Ktitarev DV, Lubeck S, Grassberger P
John von Neumann Institute fur Computing, Forschungszentrum Julich, 52425 Julich, Germany and Theoretische Physik, Gerhard-Mercator-Universitat Duisburg, 47048 Duisburg, Germany.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Jan;61(1):81-92. doi: 10.1103/physreve.61.81.
We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D>/=2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their correspondence to spanning trees, and by extensive numerical simulations. While the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for waves display clear power-law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributions of radius, area, and duration of bulk and boundary waves. Relations between them and fractal dimensions of waves are derived. We confirm that the upper critical dimension D(u) of the model is 4, and calculate logarithmic corrections to the scaling behavior of waves in D=4. In addition, we present analytical estimates for bulk avalanches in dimensions D>/=4 and simulation data for avalanches in D</=3. For D=2 they seem not easy to interpret.
我们研究了维度(D\geq2)的超立方晶格上Bak-Tang-Wiesenfeld模型中倾倒波的概率分布。波代表不包含多次倾倒事件的弛豫过程。我们通过与生成树的对应关系以及广泛的数值模拟来研究体波和边界波。虽然雪崩的标度行为很复杂，通常不受简单标度律的支配，但我们表明，波的概率分布显示出清晰的幂律渐近行为，与解析预测完全一致。获得了体波和边界波的半径、面积和持续时间分布的临界指数。推导了它们与波的分形维数之间的关系。我们证实该模型的上临界维度(D(u))为4，并计算了(D = 4)时波的标度行为的对数修正。此外，我们给出了维度(D\geq4)时体雪崩的解析估计以及维度(D\leq3)时雪崩的模拟数据。对于(D = 2)，它们似乎不容易解释。
