On the correlation between Kappa and Lévy stable distributions

Abstract

This article investigates the correlation between the Kappa and Lévy distributions via two approaches of the Klein–Kramers equation. The first approach illustrates the velocity distribution functions via the solution of the fractional Klein–Kramers equation obtained using the Riesz fractional derivative. In contrast, the second approach shows the velocity distribution functions according to steady-state Kappa distribution, which arises from the solution of the Klein–Kramers equation with variable coefficients dependent on velocity. We find a unique and straightforward formula representing the relation between the Kappa exponent and the fractality index (Lévy stable index). The results indicate a viable probability distribution obtained from the fractional equations as an alternative to the Kappa distribution. Hence, our results may shed light on the stationary power-law distribution in non extensive statistics and introduce a new correlation between the order of the fractional derivative (α) and the nonthermal index (κ) of the distribution function. Our results also show exact matching with the probability distributions illustrated in the literature.