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

TL;DR

This paper revises classical fractal laws relating conduction and diffusion on networks, showing their limitations and proposing a unifying formula that accounts for anisotropic exploration in fractal trees.

Contribution

It introduces a new formula that unifies the fractal-Einstein and Alexander-Orbach laws, clarifying their applicability and resolving longstanding discrepancies.

Findings

Classical laws are violated by certain fractal trees.

A new unifying formula is derived.

Anisotropic exploration explains the laws' failure.

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.