  Distribution functions of relative velocities among particles in a vibrated
bed of powder are studied both numerically and theoretically. In the solid
phase where granular particles remain near their local stable states, the
probability distribution is Gaussian. On the other hand, in the fluidized
phase, where the particles can exchange their positions, the distribution
clearly deviates from Gaussian. This is interpreted with two analogies;
aggregation processes and soft-to-hard turbulence transition in thermal
convection. The non-Gaussian distribution is well-approximated by the
t-distribution which is derived theoretically by considering the effect of
clustering by inelastic collisions in the former analogy.
