Confined sandpile in two dimensions: Percolation and singular diffusion

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.

Figure 1

Schematic picture of the two-dimensional confined sandpile model. The sites with discs inside are occupied, the site with a black disk inside is the source site of the jump, and the arrow points to the target site of the jump. The jump is the colored arrow. The tone gradient indicates the local potential $\mathrm{\Phi }$.

Figure 2

Occupation density as a function of $r$ for different values of $\kappa$. The points are estimated from a radial histogram of the occupation averaged over time with $N_p=8000$ particles, while the solid lines are the predictions of Eq. (7). The inset shows the occupation density calculated at $r=0$. The solid line is the analytical prediction ${\rho }_o=1-exp(-\beta \kappa N_p{\delta }^2/\pi )$.

Figure 3

Jump size distribution for different values of $\kappa$ with $N_p=8000$. The points are numerical simulations and the solid lines are the predictions of Eq. (8). Here we define $\mathcal{D}=2\sqrt{N_p/\pi }$ as the diameter of a dense disk with $N_p$ particles.

Figure 4

Mean jump energy fluctuation as function of $\kappa$ for different number of particles $N_p$. The points are the results from Eq. (9). For each value of $N_p$, the energy fluctuation exhibits a maximum at a different value ${\kappa }^{*}\left(N_p\right)$.

Figure 5

Jump size distribution for $\kappa ={\kappa }^{*}$ and different values of $N_p$. This result is obtained through Eq. (8) choosing $S_o$ in such a way that $S\left(s\right)$ is normalized. The factor $\mathcal{D}=2\sqrt{N_p/\pi }$ is the diameter of a compact cluster with all particles.

Figure 6

Snapshots of simulations of systems with 10 000 particles for different values of $\kappa$. The colors indicate different clusters. Observe the relative size of the clusters and the formation of a giant cluster when $\kappa$ increases.

Figure 7

Fraction of the largest cluster $\mathcal{M}$ computed from numerical simulations as a function of occupation density at the origin ${\rho }_o$ for different values of number of particles $N_p$. The inset shows the standard deviation of $\mathcal{M}$ as a function of ${\rho }_o$.

Figure 8

Cluster size distribution obtained through numerical simulations for $N_p=8000$ and different values of $\kappa$. The dashed line is a power law with exponent equal to $-2.35$.

Figure 9

Log-log plot of the cluster size distribution obtained through numerical simulations for different values of $N_p$ and $\kappa ={\kappa }^{\prime }$. The dashed line is a power law with exponent equal to $-2.35$.

Figure 10

Diffusion frontier with $N_p=100000$ and $\kappa =20.0$. This picture shows the results of a simulation on a two-dimensional square lattice where we put 100 000 particles following the probability given by Eq. (7). The red sites are the occupied sites that belong to the largest cluster and the black sites are the largest cluster sites that belong to the diffusion frontier. The gray sites are the occupied sites that do not belong to the largest cluster. The white sites are empty, and the blue ones correspond to the empty sites that belong to the empty cluster (connected through first or second neighbors to the more external empty sites). In this picture, there are 5960 sites at the diffusion frontier.

Figure 11

Probability of finding a diffusion frontier site at a distance $r$. In the inset, we show $\sigma$, the standard deviation of the distance of the diffusion frontier sites to the origin, and ${\mathcal{R}}_f$, the diffusion frontier radius as a function of the strength $\kappa$ of the confining potential. Here we used $N_p=10000000$.

Figure 12

Fractal dimension determination from multiple resolution analysis. The slope of the dashed line is $-d_f$ according to Eq. (10). The dashed line represents approximately the slope for most curves analyzed. The inset shows us how $d_f$ grows with $\sigma /{\mathcal{R}}_f$.