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.