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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), $ρ-ρ_0=(γ-1)^{-1}p$,\ in the two equations: $ρ=ρ_0 f(λ)$,\ and $p=ρ_0 g(λ)$, where $λ$\ is a parameter related to the kinetic energy, hence the temperature, of the gas. In the both extreme limits, they give correct EOS:\ $ρ=3p$\ in the ultra-relativistic, and\ $ρ-ρ_0=3/2p$ in the non-relativistic regime. Using these equations the adiabatic index $γ$ (=$\frac{c_p}{c_v}$) and the sound speed $a_s$ are calculated as a function of $λ$. They also satisfy the inequalities: $4/3 \le γ\le 5/3$ and $a_s \le \frac{1}{\sqrt{3}}$ perfectly.