A Lorentz invariant velocity distribution for a relativistic gas

TL;DR

This paper introduces a Lorentz invariant velocity distribution for relativistic gases, derived using relativity, the central limit theorem, and hyperbolic geometry, which converges to Maxwell-Boltzmann at low velocities.

Contribution

It presents a novel Lorentz invariant distribution for relativistic velocities, incorporating the geometry of velocity space and the rapidity variable, extending classical distributions to relativistic regimes.

Findings

Distribution reduces to Maxwell-Boltzmann at low velocities.

Mean squared velocity is always less than c^2 and bounded by 1.

Temperature estimates differ by about 10% at high temperatures.

Abstract

We derive a Lorentz invariant distribution of velocities for a relativistic gas. Our derivation is based on three pillars: the special theory of relativity, the central limit theorem and the Lobachevskyian structure of the velocity space of the theory. The rapidity variable plays a crucial role in our results. For $v^2/c^2 \ll 1$ and $1/\beta=kT/2 m_0 c^2 \ll 1$ the distribution tends to the Maxwell-Boltzmann distribution. The mean $\langle v^2 \rangle$ evaluated with the Lorentz invariant distribution is always smaller than the Maxwell-Boltzmann mean and is bounded by $\langle v^2 \rangle/c^2=1$. This implies that for a given $\langle v^2 \rangle$ the temperature is larger than the temperature estimated using the Maxwell-Boltzmann distribution. For temperatures of the order of $T \sim {10^{12}}~ K$ and $T \sim {10^{8}}~ K$ the difference is of the order of $10 \%$, respectively for particles with the hydrogen and the electron rest masses.