Mass functions of dark matter halos from maximum entropy distributions
  for self-gravitating collisionless flow
  The halo-mediated inverse mass cascade is a key feature of the intermediate
statistically steady state for self-gravitating collisionless flow (SG-CFD). A
broad spectrum of halos and halo groups are necessary to form from inverse mass
cascade for long-range interaction system to maximize its entropy. The limiting
velocity ($\textbf X$), speed ($\textbf Z$), and energy ($\textbf E$)
distributions of collisionless particles can be obtained analytically from a
maximum entropy principle. Halo mass function, i.e. the spectrum of halo mass,
is a fundamental quantity for structure formation and evolution. Instead of
basing mass functions on simplified spherical/elliptical collapse models, it is
possible to reformulate mass function as an intrinsic distribution to maximize
system entropy during the everlasting statistically steady state. Starting from
halo-based description of non-equilibrium collisionless flow, distributions of
particle virial dispersion ($\textbf H$), square of particle velocity ($\textbf
P$), and number of halos ($\textbf J$) are proposed. Their statistical
properties and connections with the limiting velocity distribution ($\textbf
X$) are well studied and established. With $\textbf H$ being essentially the
halo mass function, two limiting cases of $\textbf H$ distribution are analyzed
for large halos ($\textbf H_\infty$) and small halos ($\textbf H_s$),
respectively. For large halos, $\textbf H_\infty$ is shown to also be a maximum
entropy distribution. For small halos, $\textbf H_s$ approximates the $\textbf
P$ distribution and recovers the Press-Schechter mass function. The full
solution of $\textbf H$ distribution depends on the limiting distribution
($\textbf X$) that maximizes system entropy and the exact models of halo
velocity dispersions.
