Spectral responses in granular compaction

I study the compaction of a granular pack under periodic tapping. The magnitude of acceleration $\Gamma$ at each tap is modulated with frequency $\omega$ and amplitude $\delta\Gamma$: $\Gamma(t) = \Gamma_{\mathrm{DC}} + \delta \Gamma \sin(\omega t)$, where $t$ is time measured by the number of taps. From the temporal modulation $\delta v$ in packing volume $v$, frequency- locked to the modulated tapping input, we can define the real and imaginary volume susceptibilities $\chi_{v}' = (\delta v/\delta \Gamma) \cos \theta$ and $\chi_{v}'' = (\delta v/\delta \Gamma) \sin \theta$; here $\theta$ is the phase lag between $\Gamma(t)$ and $v(t)$. As a function of $\Gamma_{\mathrm{DC}}$, $\chi_{v}'$, $\chi_{v}''$ are peaked at low $\Gamma_{\mathrm{DC}}$, a behavior reminiscent of the temperature-dependent susceptibilities in dielectric and spin glasses. For the packing of small particles ($d = 0.5$ mm) in ambient pressure, $\chi_{v}'$ exhibits memory and rejuvenation effects under $\Gamma_{\mathrm{DC}}$ cycling, similar to that seen in the magnetic susceptibility of spin glasses when subjected to thermal cycling [1]. However this memory effect is suppressed for the packing of larger particles and in vacuum. The measurement of volume susceptibilities shows promise as a new way to study the packing of granular materials, and as an avenue to explore analogies between jammed grains and molecular and spin glasses. \\[3pt] [1] K. Jonason \textit{et al.}, Phys. Rev. Lett. \textbf{81}, 3243 (1998).