In this paper we investigate the two-dimensional Abelian sandpile model in which the conservation of sand grains is locally violated using non-conservative lattice sites. We use spatially correlated arbitrary fractal patterns to mark the non-conservative sites. We have observed a “crossover” in the scaling behavior of the distribution functions of both the avalanche areas and avalanche sizes. The pre-crossover scaling exponents are known already and are related to the embedding dimension of the model. In addition to these, we have found that new “post-crossover” scaling exponents result from fractality. In fact we have found a spectrum of values for these exponents across the fractal dimensions \(1\le {d_\mathrm{f}}\le 2\) with a dominantly declining “linear” trend.

在本文中，我们研究了二维阿贝尔沙堆模型，其中使用非保守格点局部违反了沙粒的守恒。我们使用空间相关的任意分形图案来标记非保守站点。我们已经观察到雪崩面积和雪崩大小的分布函数的缩放行为的“交叉”。预交叉缩放指数是已知的，并且与模型的嵌入维度有关。除此之外，我们发现新的“交叉后”缩放指数是由分形引起的。事实上，我们已经发现了这些指数在分形维数\(1\le {d_\mathrm{f}}\le 2\)上的一系列值，具有显着下降的“线性”趋势。