Search for a Lorentz invariant velocity distribution of a relativistic gas

TL;DR

This paper investigates the form of the velocity distribution in a relativistic gas, deriving a Lorentz-invariant distribution using relativistic principles, the central limit theorem, and the geometry of velocity space.

Contribution

It introduces a relativistic velocity distribution consistent with Lorentz invariance, extending classical results to relativistic regimes.

Findings

Distribution reduces to Maxwell-Boltzmann at low velocities

Provides a Lorentz-invariant distribution for relativistic gases

Combines analytical and numerical methods for derivation

Abstract

We examine numerically and analytically the problem of the relativistic velocity distribution in a 1-dim relativistic gas in thermal equilibrium. Our derivation is based on the special theory of relativity, the central limit theorem and the Lobachevsky structure of the velocity space of the theory, where the rapidity variable plays a crucial role. For v^2/c^2 << 1 and 1/\beta = k_B T/ m_0 c^2 << 1 the distribution tends to the Maxwell-Boltzmann distribution.