Random walks normally obey Gaussian statistics, and their average square displacement increases linearly with time; 〈r2〉 ∼ t. In many physical systems, however, it is found that diffusion follows an anomalous pattern: the mean-square displacement is 〈r2〉 ∼ t2/dw, where dw ≠ 2. Here we discuss several models of anomalous diffusion, including CTRWs (with algebraically long waiting times), Lévy flights and Lévy walks, and a variation of Mandelbrot's fractional-Brownianmotion (FBM) model. These models serve as useful, tractable approximations to the more difficult problem of anomalous diffusion in disordered media, which is discussed in subsequent chapters.

Random walks as fractal objects

The trail left by a random walker is a complicated random object. Remarkably, under close scrutiny it is found that the trail is self-similar and can be thought of as a fractal (Exercise 1). The ubiquity of diffusion in Nature makes it one of the most fundamental mechanisms giving rise to random fractals.

The fractal dimension of a random walk is called the walk dimension and is denoted by dw. If we think of the sites visited by a walker as “mass”, then the mass of the walk is proportional to time.

Diffusion in disordered, fractal structures is anomalous, different than that in regular space. Fractal structures are found everywhere in Nature, and as a consequence anomalous diffusion has far-reaching implications for a host of phenomena. We see its effects in flow within fractured and porous rocks, in the anomalous density of states in dilute magnetic systems, in silica aerogels and in glassy ionic conductors, anomalous relaxation in spin glasses and in macromolecules, conductivity of superionic conductors such as hollandite and of percolation clusters of Pb on thin films of Ge and Au, electron–hole recombination in amorphous semiconductors, and fusion and trapping of excitations in porous membrane films, polymeric glasses, and isotropic mixed crystals, to mention a few examples.

It was Pierre Gilles de Gennes who first realized the broad importance of anomalous diffusion, and who coined the term “the ant in the labyrinth”, describing the meandering of random walkers in percolation clusters. Since the pioneering work of de Gennes, the field has expanded very rapidly. The subject has been reviewed by several authors, including ourselves, and from various perspectives. This book builds upon our review on anomalous diffusion from 1987 and it covers the vast material that has accumulated since. Many questions that were unanswered then have been settled, yet, as usual, this has only brought forth a myriad of other questions. Whole new directions of research have emerged, most noticeably in the area of diffusion-limited reactions. The scope of developments is immense and cannot possibly be addressed in one volume. Neither do we have the necessary expertise. Hence, we have chosen once again to base the presentation mostly on heuristic scaling arguments.

In this book we have studied various effects produced by quenched disorder on thermodynamical properties of statistical systems. Considering different types of model, the emphasis has been made on the demonstration of the basic theoretical approaches and ideas. Although the considered systems and the corresponding problems involved (such as spin glasses, critical phenomena, directed polymers etc.) may at first look quite different, the aim of the book was to demonstrate that basically all these problems are deeply interconnected. I was trying to convince the reader that to work successfully on any particular problem in this field one needs to be familar with all the methods and ideas of statistical field theory. The physics of both the spin-glass state and critical phenomena in weakly disordered systems involve the ideas of the scaling theory of phase transitions, and the basic concepts of the replica theory of spin glasses. The aim of this book was to take the reader, starting from fundamentals and demonstrating well-established solutions of various problems, to the frontier of modern research. Here we are facing quite a few fundamental problems, both long-standing and new ones, still waiting for their solutions.

The most appealing problem in the scope of spin glasses had remained a question for almost two decades: whether or not the mean-field RSB physical picture (described in Chapters 2–6) is valid, at least at the qualitative level, for realistic spin glasses with finite-range interactions.

Random walks model a host of phenomena and find applications in virtually all sciences. With only minor adjustments they may represent the thermal motion of electrons in a metal, or the migration of holes in a semiconductor. The continuum limit of the random walk model is known as “diffusion”. It may describe Brownian motion of a particle immersed in a fluid, as well as heat propagation, the spreading of a drop of dye in a glass of still water, bacterial motion and other types of biological migration, or the spreading of diseases in dense populations. Random-walk theory is useful in sciences as diverse as thermodynamics, crystallography, astronomy, biology, and even economics, in which it models fluctuations in the stock market.

