Peculiar Velocity Decomposition, Redshift Space Distortion and Velocity
  Reconstruction in Redshift Surveys. II. Dark Matter Velocity Statistics
  Massive spectroscopic redshift surveys open a promising window to accurately
measure peculiar velocity at cosmological distances through redshift space
distortion (RSD). In paper I of this series of work we proposed to decompose
peculiar velocity into three eigen-modes (v_\delta, v_S and v_B) in order to
facilitate the RSD modeling and peculiar velocity reconstruction. In the
current paper we measure the dark matter RSD related statistics of the velocity
eigen-modes through a set of N-body simulations, including the velocity power
spectra, correlation functions, one-point probability distribution functions,
cumulants and the damping functions describing the Finger of God effect. (1)
The power spectrum measurement shows that these velocity components have
distinctly different spatial distribution and redshift evolution. In
particular, we measure the window function \tilde{W}(k,z), which describes the
impact of nonlinear evolution on the v_\delta-density relation. We confirm that
it can induce a significant systematic error of O(10%) in RSD cosmology. We
demonstrate that \tilde{W} can be accurately described by a simple fitting
formula with one or two free parameters. (2) The correlation function
measurement shows that the correlation length is O(100), O(10) and O(1) Mpc for
v_\delta, v_S and v_B respectively. These correlation lengths determine where
we can treat the velocity fields as spatially uncorrelated. (3) The velocity
PDFs and cumulants quantify non-Gaussianities of the velocity fields. We
confirm speculation in paper I that v_\delta is largely Gaussian, nevertheless
with non-negligible non-Gaussianity, v_B is significantly non-Gaussian. We also
measure the damping functions. Despite the observed non-Gaussianities, the
damping functions and hence the FOG effect are all well approximated as
Gaussian ones at scales of interest.
