Statistical mechanical description of the parking-lot model for vibrated granular materials

We apply the statistical mechanical approach proposed by Edwards and co-workers to the parking-lot model, a model that reproduces the main features of the phenomenology of vibrated granular materials. We first build the compactivity-based measure for the case of vanishingly small tapping strength and then generalize the approach to finite tapping strengths by introducing a "thermodynamic" parameter, the available volume for particle insertion, in addition to the particle density. This description is able to take into account the various memory effects observed in vibrated granular media. Although not exact, the approach gives a good description of the behavior of the parking-lot model in the regime of slow compaction.