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

Abstract

We find the probability density function $\mathcal{P}(V_{\text{r}})$ 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 $⟨V_{\text{r}}⟩$. 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.

Comments: 5 pages, 2 figures. V2: text revised. Part of this material is published in Phys. Rev. D 89, 103533 (2014) in a merged version with arXiv:1311.4508