Relativistic Gas: Invariant Lorentz Distribution for the velocities

TL;DR

This paper derives a Lorentz-invariant velocity distribution for a relativistic gas that aligns with simulations, addressing limitations of the traditional Jüttner distribution by considering the geometry of velocity space.

Contribution

It introduces a new Lorentz-invariant distribution for relativistic gases based on the geometry of velocity space, validated by extensive simulations.

Findings

The new distribution remains curvature-invariant at all energies.

Simulations confirm the new distribution's consistency with relativistic gas dynamics.

Jüttner distribution's curvature change at high energy is not observed in the new model.

Abstract

In 1911, J\"uttner proposed the generalization, for a relativistic gas, of the Maxwell-Boltzmann distribution of velocities. Here we want to discuss, among others, J\"uttner probability density function (PDF). Both the velocity space and, consequently, the momentum space are not flat in special relativity. The velocity space corresponds to the Lobachevsky one, which has a negative curvature. This curvature induces a specific power for the Lorentz factor in the PDF, affecting J\"uttner normalization constant in one, two, and three dimensions. Furthermore, J\"uttner distribution, written in terms of a more convenient variable, the rapidity, presents a curvature change at the origin at sufficiently high energy, which does not agree with our computational dynamics simulations of a relativistic gas. However, in one dimension, the rapidity satisfies a simple additivity law. This allows us to obtain, through the Central Limit Theorem, a new, Lorentz-invariant, PDF whose curvature at the origin does not change for any energy value and which agrees with our computational dynamics simulations data. Also, we perform extensive first-principle simulations of a one-dimensional relativistic gas constituted by light and heavy particles.