Trajectory entanglement in dense granular materials

The particle-scale dynamics of granular materials have commonly been characterized by the self-diffusion coefficient D. However, this measure discards the collective and topological information known to be an important characteristic of particle trajectories in dense systems. Direct measurement of the entanglement of particle space-time trajectories can be obtained via the topological braid entropy S-braid, which has previously been used to quantify mixing efficiency in fluid systems. Here, we investigate the utility of S-braid in characterizing the dynamics of a dense, driven granular material at packing densities near the static jamming point phi(J). From particle trajectories measured within a two-dimensional granular material, we typically observe that S-braid is well defined and extensive. However, for systems where phi greater than or similar to 0.79, we find that S-braid (like D) is not well defined, signifying that these systems are not ergodic on the experimental timescale. Both S-braid and D decrease with either increasing packing density or confining pressure, independent of the applied boundary condition. The related braiding factor provides a means to identify multi-particle phenomena such as collective rearrangements. We discuss possible uses for this measure in characterizing granular systems.