We present a simple model for the shape of the distribution function of galaxy peculiar velocities. We show how both nonlinear and linear theory terms combine to produce a distribution which has an approximately Gaussian core with exponential wings. The model is easily extended to study how the statistic depends on the type of particle used to trace the velocity field (dark matter particles, dark matter haloes, galaxies), and on the density of the environment in which the test particles are. Comparisons with simulations suggest that our model is accurate. We also show that the evolution of the peculiar velocities depends on the local, rather than the global density. Since clusters populate denser regions on average, using cluster velocities with the linear theory scaling may lead to an overestimate of the global value of Omega. Conversely, using linear theory with the global value of Omega to scale cluster velocities from the initial to the present time results in an underestimate of their true velocities. In general, however, the directions of motions of haloes are rather well described by linear theory. Our results help to simplify models of redshift-space distortions considerably.

18 pages, 18 figures, submitted to MNRAS 2000 July 15