Spin-Models of Granular Compaction: from One-Dimensional Models to Random Graphs

Theory and experiment have both contributed extensively to the study of compaction in granular media in recent years. The experiments of the Chicago group [18,19] have in particular inspired a large body of theorists to model the tapping of various systems. In many of these formulations, including the one we present here, an analogy is made between the volume of a granular system and the Hamiltonian of a spin system, along the lines first proposed by Edwards [11]; minimising the energy of the spin system then corresponds to minimising the volume of the granular system. Grains can interlock only in specific ways, if they are undeformable. It is thus often the case that locally compatible grain orientations could result in globally unfavourable ones, since different (and individually well-packed) clusters of grains could be unfavourably oriented with respect to each other. This frustration is therefore an essential ingredient in the modelling of granular media; it can be modelled in terms of orientational disorder of individual grains [22,15] in lattice-based models, or, as in the present work, the orientational disorder of plaquettes in either lattice or off-lattice models of granular media. The second ingredient of models of granular compaction is the dynamics designed to model a series of successive taps to the system. A tap applied to a granular assembly for a brief moment feeds kinetic energy into the system and causes