Slow time scales in a dense vibrofluidized granular material

Modeling collective motion in nonconservative systems, such as granular materials, is difficult since a general microscopic-to-macroscopic approach is not available: there is no Hamiltonian, no known stationary densities in phase space, and not a known small set of relevant variables. Phenomenological coarse-grained models are a good alternative, provided that one has identified a few slow observables and collected a sufficient amount of data for their dynamics. Here we study the case of a vibrofluidized dense granular material. The experimental study of a tracer, dispersed into the media, showed evidence of many time scales: Fast ballistic, intermediate caged, slow superdiffusive, and very slow diffusive. A numerical investigation has demonstrated that a tracer's superdiffusion is related to slow rotating drifts of the granular medium. Here we offer a deeper insight into the slow scales of the granular medium, and we propose a phenomenological model for such a "secular"dynamics. Based upon the model for the granular medium, we also introduce a model for the tracer (fast and slow) dynamics, which consists in a stochastic system of equations for three coupled variables, and is therefore more refined and successful than previous models.