The maximum entropy distributions of collisionless particle velocity,
  speed, and energy for statistical mechanics of self-gravitating collisionless
  flow (SG-CFD)
  The halo-mediated inverse mass cascade is a key feature of the intermediate
statistically steady state for self-gravitating collisionless flow (SG-CFD).
How the inverse mass cascade maximizes system entropy and develops limiting
velocity/energy distributions are fundamental questions to answer. We present a
statistical theory concerning the maximum entropy distributions of particle
velocity, speed, and energy for self-gravitating systems involving a power-law
long-range interaction with an arbitrary exponent $n$. For system with
long-range interaction ($-2<n<0$), a broad spectrum of halos and halo groups
are necessary to form from inverse mass cascade to maximize the system entropy.
While particle velocity in each halo group is still Gaussian, the velocity
distribution of entire system can be non-Gaussian. With the virial equilibrium
for local mechanical equilibrium of halo groups, the maximum entropy principle
is applied for statistical equilibrium of global system to derive the limiting
distributions. Halo mass function is not required in this formulation, but it
is a direct result of entropy maximization. The predicted velocity distribution
involves a shape parameter $\alpha$ that is dependent on the exponent $n$. The
velocity distribution approaches Laplacian with $\alpha\rightarrow0$ and
Gaussian with $\alpha\rightarrow\infty$. For intermediate $\alpha$, the maximum
entropy distribution naturally exhibits a Gaussian core at small velocity and
exponential wings at large velocity. The total energy of collisionless
particles at a given speed follows a parabolic scaling for small speed
($\epsilon \propto v^2$) and a linear scaling ($\epsilon \propto v$) for large
speed. Results are compared against a N-body simulation with good agreement.
