A coarse grained model of granular compaction and relaxation

We introduce a theoretical model for the compaction of granular materials by discrete vibrations which is expected to hold when the intensity of vibration is low. The dynamical unit is taken to be clusters of granules that belong to the same collective structure. We rigourously construct the model from first principles and show that numerical solutions compare favourably with a range of experimental results. This includes the logarithmic relaxation towards a statistical steady state, the effect of varying the intensity of vibration resulting in a so-called ``annealing'' curve, and the power spectrum of density fluctuations in the steady state itself. A mean field version of the model is introduced which shares many features with the exact model and is open to quantitative analysis.