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




respectively, with and as defined in the Details section. These functions can be viewed as a generalization of the ordinary exponential, which is recovered in the limit as , and present a power-law tail as .

The exponent quantifies the curvature (shape) of the distribution, which is less pronounced for lower values of the parameter, and more pronounced for higher values. The constant is a characteristic scale, since its value determines the scale of the probability distribution: if is small, then the distribution will be more concentrated around the mode; if is large, then it will be more spread out. Finally, the parameter measures the heaviness of the right tail: the larger its magnitude, the fatter the tail.

Assuming , the Demonstration shows the effects of different values of the parameters and on the shape of the PDF (on linear scale) and CCDF (both on a linear and log-log scale).

In the last few years the distribution has appeared in a diverse range of applications, including both physical and systems and those in other fields.


Contributed by: Fabio Clementi (October 2009)
Open content licensed under CC BY-NC-SA



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].


[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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