  The residual gas damping of the test mass (TM) in the free molecular flow
regime is studied in the finite open systems for high-precision gravity-related
experiments. Through strict derivation, we separate the damping coefficients
for two finite open systems, i.e., the bi-plate system and the sensor core
system, into base damping and diffusion damping. This elucidates the
relationship between the free damping in the infinite gas volume and the
proximity damping in the constrained volume, unifies them into one microscopic
picture, and allows us to point out three pathways of energy dissipation in the
bi-plate gap. We also provide the conditions that need to be met to achieve
this separation. In applications, for space gravitational wave detection, our
results for the residual gas damping coefficient for the 4TM torsion balance
experiment is the closest one to the experimental and simulation data compared
to previous models. For the LISA mission, our estimation for residual gas
acceleration noise at the sensitive axis is consistent with the simulation
result, within about $5\%$ difference. In addition, in the test of the
gravitational inverse-square law, our results suggest that the constraint on
the distance between TM and the conducting membrane can be reduced by about
$28\%$.
