Kaniadakis statistics

Kaniadakis statistics is a generalization of Boltzmann–Gibbs statistical mechanics, based on a relativistic generalization of the classical Boltzmann–Gibbs–Shannon 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 function is a one-parameter generalization of an exponential function, given by:
with.
The κ-exponential for can also be written in the form:
The first five terms of the Taylor expansion of are given by:

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:
For a real number, the κ-exponential has the property:

κ-logarithm function

The Kaniadakis logarithm is a relativistic one-parameter generalization of the ordinary logarithm function,
with, which is the inverse function of the κ-exponential:
The κ-logarithm for can also be written in the form:
The first three terms of the Taylor expansion of are given by:
following the rule
with, and
where and. The two first terms of the Taylor expansion of are the same as an ordinary logarithmic function.
Basic properties
The κ-logarithm function has the following properties of a logarithmic function:
For a real number, the κ-logarithm has the property:

κ-Algebra

κ-sum

For any and, the Kaniadakis sum is defined by the following composition law:
that can also be written in form:
where the ordinary sum is a particular case in the classical limit :.
The κ-sum, like the ordinary sum, has the following properties:
The κ-difference is given by.
The fundamental property arises as a special case of the more general expression below:
Furthermore, the κ-functions and the κ-sum present the following relationships:

κ-product

For any and, the Kaniadakis product is defined by the following composition law:
where the ordinary product is a particular case in the classical limit :.
The κ-product, like the ordinary product, has the following properties:
The κ-division is given by.
The κ-sum and the κ-product obey the distributive law:.
The fundamental property arises as a special case of the more general expression below:

κ-Calculus

κ-Differential

The Kaniadakis differential of is defined by:
So, the κ-derivative of a function is related to the Leibniz derivative through:
where is the Lorentz factor. The ordinary derivative is a particular case of κ-derivative in the classical limit.

κ-Integral

The Kaniadakis integral is the inverse operator of the κ-derivative defined through
which recovers the ordinary integral in the classical limit.

κ-Trigonometry

κ-Cyclic Trigonometry

The Kaniadakis cyclic trigonometry is based on the κ-cyclic sine and κ-cyclic cosine functions defined by:
where the κ-generalized Euler formula is
The κ-cyclic trigonometry preserves fundamental expressions of the ordinary cyclic trigonometry, which is a special case in the limit κ → 0, such as:
The κ-cyclic tangent and κ-cyclic cotangent functions are given by:
The κ-cyclic trigonometric functions become the ordinary trigonometric function in the classical limit.
κ-Inverse cyclic function
The Kaniadakis inverse cyclic functions are associated to the κ-logarithm:

κ-Hyperbolic Trigonometry

The Kaniadakis hyperbolic trigonometry is based on the κ-hyperbolic sine and κ-hyperbolic cosine given by:
where the κ-Euler formula is
The κ-hyperbolic tangent and κ-hyperbolic cotangent functions are given by:
The κ-hyperbolic trigonometric functions become the ordinary hyperbolic trigonometric functions in the classical limit.
From the κ-Euler formula and the property the fundamental expression of κ-hyperbolic trigonometry is given as follows:
κ-Inverse hyperbolic function
The Kaniadakis inverse hyperbolic functions are associated to the κ-logarithm:
in which are valid the following relations:
The κ-cyclic and κ-hyperbolic trigonometric functions are connected by the following relationships:

Kaniadakis entropy

The Kaniadakis statistics is based on the Kaniadakis κ-entropy, which is defined through:
where is a probability distribution function defined for a random variable, and 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 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 is a κ-deformed integral transform of the ordinary Laplace transform. The κ-Laplace transform converts a function of a real variable to a new function in the complex frequency domain, represented by the complex variable. This κ-integral transform is defined as:
The inverse κ-Laplace transform is given by:
The ordinary Laplace transform and its inverse transform are recovered as.
Properties
Let two functions and, and their respective κ-Laplace transforms and, the following table presents the main properties of κ-Laplace transform:

Property
Linearity
Time scaling
Frequency shifting
Derivative
Derivative
Time-domain integration


Dirac delta-function
Heaviside unit function
Power function
Power function
Power function

The κ-Laplace transforms presented in the latter table reduce to the corresponding ordinary Laplace transforms in the classical limit.

κ-Fourier Transform

The Kaniadakis 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:
which can be rewritten as
where and. The κ-Fourier transform imposes an asymptotically log-periodic behavior by deforming the parameters and in addition to a damping factor, namely .
The kernel of the κ-Fourier transform is given by:
The inverse κ-Fourier transform is defined as:
Let, the following table shows the κ-Fourier transforms of several notable functions:


Step function
Modulation
Causal -exponential
Symmetric -exponential
Constant
-Phasor
Impuslse
Signum	Sgn
Rectangular

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


Linearity
Scaling	where and
-Scaling
Complex conjugation
Duality
Reverse
-Frequency shift
-Time shift
Transform of -derivative
-Derivative of transform
Transform of integral
-Convolution	where
Modulation

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