  We present numerical studies of the properties of the stellar velocity
distribution in galactic disks which have developed a saturated, two-armed
spiral structure. In previous papers we used the Boltzmann moment equations
(BME) up to second order for our studies of the velocity structure in
self-gravitating stellar disks. A key assumption of our BME approach is the
zero-heat flux approximation, i.e. the neglection of third order velocity
terms. We tested this assumption by performing test particle simulations for
stars in a disk galaxy subject to a rotating spiral perturbation. As a result
we corroborated qualitatively the complex velocity structure found in the BME
approach. It turned out that an equilibrium configuration in velocity space is
only slowly established on a typical timescale of 5 Gyrs or more. Since many
dynamical processes in galaxies (like the growth of spirals or bars)act on
shorter timescales, pure equilibrium models might not be fully appropriate for
a detailed comparison with observations like the local Galactic velocity
distribution. Third order velocity moments were typically small and
uncorrelated over almost all of the disk with the exception of the 4:1
resonance region (UHR). Near the UHR (normalized) fourth and fifth order
velocity moments are still of the same order as the second and third order
terms. Thus, at the UHR higher order terms are not negligible.
