Search for a Lorentz invariant velocity distribution of a relativistic gas

We examine numerically and analytically the problem of the relativistic velocity distribution in a 1-dim relativistic gas in thermal equilibrium. Our derivation is based on the special theory of relativity, the central limit theorem and the Lobachevsky structure of the velocity space of the theory, where the rapidity variable plays a crucial role. For v^2/c^2 << 1 and 1/\beta = k_B T/ m_0 c^2 << 1 the distribution tends to the Maxwell-Boltzmann distribution.