We study agitated frictional disks in two dimensions with the aim of developing a scaling theory for their diffusion over time. As a function of the area fraction $\varphi$ and mean-square velocity fluctuations $\langle v^2\rangle$ the mean-square displacement of the disks $\langle d^2\rangle$ spans four to five orders of magnitude. The motion evolves from a subdiffusive form to a complex diffusive behavior at long times. The statistics of $\langle d^n\rangle$ at all times are multiscaling, since the probability distribution function (PDF) of displacements has very broad wings. Even where a diffusion constant can be identified it is a complex function of $\varphi$ and $\langle v^2\rangle$. By identifying the relevant length and time scales and their interdependence one can rescale the data for the mean-square displacement and the PDF of displacements into collapsed scaling functions for all $\varphi$ and $\langle v^2\rangle$. These scaling functions provide a predictive tool, allowing one to infer from one set of measurements (at a given $\varphi$ and $\langle v^2\rangle$) what are the expected results at any value of $\varphi$ and $\langle v^2\rangle$ within the scaling range.

• Received 21 May 2019
• Revised 6 August 2019

DOI:https://doi.org/10.1103/PhysRevE.100.042902