Avalanche behavior in an absorbing state Oslo model

Self-organized criticality can be translated into the language of absorbing state phase transitions. Most models for which this analogy is established have been investigated for their absorbing state characteristics. In this paper, we transform the self-organized critical Oslo model into an absorbing state Oslo model and analyze the avalanche behavior. We find that the resulting gap exponent $D$ is consistent with its value in the self-organized critical model. For the avalanche size exponent $\tau$ an analysis of the effect of the external drive and the boundary conditions is required.

Figure 1

The logarithm of the rescaled moments $⟨s^n⟩(\zeta ;L)∕L^{{\gamma }_n}$ versus the slope density $\zeta$ for $n=1$, 2. The exponential interpolations between the data points for increasing system sizes are marked with lines of increasing dash length. For the triplet $L=2536$, 5072, 10 144, the lower set of graphs for $n=1$ intersect in a single point at ${\zeta }_c=1.732608$ with ${\gamma }_1=2.064$ and the upper set of graphs for $n=2$ intersect at ${\zeta }_c=1.732609$ with ${\gamma }_2=4.342$.

Figure 2

The estimated exponents ${\gamma }_n$ vs the order $n$ of the moment for the triplets $L=1268$, 2536, 5072 (open squares) and $L=2536$, 5072, 10 144 (open circles). Assuming simple finite-size scaling it follows from Eq. (3) that the slope of the graphs is the gap exponent $D$ while the intersection with the $n$ axis is $\tau -1$; see inset (dotted lines indicate magnified region). Linear regression yields $D=2.22\left(23\right),\tau =1.07\left(18\right)$ for the first triplet (dashed line) and $D=2.28\left(23\right),\tau =1.09\left(13\right)$ for the second triplet (solid line).