An orbit-averaged generalized-Landau kinetic equation for the relaxation
  evolution of finite weakly-coupled star clusters: `Discreteness' stochastic
  acceleration and anti-normalization
  In the relaxation evolution of finite weakly-coupled star clusters, stars
undergo stochastic acceleration due to the `discreteness' of the clusters (the
finiteness of the total stellar number), in addition to fundamental two-body
relaxation processes. The acceleration is the essential non-collective
many-body relaxation process. However, existing works have never detailed the
`discreteness' stochastic acceleration and the corresponding mathematical
model, i.e., the generalized-Landau (g-Landau) kinetic equation for the stellar
distribution function. The present paper shows the kinetic formulation of an
orbit-averaged g-Landau equation in action-angle coordinates, beginning with
Bogoliubox-Born-Green-Kirkwood-Yvon hierarchy. We show that only the g-Landau
equation can satisfy the anti-normalization condition among existing
approximated star-cluster kinetic equations if gravitational polarization is
neglected. Accordingly, only the equation can correctly define the total energy
and the total number of stars in phase space. It also holds the conservation
laws and the H-theorem. We further show that stars undergoing the discreteness
stochastic acceleration can hold these physical properties independently of
stars experiencing the two-body relaxation. Lastly, we suggest that the
stochastic acceleration tends to isotropize the stellar DF and to drive star
clusters into a stationary state in the early relaxation-evolution stage before
the two-body relaxation thermalizes stars.
