Slow relaxation in a diffusional model for granular compaction

We investigate the behaviour of a lattice model based on the ideas of random sequential adsorption and diffusional relaxation, for the compaction of a granular system subject to random perturbation and gravity. The lattice is composed by a given number of horizontal layers where non overlapping particles diffuse. Besides diffusion within its own layer a particle suffers a downfall to the layer below whenever there is enough space there. We restrict at first to the case of one-dimensional layers, and observe, both numerically and by a mean field approximation, an algebraic growth of the particle density that becomes slower and slower as the particle size increases with respect to the lattice spacing, both numerically and by a mean field approximation. We then consider a modification of the model that allows for a nearly exact treatment in the limit of low densities, and show that the same algebraic decay takes place in the general d dimensional case suggesting a logarithmic-like behaviour in the continuum limit.