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) h₀. During the first stage of the evolution, the density of particles increases linearly. Reaching a critical density ρc(h₀), the system changes its behavior and relaxes exponentially to the stationary state of the ASM with density ρₛ. Considering initial heights -1≤h₀≤4, we observe a dramatic decrease of the difference ρc(h₀)-ρₛ when h₀ 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 ρₜₕ(h₀) of the FES converges rapidly to ρₛ for h₀<1.