Generalisation of the fractal Einstein law relating conduction and diffusion on networks

Anthony P. Roberts; Christophe P. Haynes

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arXiv · 0903.3279

Generalisation of the fractal Einstein law relating conduction and diffusion on networks

Anthony P. Roberts·Christophe P. Haynes

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Abstract

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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Anthony P. Roberts, Christophe P. Haynes. 2009-03-19. Generalisation of the fractal Einstein law relating conduction and diffusion on networks. https://arxiv.org/abs/0903.3279

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