Determining the Critical Temperature and Number of Frozen Layers in a Two-Dimensional Bed of Vibrating Hard Spheres Using a Global Equation of State

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.