噪声对沙堆生长形态的影响
Effect of Noise on Patterns Formed by Growing Sandpiles
摘要：
我们考虑了在具有周期性初始构形的阿贝尔沙堆模型中，在单个位置上加入大量沙粒，然后松弛所产生的图案。这些模式显示了成比例的增长。我们研究了这些模式对不同类型的噪声，\textit{viz.}，在添加点的随机性，在初始周期性配置的混乱，和混乱的底层晶格的连接的鲁棒性。我们发现，在每种情况下，模式显示出不同程度的鲁棒性，除了少量的噪声。然而，在倾倒规则中引入随机性似乎完全破坏了渐近模式，即使是弱噪声。我们还讨论了一个变分制定的模式选择问题的增长阿贝尔沙堆。
英文摘要：
We consider patterns generated by adding large number of sand grains at a single site in an abelian sandpile model with a periodic initial configuration, and relaxing. The patterns show proportionate growth. We study the robustness of these patterns against different types of noise, \textit{viz.}, randomness in the point of addition, disorder in the initial periodic configuration, and disorder in the connectivity of the underlying lattice. We find that the patterns show a varying degree of robustness to addition of a small amount of noise in each case. However, introducing stochasticity in the toppling rules seems to destroy the asymptotic patterns completely, even for a weak noise. We also discuss a variational formulation of the pattern selection problem in growing abelian sandpiles.
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