Time Dependent Behavior of Granular Material in a Vibrating Box

Using numerical and analytic methods, we study the time dependent behavior of granular material in a vibrating box. We find, by molecular dynamics simulation, that the temporal fluctuations of the pressure and the height expansion scale in $A f$, where $A$ ($f$) is the amplitude (frequency) of the vibration. On the other hand, the fluctuations of the velocity and the granular temperature do not scale in any simple combination of $A$ and $f$. Using the kinetic theory of Haff, we study the temporal behaviors of the hydrodynamic quantities by perturbing about their time averaged values in the quasi-incompressible limit. The results of the kinetic theory disagree with the numerical simulations. The kinetic theory predicts that the whole material oscillates roughly as a single block. However, the numerical simulations show that the region of active particle movement is localized and moves with time, behavior very similar to the propagation of a sound wave.