The Relativistic Maxwell-J\"uttner Velocity Distribution Function

TL;DR

This paper derives the relativistic Maxwell-Jüttner velocity distribution, compares it with the classical Maxwell-Boltzmann distribution, and demonstrates its importance in high-temperature astrophysical and plasma contexts.

Contribution

It provides a detailed derivation of the relativistic distribution and compares its applicability to classical models in various high-temperature scenarios.

Findings

Relativistic effects are significant when the particle rest mass to temperature ratio is low.

The Maxwell-Jüttner distribution differs notably from Maxwell-Boltzmann at high temperatures.

Relativistic treatment is necessary for accurate velocity distributions in astrophysical plasmas.

Abstract

In this paper, the expression of the relativistic Maxwell-J\"uttner velocity distribution was deduced in a detailed and didactic way. This distribution was compared with the well-known Maxwell-Boltzmann distribution, which does not take into account relativistic effects. Applications and comparisons between the two distributions were carried out in three situations: i) protons at solar surface temperature, ii) electrons at Lawson's criterion temperature, and iii) protons at quasar temperatures. It was found that the value of the ratio between the rest mass of the particle considered and the temperature of the system determines the need to consider or not a relativistic treatment for the velocity distribution function of the system.