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

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.

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