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

TL;DR

This paper uses a global equation of state to model a 2D hard sphere system under gravity, determining the critical temperature for condensation and the number of frozen layers, validated against molecular dynamics simulations.

Contribution

It introduces a method to calculate the critical temperature and frozen layers in a 2D vibrated hard sphere system using an empirical global equation of state.

Findings

Accurate density profile modeling with the global equation of state.

Derived formula for critical temperature T_c and frozen layers .

Good agreement between theoretical predictions and MD simulation data.

Abstract

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.