# The Kappa Distribution

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This Demonstration shows the probability density function (PDF) and the complementary cumulative distribution function (CCDF) of the distribution, given by

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Contributed by: Fabio Clementi (October 2009)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The distribution presented in this Demonstration is based on the following one-parameter deformation of the exponential function proposed by Kaniadakis [1-3]:

,

with and . The Kaniadakis exponential can be inverted easily and the Kaniadakis logarithm is defined by

,

with and .

The above functions—shown in the upper-left corner plot of the window for different values of —have many very interesting properties, some being identical to those of the ordinary exponential and logarithm that are recovered in the limit as . For applications in statistics, the most interesting property is their power-law asymptotic behavior

,

,

.

The deformation mechanism introduced by emerges naturally within Einstein's theory of special relativity, and ultimately comes from the Lorentz transformations. The value of is proportional to the reciprocal of the light speed and tends to zero as , recovering in this way ordinary statistical mechanics and thermodynamics.

The particularly interesting mathematical properties of the Kaniadakis exponential and logarithm functions are also a very flexible mathematical tool for the efficient study of dynamical systems. Indeed, in the literature these functions have been used extensively in several fields beyond relativity, for example, in systems at the edge of chaos, fractal systems, game theory, error theory, economics, and so on. In particular, in economics the deformation has been recently employed for modeling personal income distributions [4-6].

References:

[1] G. Kaniadakis, "Non-Linear Kinetics Underlying Generalized Statistics," *Physica A*, 296(3-4), 2001 pp. 405–425.

[2] G. Kaniadakis, "Statistical Mechanics in the Context of Special Relativity," *Phys. Rev. E*, 66(5), 2002.

[3] G. Kaniadakis, "Statistical Mechanics in the Context of Special Relativity. II," *Phys. Rev. E*, 72(3), 2005.

[4] F. Clementi, M. Gallegati, and G. Kaniadakis, "κ-Generalized Statistics in Personal Income Distribution," *Eur. Phys. J. B*, 57(2), 2007 pp. 187–193.

[5] F. Clementi, T. Di Matteo, M. Gallegati, and G. Kaniadakis, "The κ-Generalized Distribution: A New Descriptive Model for the Size Distribution of Incomes," *Physica A*, 387(13), 2008 pp. 3201–3208.

[6] F. Clementi, M. Gallegati, and G. Kaniadakis, "A κ-Generalized Statistical Mechanics Approach to Income Analysis," *J. Stat. Mech.*, 2009.

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