Numerical solution of the Smoluchowski equation for a vibrofluidized granular bed

A stochastic approach, similar to that used to describe Brownian motion, was used to model the displacement probability of grains in a three-dimensional vibrofluidized granular bed. As neither an analytical description nor measurements of the diffusion coefficients were available, the governing partial differential equation, namely, the Smoluchowski equation, was solved numerically using an iterative procedure, modifying the granular temperature profile at each step. The results of this stochastic model were compared to experimental measurements of the displacement probability density made using positron emission particle tracking. The results indicate that methods based on hard elastic systems such as the Smoluchowski equation are appropriate to granular systems, particularly over timescales greater than the mean collision time.