Numerical study of the correspondence between the dissipative and fixed-energy Abelian sandpile models.
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 ρ(c)(h(0)), the system changes its behavior and relaxes exponentially to the stationary state of the ASM with density ρ(s). Considering initial heights -1 ≤ h(0) ≤ 4, we observe a dramatic decrease of the difference ρ(c)(h(0)) - ρ(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 ρ(th)(h(0)) of the FES converges rapidly to ρ(s) for h(0) < 1.
Institute for Informatics and Automation Problems NAS of Armenia, 375044 Yerevan, Armenia.
Su S
SS
Poghosyan
V S
VS
Poghosyan
V B
VB
Priezzhev
P
P
Ruelle