Part two - Anomalous diffusion

Summary

This part of the book explores the subject of anomalous diffusion in fractals and disordered media. Our goal here is twofold: to introduce various approaches to the problem – exact, as well as approximate; and to become intimately acquainted with the phenomenon of anomalous diffusion, its characteristics, and its causes.

Chapter 5 discusses diffusion in the Sierpinski gasket. Anomalous diffusion is demonstrated through simulations and exact enumerations, and through an exact renormalization of the first-passage time. The important relation between diffusion and conductivity (the Einstein relation), introduced in Section 3.4, is used to rederive the anomalous diffusion exponent in yet another way. The chapter closes with a discussion of probability-density-distribution functions and of fractons and spectral dimensions.

In Chapter 6 we present a summary of diffusion in percolation clusters. The question of diffusion in the incipient infinite cluster versus diffusion in all of the clusters is analyzed through scaling and simulations. The chapter describes also the attempt of the Alexander–Orbach conjecture to connect between static and dynamic exponents, diffusion in chemical space, and the multifractality of conductivity and diffusion in percolation clusters.

Chapter 7 discusses diffusion in loopless fractals: Eden trees, combs, etc. In this important case some exact results may be derived, including general relations between static and dynamic exponents, which shed some light on the more general situation in which loops are relevant.