  We write the correlation function of dark matter particles, xi(r), as the sum
of two terms - one which accounts for nonlinear evolution, and dominates on
small scales, and another which is essentially the term from linear theory, and
dominates on large scales. We use models of the number and spatial distribution
of haloes and halo density profiles to describe the nonlinear term and its
evolution. The result provides a good description of the evolution of xi(r) in
simulations. We then use this decomposition to provide simple and accurate
models of how the single particle velocity dispersion evolves with time, and
how the first and second moments of the pairwise velocity distribution depend
on scale. The key idea is to use the simple physics of linear theory on large
scales, the simple physics of the virial theorem on small scales, and our model
for the correlation function to tell us how to weight the two types of
contributions (linear and nonlinear) to the pairwise velocity statistics. When
incorporated into the streaming model, our results will allow a simple accurate
description of redshift-space distortions over the entire range of linear to
highly nonlinear regimes.
