  The halo-mediated inverse mass cascade is a key feature of the intermediate
statistically steady state for self-gravitating collisionless dark matter 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, the distribution of total mass
in halos, 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 dark
matter 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 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 is determined by the
velocity distribution ($\textbf X$) that maximizes system entropy and the exact
model of halo velocity dispersion.
