Granular relaxation under tapping and the traffic problem

We study the relaxation of a one-dimensional granular pile of height L in a confined geometry under repeated tapping within the context of the diffusing void model. The reduction of height as a function of the number of taps is proportional to the accumulated void density at the top layer. The relaxation process is characterized by the two dynamic exponents z and z′ which describe the time dependence of the height reduction Δh(t)≈${\mathrm{t}}^{\mathrm{z}}$ and the total relaxation time T(L)≈${\mathrm{L}}^{\mathrm{z}\prime }$. While the governing equation is nonlinear, we find numerically that z=z′=1, which is robust against perturbations and independent of the initial void distributions. We then show that the existence of a steady state traveling wave solution is responsible for such a linear behavior. Next, we examine the case where each void is able to maintain its overall topology as a round object that can subject itself to compression. In this regime, the governing equations for voids reduce to traffic equations and numerical solutions reveal that a cluster of voids arrives at the top periodically, which is manifested by the appearance of periodic solutions in the density at the top. In this case, the relaxation proceeds via a stick-slip process and the reduction of the height is sudden and discontinuous.