# Confined Sandpile in two Dimensions: Percolation and Singular Diffusion

## TL;DR

This paper studies a two-dimensional confined sandpile model, revealing two scale-invariant regimes related to percolation and giant cluster formation, with implications for understanding critical phenomena in confined granular systems.

## Contribution

It derives a diffusion equation from microdynamics and identifies two distinct scale-invariant regimes in a confined sandpile model, linking them to percolation and cluster coalescence phenomena.

## Key findings

- Power-law cluster size distribution near percolation threshold
- Evidence of gradient percolation characteristics
- Power-law jump size distribution in giant cluster regime

## Abstract

We investigate the properties of a two-state sandpile model subjected to a confining potential in two dimensions. From the microdynamical description, we derive a diffusion equation, and find a stationary solution for the case of a parabolic confining potential. By studying the systems at different confining conditions, we observe two scale-invariant regimes. At a given confining potential strength, the cluster size distribution takes the form of a power law. This regime corresponds to the situation in which the density at the center of the system approaches the critical percolation threshold. The analysis of the fractal dimension of the largest cluster frontier provides evidence that this regime is reminiscent of gradient percolation. By increasing further the confining potential, most of the particles coalesce in a giant cluster, and we observe a regime where the jump size distribution takes the form of a power law. The onset of this second regime is signaled by a maximum in the fluctuation of energy.