Entropy for relaxation dynamics in granular media

We investigate the role of entropic concepts for the relaxation dynamics in granular systems. In these systems the existence of a geometrical frustration induces a drastic modification of the allowed phase space, which in its turn induces a dynamic behavior characterized by hierarchical relaxation phenomena with several time scales associated. In particular we show how, in the framework of a mean-field model introduced for the compaction phenomenon, there exists a free-energy-like functional which decreases along the trajectories of the dynamics and which allows to account for the asymptotic behavior: e.g. density profile, segregation phenomena. Also we are able to perform the continuous limit of the above mentioned model which turns out to be a diffusive limit. In this framework one can single out two separate physical ingredients: the free-energy-like functional that defines the phase-space and the asymptotic states and a diffusion coefficient $D(\rho)$ accounting for the velocity of approach to the asymptotic stationary states