Trajectory entanglement in dense granular materials

TL;DR

This paper explores the use of topological braid entropy to characterize particle trajectory entanglement in dense granular materials, revealing insights into collective dynamics and ergodicity near the jamming transition.

Contribution

It demonstrates the applicability of topological braid entropy in dense granular systems and compares it with traditional diffusion measures, highlighting its ability to detect collective phenomena.

Findings

$ ext{S}_{ ext{braid}}$ is well-defined in less dense systems

Both $ ext{S}_{ ext{braid}}$ and $D$ decrease with increasing density and pressure

$ ext{S}_{ ext{braid}}$ indicates non-ergodic behavior near $ ext{density} \, ext{gtrsim} \, 0.79$

Abstract

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 $\Sbraid$, which has previously been used to quantify mixing efficiency in fluid systems. Here, we investigate the utility of $\Sbraid$ 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 $\Sbraid$ is well-defined and extensive. However, for systems where $\phi \gtrsim 0.79$, we find that $\Sbraid$ (like $D$) is not well-defined, signifying that these systems are not ergodic on the experimental timescale. Both $\Sbraid$ 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.