网络上分形爱因斯坦传导扩散定律的推广
Generalisation of the fractal Einstein law relating conduction and diffusion on networks
摘要：
在20世纪80年代，分形新兴领域的一个重要目标是确定它们的物理和几何属性之间的关系。分形-爱因斯坦定律和亚历山大-奥巴赫定律是这种类型的两个关键理论，它们将电学、扩散和分形性质联系起来。在这里，我们解决了一个长期存在的争议，他们的准确性表明，一类分形树的性质违反这两个法律。一个新的公式推导出统一的两个经典的结果证明，如果一个持有，那么必须其他的，并解决了令人困惑的差异的性质的伊甸园树和扩散限制聚集。经典定律的失败是由于网络的各向异性探索的随机步行者。这种新发现的行为的发生意味着许多理论，例如最近的首次通过时间结果，被限制在比以前认为的更窄的网络范围内。
英文摘要：
In the 1980s an important goal of the emergent field of fractals was to determine the relationships between their physical and geometrical properties. The fractal-Einstein and Alexander-Orbach laws, which interrelate electrical, diffusive and fractal properties, are two key theories of this type. Here we settle a long standing controversy about their exactness by showing that the properties of a class of fractal trees violate both laws. A new formula is derived which unifies the two classical results by proving that if one holds, then so must the other, and resolves a puzzling discrepancy in the properties of Eden trees and diffusion limited aggregates. The failure of the classical laws is attributed to anisotropic exploration of the network by a random walker. The occurrence of this newly revealed behaviour means that numerous theories, such as recent first passage time results, are restricted to a narrower range of networks than previously thought.
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