Power-law tailed statistical distributions and Lorentz transformations

The present Letter, deals with the statistical theory [G. Kaniadakis, Phys. Rev. E 66 (2002) 056125; G. Kaniadakis, Phys. Rev. E 72 (2005) 036108], which predicts the probability distribution p(E) proportional to exp,(-1), where, I proportional to beta E - beta mu, is the collision invariant, and exp(kappa)(chi) = (root 1 + kappa(2)chi(2))(1/kappa), with kappa(2) < 1. This, experimentally observed distribution, at low energies behaves as the Maxwell-Boltzmann exponential distribution, while at high energies presents power law tails. Here we show that the function exp kappa(chi) and its inverse In-kappa(chi), can be obtained within the one-particle relativistic dynamics, in a very simple and transparent way, without invoking any extra principle or assumption, starting directly from the Lorentz transformations. The achievements support the idea that the power law tailed distributions are enforced by the Lorentz relativistic microscopic dynamics, like in the case of the exponential distribution which follows from the Newton classical microscopic dynamics. (C) 2010 Elsevier BM. All rights reserved.