Collision-free gases in Bianchi space-times

We study collision-free gases in Bianchi space-times. Spatially homogeneous distribution functions are found for all Bianchi types by supposing that the distribution function f(x, p) is a function of the Killing vector constants of the motion only. Bianchi types I, VIII and IX only, lead to physical distributions. In types VIII and IX the average behaviour of the gas is that of a nonrotating viscous fluid. In an attempt to obtain physical spatially homogeneous distribution functions for all Bianchi types, we write the Liouville equation in a spatially homogeneous orthonormal tetrad. Furthermore, the general inhomogeneous solution of Liouville's equation in Bianchi type I is obtained, depending on constants of the motion that generalise the conserved quantities generated by Lorentz boosts in flat space-time.