Two-thirds law for pairwise velocity and origin of critical MOND
  acceleration from distributions of density, velocity, and acceleration in
  dark matter flow
  A halo-based non-projection approach is proposed to study scale/redshift
dependence of distributions (PDF) in dark matter flow. All particles are
divided into halo and out-of-halo particles such that PDF can be studied
separately. Without projecting particle fields onto grid, scale dependence is
analyzed by counting all pairs on different scales $r$. Redshift dependence is
studied via generalized kurtosis. From this analysis, we demonstrate: i)
Root-mean-square acceleration (~$10^{-10}m/s^2$) in halos matches critical
acceleration $a_0$ in MOND that can be determined by the rate of energy
transfer $\epsilon_u$, where $a_0=-(3\pi)^2\epsilon_u/u$ and $u^2(a=1)=u^2_0$
is velocity dispersion. Particle energy is dependent on its speed as
$\propto{v^2}$ if its acceleration $a_p$>>$a_0$ and $\propto{v^1}$ if
$a_p$<<$a_0$. Both deep-MOND and Newtonian behavior can be recovered in dark
matter flow; ii) $m$th moment of pairwise velocity $\langle(\Delta
u_L)^m\rangle$ is analytically modelled. On small scale, even moments can be
modelled by a two-thirds law $\propto{(-\epsilon_ur)}^{2/3}$, odd moments
$\propto{r}$ and satisfy GSCH; iii) Scale dependence is studied for
longitudinal velocity $u_L$ or $u_L^{'}$, pairwise velocity $\Delta
u_L$=$u_L^{'}$-$u_L$ and sum velocity $\Sigma u_L$=$u^{'}_L$+$u_L$. Fully
developed velocity fields are never Gaussian on any scale. On small scale, both
$u_L$ and $\Sigma u_L$ can be modelled by a X distribution for maximum entropy.
PDF of $\Delta u_L$ is different with its moments derived. On large scale, both
$\Delta u_L$ and $\Sigma u_L$ can be modelled by a logistic function. Time
evolution of velocity distributions follows prediction of X distribution with a
decreasing shape parameter $\alpha$ to maximize system entropy; iv) Delaunay
tessellation is used to reconstruct density field. Density
correlations/spectrum are obtained, modeled and compared with theory.
