Aging dynamics and density relaxation in kinetic lattice gases under gravity

– We present an analytical approach to the out of equilibrium dynamics of a class of kinetic lattice gases under gravity. The location of the jamming transition, the critical exponents, and the scaling functions characterizing the relaxation processes are determined. In particular, we find that logarithmic compaction and simple aging are intimately related to the Vogel-Fulcher law, while power-law compaction and super-aging behavior occur in presence of a power-law diffusion. Granular materials set in rapid motion by vibration [1] exhibit features such as universal velocity distribution [2], and " de Gennes narrowing " — a physical signature of the cage effect [3] — which in spite of the non-equilibrium nature of the stationary state, closely resemble those observed in simple liquids at thermal equilibrium [4]. In the opposite, quasi-static flow limit, slow compaction phenomena appear [5]. During compaction, the free volume available to grains decreases, and the mobility steeply falls to zero, hence aging phenomena are expected to occur [6–8], as is confirmed in several numerical simulations [9–12]. It has been suggested that in this regime a granular material should resemble a highly viscous liquid or a glass [13], and several approaches have been proposed to describe different aspects of the granular compaction dynamics. These are mainly based on Langevin [14] and Fokker-Planck equation [15], fluctuating nonlinear hydrodynamics [16], and mode-coupling theory [17]. Two effects are responsible for the unusual behavior of a compacting granular material. First, collisions between the particles are inelastic, and energy has to be constantly pumped into the system. Second, at high packing density, steric hindrance, and the associated cage effect, play a crucial role very similar to the one observed in amorphous systems. In this letter we shall be concerned precisely with this second effect, which in the case of gentle shaking is the dominant one. A number of lattice-gas models have been introduced to study numerically the slow dynamics induced by this effect [18–20]. Our approach allows to characterize in a