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:
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 (or κ-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 (or κ-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: associativity: neutral element: opposite element: commutativity: 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 (or κ-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: associativity: neutral element: inverse element: commutativity: 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:
Furthermore, the κ-functions and the κ-product present the following relationships:
κ-Calculus
κ-Differential The Kaniadakis differential (or κ-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 (or κ-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 (or κ-cyclic trigonometry) is based on the κ-cyclic sine (or κ-sine) and κ-cyclic cosine (or κ-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 (or κ-inverse cyclic functions) are associated to the κ-logarithm:
κ-Hyperbolic Trigonometry The Kaniadakis hyperbolic trigonometry (or κ-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 (or κ-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 (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 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: + Properties of the κ-Laplace transformProperty 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 (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:
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: + κ-Fourier transform of several 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.
+ κ-Fourier properties 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 .
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
External links Giorgio Kaniadakis Google Scholar page Kaniadakis Statistics on arXiv.org