Power-Law tailed statistical distributions and Lorentz transformations

TL;DR

This paper demonstrates that power-law tailed distributions observed in relativistic systems naturally emerge from Lorentz transformations, linking microscopic relativistic dynamics to macroscopic statistical distributions.

Contribution

It shows that the $ ext{exp}_ ext{kappa}$ distribution and its inverse can be derived directly from Lorentz transformations without additional assumptions.

Findings

Power-law tails arise from relativistic Lorentz transformations.

The $ ext{exp}_ ext{kappa}$ distribution is derived from relativistic dynamics.

Relativistic effects explain the emergence of power-law distributions.

Abstract

The present Letter, deals with the statistical theory [Phys. Rev. E {\bf 66}, 056125 (2002) and Phys. Rev E {\bf 72}, 036108 (2005)], which predicts the probability distribution $p(E) \propto \exp_{\kappa} (-I)$, where, $I \propto \beta E -\beta \mu$, is the collision invariant, and $\exp_{\kappa}(x)=(\sqrt{1+ \kappa^2 x^2}+\kappa x)^{1/\kappa}$, 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 $\exp_{\kappa}(x)$ and its inverse $\ln_{\kappa}(x)$, 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.