The statistical theory of self-gravitating collisionless dark matter
  flow and the correlation, structure, and dispersion functions for velocity,
  density, and potential fields
  N-body simulation is an invaluable tool to understand cosmic velocity field.
However, simulation samples velocity only at particle locations that leads to
information loss when projecting velocity fields onto structured grid. Here we
extract two-point statistics without field projection. These statistics, i.e.
correlation/structure/dispersion functions in real space and spectrum functions
in Fourier space, are modeled on both small and large scales. Kinematic
relations between statistical measures are fully developed for incompressible,
constant divergence, and irrotational flow. The nature of flow can be
identified by these relations. Much more complex than incompressible flow,
peculiar velocity of dark matter flow is of constant divergence on small scale
and irrotational on large scale. Incompressible and constant divergence flow
share same kinematic relations for even order correlations. The limiting
correlation of velocity $\rho_L=1/2$ on the smallest scale ($r=0$) is a unique
feature of collisionless flow ($\rho_L=1$ for incompressible flow). Assuming
gravity is the only interaction and no radiation produced, this feature leads
to an increase in particle mass converted from kinetic energy upon
"annihilation". On large scale, transverse velocity correlation has an
exponential form with a comoving scale $r_2$=21.3Mpc/h that maybe related to
the size of sound horizon. All other correlation/structure/dispersion/spectrum
functions for velocity/density/potential are derived analytically from
kinematic relations for irrotational flow. On small scale, longitudinal
structure function follows one-fourth law of $S^l_2\propto r^{1/4}$. All other
statistical measures are obtained from kinematic relations for constant
divergence flow. Vorticity is negatively correlated for $r$ between 1 and
7Mpc/h. Divergence is negatively correlated for $r$>30Mpc/h that leads to
negative density correlation.