The simple random walk

A random walk is a stochastic process defined on the points of a lattice. Usually, the time variable is considered discrete. At each time unit the “walker” steps from its present position to one of the other sites of the lattice according to a prescribed random rule. This rule is independent of the history of the walk, and so the process is Markovian.

In the simplest version of a random walk, the walk is performed in a hypercubic d-dimensional lattice of unit lattice spacing. At each time step the walker hops to one of its nearest-neighbor sites, with equal probabilities.

In an insightful pioneering work de Gennes (1976a; 1976b) pondered the problem of a random walker in percolation clusters, which he described as “the ant in the labyrinth”. Similar ideas were presented at the time by Brandt (1975), Kopelman (1976), and Mitescu and Roussenq (1976). de Gennes' “ants” triggered intensive research on diffusion in disordered media. Here we describe the more important aspects of the subject. A brief account of percolation theory has been presented in Chapter 2.

The analogy with diffusion in fractals

As discussed in Chapter 2, percolation clusters may be regarded as random fractals. Below the critical threshold, p < pc the clusters are finite. The largest clusters have a typical size of the order of the (finite) correlation length ξ(p), and they possess a fractal dimension df = d − β/ν (Eq. (2.8)).

At criticality, p = pc, there emerges an infinite percolation cluster that may be described as a random fractal with the same dimension df = d − β/ν. The inception of the infinite cluster coincides with the divergence of the correlation length, ξ ∼ |p − pc|−ν. Along with the incipient infinite cluster there exist clusters of finite extent. The finite clusters may be regarded as fractals possessing the usual percolation dimension df. They are not different, practically, than the clusters that form below the percolation threshold, other than that there is no limit to their typical size – since the global correlation length diverges.

When the rate of coalescence is finite the typical time for reaction competes with the typical diffusion time, and a crossover between the reaction-limited regime and the diffusion-limited regime is observed. The model cannot be solved exactly, but it can be approached through an approximation based on the IPDF method. The kinetic crossover is well captured by this approximation.

A model for finite coalescence rates

Until now we have dealt with infinite coalescence rates: when a particle hops into an occupied site the coalescence reaction is immediate. Thus, the typical reaction time is zero and the process is diffusion-limited. We want now to discuss the case in which the coalescence rate (and the typical time for the coalescence reaction) is finite. In this case a competition arises between the typical transport time and the typical reaction time.

The model we have in mind is the following: when a particle attempts to hop onto a site that is already occupied, the move is allowed and coalescence takes place with probability k (0 ≤ k ≤ 1). The attempt is rejected, and the state of the system remains unchanged, with probability (1―k). The case of k = 1 corresponds to an infinite coalescence rate, which we have studied so far. The opposite limit, of k-0, describes diffusion of the particles with hard-core repulsion, but no reactions take place.

Suppose that 0 < k « 1. Reactions then require a large number of collisions, and the kinetics is dominated by the long reaction times (the reaction-limited regime).

In previous chapters, we have seen examples of anomalous diffusion in the CTRW model, in Lévy flights, and in long-range correlated walks. Diffusion in fractal lattices is also anomalous. Here we consider nearest-neighbor random walks in the Sierpinski gasket, for which an exact solution is possible. We analyze the problem and solve it following several different approaches. The analysis not only illustrates anomalous diffusion in a simple way, but also stresses important aspects of diffusion theory, such as the relation to conductivity and elasticity.

Anomalous diffusion

Imagine a random walk in the Sierpinski gasket. At each step the walker moves randomly to one of the four nearest-neighbor sites on the gasket, with equal probabilities. We require the mean-square displacement after n steps, 〈r2(n)〉.

Naively, one should think that, since diffusion is regular (〈r2(n)〉 ∼ n) in all integer dimensions, so would be the case for fractals, since fractals may be regarded as mere extrapolations of regular space to noninteger dimensions. Surprisingly, this turns out to be wrong! Perhaps the best way to convince oneself of this fact is to perform numerical simulations of random walks on the Sierpinski gasket. The technique is described in Appendix A.

In Fig. 5.1 we show results obtained from exact enumeration. In fractals, however, the different sites are not equivalent. Regard the Sierpinski gasket as embedded in a regular two-dimensional triangular lattice. At each node of the gasket two bonds of the embedding triangular lattice are missing, but which bonds are missing varies from one gasket site to the next.