# Diffusion in Agitated Frictional Granular Matter Near the Jamming Transition

## TL;DR

This study develops a scaling theory for the diffusion of agitated frictional disks near the jamming transition, revealing complex behaviors and broad displacement distributions that can be collapsed into universal scaling functions.

## Contribution

It introduces a comprehensive scaling framework that unifies diffusion behavior across different area fractions and velocity fluctuations in granular matter.

## Key findings

- Displacement statistics are multiscaling with broad wings.
- Diffusion constant depends complexly on area fraction and velocity fluctuations.
- Scaling functions enable prediction of diffusion behavior across parameters.

## Abstract

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 $\phi$ and mean-square velocity fluctuations $\langle v^2\rangle$ the mean-square displacement of the disks $\langle d^2\rangle$ spans 4-5 orders of magnitude. The motion evolves from a subdiffusive form to a complex diffusive behabvior 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 $\phi$ 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 $\phi$ and $\langle v^2\rangle$. These scaling functions provide a predictive tool, allowing to infer from one set of measurements (at a given $\phi$ and $\langle v^2\rangle$) what are the expected results at any value of $\phi$ and $\langle v^2\rangle$.