A generalized Landau kinetic equation for weakly-coupled probability
  distribution of N-stars in dense star cluster
  The secular evolution of a collisional star cluster of $N$-'point' stars have
been conventionally discussed based on cumulative two-body relaxation process.
The relaxation process requires a cut-off on the range of two-body encounter
between stars in physical space and the relaxation time is characterized by
Coulomb logarithm $\ln[N]$; the conventional cut-off on the encounter distance
in the literature gives "dominant" effect. In addition, incorrect cut-offs
exposed a mathematical "infinite-density" problem in the late stage of
core-collapse.
  The present paper shows these are merely the results due to incorrect cut-off
process. If one correctly constrains the cut-off on interaction range between
stars based on truncated BBGKY hierarchy, one must introduce a self-consistent
'truncated' Newtonian mean-field (m.f.) acceleration of star. The present paper
shows the effect of total number on the structure of finite star cluster (in
which stars undergo only two-body encounters) for the first time after
establishing a mathematical formulation of a generalized Landau kinetic
equation that includes the cut-off effect on both of collision term and m.f.
potential by employing a BBGKY hierarchy for truncated DF stars. The cut-off
effect increases the typical relaxation time by a few of percentage, which
means the effect of cut-off itself is "non-dominant" on the relaxation time. On
the other hand, the cut-off on m.f. acceleration is necessary to avoid
"infinite-density" problem at the center of the system; the effect of total
number $N$ on density profile and m.f. acceleration are shown by applying the
truncated-DF BBGKY hierarchy to a toy model (a quasi-static modified Hubble
density profile) for a core-halo structure of a star cluster at the late stage
of evolution.
