Absorbing phase transitions in deterministic fixed-energy sandpile models

We investigate the origin of the difference, which was noticed by Fey et al. [Phys. Rev. Lett. 104, 145703 (2010)], between the steady state density of an Abelian sandpile model (ASM) and the transition point of its corresponding deterministic fixed-energy sandpile model (DFES). Being deterministic, the configuration space of a DFES can be divided into two disjoint classes such that every configuration in one class should evolve into one of absorbing states, whereas no configurations in the other class can reach an absorbing state. Since the two classes are separated in terms of toppling dynamics, the system can be made to exhibit an absorbing phase transition (APT) at various points that depend on the initial probability distribution of the configurations. Furthermore, we show that in general the transition point also depends on whether an infinite-size limit is taken before or after the infinite-time limit. To demonstrate, we numerically study the two-dimensional DFES with Bak-Tang-Wiesenfeld toppling rule (BTW-FES). We confirm that there are indeed many thresholds. Nonetheless, the critical phenomena at various transition points are found to be universal. We furthermore discuss a microscopic absorbing phase transition, or a so-called spreading dynamics, of the BTW-FES, to find that the phase transition in this setting is related to the dynamical isotropic percolation process rather than self-organized criticality. In particular, we argue that choosing recurrent configurations of the corresponding ASM as an initial configuration does not allow for a nontrivial APT in the DFES.

Figure 1

Plot of ${\zeta }_c$ vs $\lambda$ for the TM (14) with the initial condition (18) together with the distribution (23).

Figure 2

Time evolution of the DFES with the TM (16) for the initial configuration (32) with $V=80$. The PU is employed. At every time step, the length of the middle string of 2's (underlined) decreases by 2.

Figure 3

Semilogarithmic plots of probability density of the reduced parameter $x=(e-{\zeta }_{h_o,h_e})/{\sigma }_{h_o,h_e}$ for $h_o=0$, $h_e=1$ (open symbols) and $h_o=0$, $h_e=2$ (solid symbols). The system sizes are $L=2^7$ (squares), $2^8$ (circles), and $2^9$ (triangles). For comparison, the Gaussian distribution with zero mean and unit variance is also drawn by a dashed curve. Inset: Plots of normalized probability density for $h_o=h_e=2$ on a semilogarithmic scale. The system sizes are $L=2^7$ (squares) and $2^9$ (triangles). The Gaussian distribution which is depicted by a dashed curve deviates significantly from $P\left(x\right)$.

Figure 4

Plots of the effective exponent $-{\delta }_{\text{eff}}\left(t\right)$ around the critical point as a function of $t^{-0.3}$ for (a) $h_o=0$, $h_e=1$, (b) $h_o=h_e=1$, and (c) $h_o=0$, $h_e=2$. Although the transition points are different, the decay exponent for every case is $0.38\pm 0.01$, which supports that all belong to the same universality class.

Figure 5

Plot of ${\rho }_a$ vs $t$ for $h_o=h_e=2$ and $\zeta =2.07$.

Figure 6

Plots of (a) ${\eta }_{\text{eff}}$ vs $1/t$, (b) $-{\theta }_{\text{eff}}$ vs $1/t$, and (c) $z_{\text{eff}}$ vs $1/\sqrt{t}$ for the avalanche dynamics with Eq. (48). The values of $p_3$ are 0.580 59 (top curve), 0.580 585 (middle curve), and 0.580 58 (bottom curve). The critical exponents are consistent with those of the dynamical isotropic percolation, given in Eq. (46).