Effect of Rare Events on Out-of-Equilibrium Relaxation

This Letter reports experimental and numerical results on particle dynamics in an out-of-equilibrium granular medium. We observed two distinct types of grain motion: the well known cage motion, during which a grain is always surrounded by the same neighbors, and low probability “jumps,” during which a grain moves significantly more relative to the others. These observations are similar to the results obtained for other out-of-equilibrium systems (glasses, colloidal systems, etc.). Although such jumps are extremely rare, by inhibiting them in numerical simulations we demonstrate that they play a significant role in the relaxation of out-of-equilibrium systems.

Figure 1

Experimental observation of a jump and distribution of bead displacements obtained by confocal imaging in an index matching fluid. The two pictures correspond to a horizontal cross section at the same altitude of the granular sample before and after the 126th tap of intensity $\Gamma =6$. All the beads remain in their original layer except one bead (pointed out by a white arrow) which falls from the upper to the lower layer. The distribution of bead displacements computed for 10 taps, 5 before and 5 after the 126th tap, reveals the two scales of displacement.

Figure 2

The histogram of displacements in a horizontal direction ($\delta x$) for our 3D experiment ($\Gamma =6$) and for our corresponding numerical simulation ($\epsilon =0.05$). The lines correspond to Gaussian fits. We observe deviations from the fits in the tails of the distributions.

Figure 3

Trajectory of a bead, exhibiting one jump, during the 100 first taps. The position of the bead center is plotted after each tap during a simulation of linear dilation parameter $\epsilon =0.05$. Between the 46th and the 47th taps, a jump is visible corresponding to a large displacement. The units are in bead diameter.

Figure 4

Evolution of the packing fraction with and without jumps for the same dilation parameter $\epsilon =0.05$. For the latter case, the increase of the packing fraction is slower at the beginning, as the number of inhibited events is greater at the beginning. Interestingly, the packing fraction reached by the sample after a long time is lower without jumps than with jumps.

Figure 5

Evolution of $⟨\Delta r^2⟩$ with $t$ for simulations with jumps and with inhibited jumps but for the same dilation parameter $\epsilon =0.05$. The random walk created correctly describes the behavior without jumps but fails to reproduce the behavior observed with jumps.