We consider the Abelian sandpile model (ASM) on the square lattice with a single dissipative site (sink). Particles are added one by one per unit time at random sites and the resulting density of particles is calculated as a function of time. We observe different scenarios of evolution depending on the value of initial uniform density (height) $h_0$. During the first stage of the evolution, the density of particles increases linearly. Reaching a critical density ${\rho }_c\left(h_0\right)$, the system changes its behavior and relaxes exponentially to the stationary state of the ASM with density ${\rho }_s$. Considering initial heights $-1\le h_0\le 4$, we observe a dramatic decrease of the difference ${\rho }_c\left(h_0\right)-{\rho }_s$ when $h_0$ is zero or negative. In parallel with the ASM, we consider the conservative fixed energy sandpile (FES). The extensive Monte Carlo simulations show that the threshold density ${\rho }_{\mathrm{th}}\left(h_0\right)$ of the FES converges rapidly to ${\rho }_s$ for $h_0<1$.

• Received 27 May 2011

DOI:https://doi.org/10.1103/PhysRevE.84.066119