The statistical theory of self-gravitating collisionless dark matter
  flow and high order kinematic and dynamic relations for velocity correlations
  on small and large scales
  Statistical theory for self-gravitating collisionless dark matter flow is not
fully developed because of 1) intrinsic complexity involving constant
divergence flow on small scale and irrotational flow on large scale; 2) lack of
self-closed description for peculiar velocity; and 3) mathematically
challenging. To better understand dark matter flow, kinematic and dynamic
relations among different statistical measures of velocity must be developed
for different types of flow. In this paper, a compact derivation is presented
to formulate general kinematic relations on any order for incompressible,
constant divergence, and irrotational flow. Results are validated by N-body
simulation. Dynamic relations can only be determined from self-closed
description of velocity evolution. On large scale, we found i) third order
correlation can be related to density correlation or pairwise velocity; ii)
effective viscosity in adhesion model originates from velocity fluctuations;
iii) negative viscosity is due to inverse energy cascade; iv) velocity
correlations follow $\propto a^{(q+2)/2}$ for odd q and $\propto a^{q/2}$ for
even q; v) overdensity is proportional to density correlation on the same
scale, $\langle\delta\rangle\propto\langle\delta\delta'\rangle$; vi) (reduced)
velocity dispersion is proportional to density correlation on the same scale;
vii) combined kinematic and dynamic relations determines exponential velocity
correlations on large scale. On small scale, self-closed description for
velocity evolution is developed by decomposing velocity into motion in halo and
motion of halo. Vorticity, enstrophy, and energy evolution can all be derived
from self-closed equation for velocity. Dynamic relation is derived to relate
third and second order correlations. Third moment of pairwise velocity is
determined by energy production rate $\epsilon_u$ or $\langle(\Delta
u_L)^3\rangle\propto\epsilon_uar$.
