  To better understand the collisionless dark matter flow on different scales,
statistical theory involving kinematic and dynamic relations must be developed
for different types of flow, e.g. incompressible, constant divergence, and
irrotational flow. This paper extends our previous work on the second-order
statistics (Phys. Fluids 35, 077105) to high order statistics. Kinematic and
dynamic relations were developed for dark matter flow on different scales. The
results were validated by N-body simulations. On large scales, we found i)
third-order velocity correlations can be related to density correlation or
pairwise velocity; ii) the $p$th-order velocity correlations follow $\propto
a^{(p+2)/2}$ for odd $p$ and $\propto a^{p/2}$ for even $p$, where $a$ is the
scale factor; iii) the overdensity $\delta$ is proportional to density
correlation on the same scale; iv) velocity dispersion on a given scale $r$ is
proportional to the overdensity on the same scale. On small scales, i) a
self-closed velocity evolution is developed by decomposing the velocity into
motion in haloes and motion of haloes; ii) the evolution of vorticity and
enstrophy are derived from the evolution of velocity; iii) dynamic relations
are derived to relate second- and third-order correlations; iv) while the first
moment of pairwise velocity follows $\langle\Delta u_L\rangle=-Har$ ($H$ is the
Hubble parameter), the third moment follows $\langle(\Delta
u_L)^3\rangle\propto\varepsilon_uar$ that can be directly compared with
simulations and observations, where $\varepsilon_u\approx10^{-7}$m$^2$/s$^3$ is
the constant rate for energy cascade; v) the $p$th order velocity correlations
follow $\propto a^{(3p-5)/4}$ for odd $p$ and $\propto a^{3p/4}$ for even $p$.
Finally, the combined kinematic and dynamic relations lead to exponential and
one-fourth power-law velocity correlations on large and small scales,
respectively.
