4-Velocity distribution function using Maxwell-Boltzmann's original approach and a new form of the relativistic equation of state

TL;DR

This paper derives a relativistic 4-velocity distribution function based on Maxwell-Boltzmann's original approach, introduces a new relativistic equation of state, and calculates related thermodynamic properties, ensuring consistency with known limits.

Contribution

It presents a novel derivation of the relativistic 4-velocity distribution and a new form of the relativistic equation of state, extending Maxwell-Boltzmann's approach to relativistic gases.

Findings

Distribution reduces to Maxwell-Boltzmann in non-relativistic limit

Correct ultra-relativistic and non-relativistic equations of state

Adiabatic index and sound speed bounds are satisfied

Abstract

Following the original approach of Maxwell-Boltzmann(MB), we derive a 4-velocity distribution function for the relativistic ideal gas. This distribution function perfectly reduces to original MB distribution in the non-relativistic limit. We express the relativistic equation of state(EOS), $\rho-\rho_0=(\gamma-1)^{-1}p$,\ in the two equations: $\rho=\rho_0 f(\lambda)$,\ and $p=\rho_0 g(\lambda)$, where $\lambda$\ is a parameter related to the kinetic energy, hence the temperature, of the gas. In the both extreme limits, they give correct EOS:\ $\rho=3p$\ in the ultra-relativistic, and\ $\rho-\rho_0=3/2p$ in the non-relativistic regime. Using these equations the adiabatic index $\gamma$ (=$\frac{c_p}{c_v}$) and the sound speed $a_s$ are calculated as a function of $\lambda$. They also satisfy the inequalities: $4/3 \le \gamma \le 5/3$ and $a_s \le \frac{1}{\sqrt{3}}$ perfectly.