Numerical Study of the Correspondence Between the Dissipative and Fixed Energy Abelian Sandpile Models

We consider the Abelian sandpile model (ASM) on the large square lattice with a single dissipative site (sink). Particles are added 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) $h_0=0,1,2,3$. During the first stage of the evolution, the density of particles increases linearly. Reaching a critical density $\rho_c(h_0)$, the system changes its behavior sharply and relaxes exponentially to the stationary state of the ASM with $\rho_s=25/8$. We found numerically that $\rho_c(0)=\rho_s$ and $\rho_c(h_0>0) \neq \rho_s$. Our observations suggest that the equality $\rho_c=\rho_s$ holds for more general initial conditions with non-positive heights. In parallel with the ASM, we consider the conservative fixed-energy Abelian sandpile model (FES). The extensive Monte-Carlo simulations for $h_0=0,1,2,3$ have confirmed that in the limit of large lattices $\rho_c(h_0)$ coincides with the threshold density $\rho_{th}(h_0)$ of FES. Therefore, $\rho_{th}(h_0)$ can be identified with $\rho_s$ if the FES starts its evolution with non-positive uniform height $h_0 \leq 0$.