Scaling and universality in avalanches

We have studied various one- and two-dimensional models in order to simulate the behavior of avalanches. The models are based on cellular automata and were intended to have the property of self-organized criticality proposed by Bak, Tang, and Wiesenfeld [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 384 (1988)]. By varying the sizes of the systems, we have investigated the scaling properties of these models. In particular, we have addressed the question as to whether simple finite-size scaling or multifractal analysis is more suited to fitting the data on the distribution of avalanche sizes. By varying the underlying microscopic rules that describe how an avalanche is generated, we have also studied whether different models have the same, universal properties. In our one-dimensional models we find that the multifractal analysis is much better than the analysis based on simple finite-size scaling. We also find that there are several different universality classes. Nevertheless, certain models with similar rules appear to belong to the same class. In two dimensions, we find that the simple finite-size scaling works quite well and that the distribution functions can be fit over wide ranges by a simple power law. The multifractal analysis also works well and it is difficult to tell which form is a better fit to the data. Again, as in one dimension, there are several different universality classes and different models with similar rules belong to the same class.