The mean flow, velocity dispersion, and evolution of rotating and
  growing dark matter halos and their effects on the energy transfer in
  self-gravitating collisionless flow
  By decomposing velocity dispersions into nonspin- and spin-induced, mean flow
and dispersions are analytically solved for axisymmetric rotating and growing
halos. The polar flow can be neglected and azimuthal flow is directly related
to dispersions. The fictitious ("Reynolds") stress acts on radial flow to
enable energy transfer from mean flow to random motion and maximize system
entropy. For large halos (high peak height $\nu$ at early stage of halo life)
with constant concentration, there exists a self-similar radial flow (outward
in core and inward in outer region). Halo mass, size and specific angular
momentum increase linearly with time via fast mass accretion. Halo core spins
faster than outer region. Large halos rotate with an angular velocity
proportional to Hubble parameter and spin-induced dispersion is dominant. All
specific energies (radial/rotational/kinetic/potential) are time-invariant.
Both halo spin (~0.031) and anisotropic parameters can be analytically derived.
For "small" halos with stable core and slow mass accretion (low peak height
$\nu$ at late stage of halo life), radial flow vanishes. Small halos rotate
with constant angular velocity and non-spin dispersion is dominant. Small halos
are more spherical in shape, incompressible, and isotropic. Radial and
azimuthal dispersions are comparable and greater than polar dispersion. Due to
finite spin, kinetic energy is not equipartitioned with the greatest energy
along azimuthal direction. Halos are hotter with faster spin. Halo relaxation
(evolution) from early to late stage involves continuous variation of shape,
density, mean flow, momentum, and energy. During relaxation, halo isotopically
"stretches" with conserved specific angular kinetic energy, increasing halo
concentration and momentum of inertial. Therefore, halo "stretching" leads to
decreasing angular velocity and increasing angular momentum and spin parameter.
