  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.
