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

TL;DR

This study numerically investigates the relationship between dissipative and fixed-energy Abelian sandpile models, revealing that their critical densities coincide under certain initial conditions and large lattice limits.

Contribution

It demonstrates the numerical equivalence of critical densities in dissipative and fixed-energy ASM models for specific initial conditions and lattice sizes.

Findings

Critical density equals stationary density for zero initial height.

Critical density differs from stationary density for positive initial heights.

Threshold density in fixed-energy models matches critical density in dissipative models for large lattices.

Abstract

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 $h_0 \leq 0$.