Probability distribution for the relative velocity of colliding particles in a relativistic classical gas

We find the probability density function $\mathcal{P}(V_{\texttt{r}})$ of the relativistic relative velocity for two colliding particles in a non-degenerate relativistic gas. The distribution reduces to Maxwell distribution for the relative velocity in the non-relativistic limit. We find an exact formula for the mean value $\langle V_{\texttt{r}}\rangle$. The mean velocity tends to the Maxwell's value in the non-relativistic limit and to the velocity of light in the ultra-relativistic limit. At a given temperature $T$, when at least for one of the two particles the ratio of the rest energy over the thermal energy $m c^2/k_B T$ is smaller than 40 the Maxwell distribution is inadequate.

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