L.; Silva, R.; Vasconcelos, M. S.; Mello, V. D. (2014). "A κ -statistical analysis of the Y-chromosome". EPL (Europhysics Letters). 108 (3): 38004. doi:10.1209/0295-5075/108/38004. ISSN 0295-5075. S2CID 122456729.

[15] Costa, M. O.; Silva, R.; Anselmo, D. H. A. L.; Silva, J. R. P. (2019). "Analysis of human DNA through power-law statistics". Physical Review E. 99 (2): 022112. Bibcode:2019PhRvE..99b2112C. doi:10.1103/PhysRevE.99.022112. ISSN 2470-0045. PMID 30934358. S2CID 91186653.

[16] Clementi, Fabio; Gallegati, Mauro; Kaniadakis, Giorgio (2012). "A new model of income distribution: the κ-generalized distribution". Journal of Economics. 105 (1): 63–91. doi:10.1007/s00712-011-0221-0. hdl:11393/73598. ISSN 0931-8658. S2CID 155080665.

[17] Trivellato, Barbara (2013). "Deformed Exponentials and Applications to Finance". Entropy. 15 (12): 3471–3489. Bibcode:2013Entrp..15.3471T. doi:10.3390/e15093471. ISSN 1099-4300.

[18] Kaniadakis, Giorgio; Baldi, Mauro M.; Deisboeck, Thomas S.; Grisolia, Giulia; Hristopulos, Dionissios T.; Scarfone, Antonio M.; Sparavigna, Amelia; Wada, Tatsuaki; Lucia, Umberto (2020). "The κ-statistics approach to epidemiology". Scientific Reports. 10 (1): 19949. arXiv:2012.00629. Bibcode:2020NatSR..1019949K. doi:10.1038/s41598-020-76673-3. ISSN 2045-2322. PMC 7673996. PMID 33203913.

[19] Abe, S.; Kaniadakis, G.; Scarfone, A. M. (2004) [2004]. "Stabilities of generalized entropies". Journal of Physics A: Mathematical and General. 37 (44): 10513–10519. arXiv:cond-mat/0401290. Bibcode:2004JPhA...3710513A. doi:10.1088/0305-4470/37/44/004. S2CID 16080176.

[20] Kaniadakis, G. (2001). "H-theorem and generalized entropies within the framework of nonlinear kinetics". Physics Letters A. 288 (5–6): 283–291. arXiv:cond-mat/0109192. Bibcode:2001PhLA..288..283K. doi:10.1016/S0375-9601(01)00543-6. S2CID 119445915.

[:1-21] 1 2 Kaniadakis, Giorgio (2013-09-25). "Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions". Entropy. 15 (12): 3983–4010. arXiv:1309.6536. Bibcode:2013Entrp..15.3983K. doi:10.3390/e15103983. ISSN 1099-4300.

[:2-22] 1 2 3 Scarfone, A.M. (2017). "κ -deformed Fourier transform". Physica A: Statistical Mechanics and Its Applications. 480: 63–78. arXiv:2206.06869. Bibcode:2017PhyA..480...63S. doi:10.1016/j.physa.2017.03.036. S2CID 126079408.

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