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Theoretical studies of self-organized criticality

机译：自组织临界的理论研究

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These notes are intended to provide a pedagogical introduction to the abelian sandpile model of self-organized criticality, and its related models. The abelian group, the algebra of particle addition operators, the burning test for recurrent states, equivalence to the spanning trees problem are described. The exact solution of the directed version of the model in any dimension is explained. The model's equivalence to Scheidegger's model of river basins, Takayasu's aggregation model and the voter model is discussed. For the undirected case, the solution for one-dimensional lattices and the Bethe lattice is briefly described. Known results about the two dimensional case are summarized. Generalization to the abelian distributed processors model is discussed. Time-dependent properties and the universality of critical behavior in sandpiles are briefly discussed. I conclude by listing some still-unsolved problems. (c) 2006 Elsevier B.V. All rights reserved.
机译：这些说明旨在为自组织临界的阿贝尔沙堆模型及其相关模型提供教学上的介绍。描述了阿贝尔群，粒子加法算子的代数，递归状态的燃烧测试，与生成树问题的等价性。解释了模型在任何维度上的有向版本的精确解。讨论了该模型与Scheidegger流域模型，Takayasu的聚集模型和选民模型的等效性。对于无向情况，将简要描述一维晶格和Bethe晶格的解。总结了有关二维情况的已知结果。讨论了对abelian分布式处理器模型的推广。简要讨论了随时间变化的特性和沙堆中关键行为的普遍性。最后，我列出了一些尚未解决的问题。 （c）2006 Elsevier B.V.保留所有权利。

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