  This study theoretically considers the motion of N identical inelastic
particles between two oscillating walls. The particles' average energy
increases abruptly at certain critical filling fractions, wherein the system
changes into a solid-like phase with particles clustered in their compact form.
Molecular dynamics simulations of the system show that the critical filling
fraction is a decreasing function of vibration amplitude independent of
vibration frequency, which is consistent with previous experimental results.
This study considers the entire group of particles as a giant pseudo-particle
with an effective size and an effective coefficient of restitution. The
N-particles system is then analytically treated as a one-particle problem. The
critical filling fraction's dependence on vibration amplitude can be explained
as a necessary condition for a stable resonant solution. The fluctuation to the
system's mean flow energy is also studied to show the relation between the
granular temperature and the system phase.
