Fractional kinetic model for granular compaction

We present an approach to granular compaction based on subordination of stochastic processes. In order to imitate, in a very simplified way, the compaction dynamics of granular material under tapping, we impose that particles switch stochastically between the two possible orientational states characterizing the average volumes of the grain in the presence of other grains. The main physical idea of our approach is that the interaction of grains with their environment is taken into account with the aid of the temporal subordination. Accordingly, we assume that the time intervals between the consecutive grain’s reorientations are governed by a certain waiting-time distribution ψ ( t ). It is demonstrated how the presence of the trapping events leads to the macroscopic observation of slow compaction dynamics, described by an exact fractional kinetic equation. We also perform numerical simulations to examine our analytical result. In addition, we reproduce the memory effects numerically by considering the response of the system to the abrupt change in the external excitation.