4-Velocity distribution function using Maxwell-Boltzmann's original   approach and a new form of the relativistic equation of state
  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. 
Prasad Basu and Soumen Mondal
astro-ph.SR