  We study the velocity distribution in spherical collapses and cluster-pair
collisions by use of N-body simulations. Reflecting the violent gravitational
processes, the velocity distribution of the resultant quasi-stationary state
generally becomes non-Gaussian. Through the strong mixing of the violent
process, there appears a universal non-Gaussian velocity distribution, which is
a democratic (equal-weighted) superposition of many Gaussian distributions (DT
distribution). This is deeply related with the local virial equilibrium and the
linear mass-temperature relation which characterize the system. We show the
robustness of this distribution function against various initial conditions
which leads to the violent gravitational process. The DT distribution has a
positive correlation with the energy fluctuation of the system. On the other
hand, the coherent motion such as the radial motion in the spherical collapse
and the rotation with the angular momentum suppress the appearance of the DT
distribution.
