Slow dynamics under gravity: a nonlinear diffusion model

We present an analytical and numerical study of a nonlinear diffusion model which describes density relaxation of densely packed particles under gravity and weak random (thermal) vibration, and compare the results with Monte Carlo simulations of a lattice gas under gravity. The dynamical equation can be thought of as a local density functional theory for a class of lattice gases used to model slow relaxation of glassy and granular materials. The theory predicts a jamming transition line between a low-density fluid phase and a high-density glassy regime, characterized by diverging relaxation time and logarithmic or power-law compaction according to the specific form of the diffusion coefficient. In particular, we show that the model exhibits history-dependent properties, such as quasi-reversible–irreversible cycle and memory effects—as observed in recent experiments, and dynamical heterogeneities.