Finite-size scaling of critical avalanches

We examine probability distribution for avalanche sizes observed in self-organized critical systems. While a power-law distribution with a cutoff because of finite system size is typical behavior, a systematic investigation reveals that it may also decrease with increasing the system size at a fixed avalanche size. We implement the scaling method and identify scaling functions. The data collapse ensures a correct estimation of the critical exponents and distinguishes two exponents related to avalanche size and system size. Our simple analysis provides striking implications. While the exact value for avalanches size exponent remains elusive for the prototype sandpile on a square lattice, we suggest the exponent should be 1. The simulation results represent that the distribution shows a logarithmic system size dependence, consistent with the normalization condition. We also argue that for the train or Oslo sandpile model with bulk drive, the avalanche size exponent is slightly less than 1, which differs significantly from the previous estimate of 1.11.

Figure 1

The curves show area distribution for the BTW sandpile model, where (a) $P\left(a\right)$ and (b) $ln\left(N\right)P\left(a\right)$. The system size $N$ varies from $2^4,2^6,\cdots$ to $2^{14}$. Throughout our simulation, we use $\mathcal{M}={10}^8$. For comparison, we draw lines (thick dashed) with respective slopes. (c) The variation of $P(a=1)$ with $N$ along with best fit curve $\sim 1/ln\left(N\right)$.

Figure 2

For the BTW sandpile model, the plot of (a) $P\left(s\right)$ and (b) $ln\left(N\right)P\left(s\right)$, with different system size $N$, varying from $2^6,2^8\cdots$ to $2^{14}$.

Figure 3

The data collapses correspond to Fig. 1. The scaling exponent is $\theta \approx 0.17$.

Figure 4

Bulk driven train model: The avalanche size distribution (a) $P\left(s\right)$ (b) $N^{0.17}P\left(s\right)$ for different $N$ values $2^4,2^6,2^8$, and $2^{10}$.