  In 1911, J\"uttner proposed the generalization, for a relativistic gas, of
the Maxwell-Boltzmann distribution of velocities. Here we want to discuss,
among others, J\"uttner probability density function (PDF). Both the velocity
space and, consequently, the momentum space are not flat in special relativity.
The velocity space corresponds to the Lobachevsky one, which has a negative
curvature. This curvature induces a specific power for the Lorentz factor in
the PDF, affecting J\"uttner normalization constant in one, two, and three
dimensions. Furthermore, J\"uttner distribution, written in terms of a more
convenient variable, the rapidity, presents a curvature change at the origin at
sufficiently high energy, which does not agree with our computational dynamics
simulations of a relativistic gas. However, in one dimension, the rapidity
satisfies a simple additivity law. This allows us to obtain, through the
Central Limit Theorem, a new, Lorentz-invariant, PDF whose curvature at the
origin does not change for any energy value and which agrees with our
computational dynamics simulations data. Also, we perform extensive
first-principle simulations of a one-dimensional relativistic gas constituted
by light and heavy particles.
