Logarithmic density relaxation in compaction of granular materials

On the basis of physical considerations we propose a one-dimensional discrete lattice model for the density relaxation of granular materials under tapping. Solving the difference equation numerically, we find a logarithmic time dependence of the density relaxation. This is in agreement with experimental results of Knight et al. [Phys. Rev. E 51, 3957 (1995)]. The origin of this anomalous relaxation is elucidated analytically by solving the equation of its continuum version asymptotically in time.