Cages and Anomalous Diffusion in Vibrated Dense Granular Media

A vertically shaken granular medium hosts a blade rotating around a fixed vertical axis, which acts as a mesorheological probe. At high densities, independently of the shaking intensity, the blade’s dynamics shows strong caging effects, marked by transient subdiffusion and a maximum in the velocity power density spectrum, at a resonant frequency $\sim 10\mathrm{Hz}$. Interpreting the data through a diffusing harmonic cage model allows us to retrieve the elastic constant of the granular medium and its collective diffusion coefficient. For high frequencies $f$, a tail $\sim 1/f$ in the velocity power density spectrum reveals nontrivial correlations in the intracage microdynamics. At very long times (larger than 10 s), a superdiffusive behavior emerges, ballistic in the most extreme cases. Consistently, the distribution of slow velocity inversion times $\tau$ displays a power-law decay, likely due to persistent collective fluctuations of the host medium.

Figure 1

(a) Schematic of the experimental setup. (b),(c) Mean squared angular velocity of the blade in the various experiments.

Figure 2

Power density spectra of the blade’s angular velocity for the two series. (a) The number of beads is fixed ($N=2600$) and the shaking intensity varies. (b) The shaking intensity is fixed to $\mathrm{\Gamma }=39.8$ and $N$ changes. The frequencies of vibration are marked by the yellow bar. Dashed lines are fits of Eq. (2) to the experimental data in regions II and III. Thick-dashed lines show the limit behaviors $f^{-2}$ and $f^{-1}$.

Figure 3

Mean-squared displacement $⟨[\mathrm{\Delta }\theta (t){]}^2⟩$ after a time $t$ for the two series. (a) $N=2600$ and $\mathrm{\Gamma }$ varies. (b) The shaking intensity is fixed to $\mathrm{\Gamma }=39.8$ and $N$ changes. Black dashed lines are guides for the eye.

Figure 4

(a),(b) Angular velocity $\omega (t)$ and filtered signal ${\omega }_s(t)$ (with a running average over a time $\tau =2\mathrm{s}$), for $N=2600$ and $\mathrm{\Gamma }=26.8$. (c)–(f) PDF of the inversion times calculated (with $\tau =1$ and 2 s) in different experiments. From (c)–(f) the following configurations are displayed: very dilute, intermediate, dense at high energy, dense at low energy. Colored dashed lines are guides for the eye.