Power-law tailed statistical distributions and Lorentz transformations

Abstract 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 ) ∝ exp κ ( − I ) , where, I ∝ β E − β μ , is the collision invariant, and exp κ ( x ) = ( 1 + κ 2 x 2 + κ x ) 1 / κ , with κ 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 κ ( x ) and its inverse ln κ ( x ) , 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.