  Using a global equation of state, empirically derived by Luding, we
accurately model the density profile of a two-dimensional hard sphere system
with diameter D and mass m under gravity with a given temperature T [Physica A,
271, 192 (1999)]. We then compare our solutions to MD simulated data. From the
density profile, we can then solve for the critical temperature T_c, which we
define as the temperature at which the system begins to condensate. Then, if T
is below T_c, there is some frozen portion of the system. We derive a formula
for the number of frozen layers \mu_f, and compare our solution to the number
of frozen layers in our simulated data.
