ABSTRACT
An intuitive picture for light diffusion in disordered materials is the Brownian random walk model, in which multiply-scattered light is described as particles performing steps of varying length in random directions. By virtue of the Central Limit Theorem, when the variance of the step-length distribution is nite and the steps are independent, the mean step-length distribution after many steps becomes approximately normally distributed.7 The process therefore becomes diffusive irrespective of the microscopic transport mechanism. The ubiquity of diffusive processes in nature is inherent to the fact that the normal distribution is the attractor for random variables with nite variance. When, however, the step-length distribution of the random walk has a diverging variance, that is, the probability to perform arbitrary long steps becomes signicant, the limit distribution becomes a so-called α-stable Lévy distribution and the transport process no longer obeys standard diffusion dynamics, that is, it becomes anomalous.8,9
The α-stable Lévy distribution, named after Paul Pierre Lévy for his work on the problem of sums of random variables,10 constitutes the whole stable distribution family. Its probability density function has no general analytical expression but can be dened through the Fourier transform of its characteristic function. For symmetric and centered distributions, it is given by:11
f x kck ikx( ) = − −
∞∫12pi αe e d
(3.4.1)
where α ∈ (0,2] is the so-called stability index and c > 0 is a scale parameter that indicates the width of the distribution. One recognizes easily the normal and the Cauchy distributions for α = 2 and α = 1, respectively. The importance of the α parameter comes from the generalized form of the Central Limit Theorem,12 which states that the sum of many random variables, whose distribution asymptotically decays as |x|−(α+1) with 0 < α ≤ 2, converges to a distribution with stability index α. The heavy tail of α-stable Lévy distributions entails that transport following Lévy statistics is dominated by rare but very long steps, and thus appears radically different from that of a Brownian motion, as shown in Figure 3.4.1. The power-law decay of the step-length distribution is such that Lévy-type motions have the property of scale invariance. As a matter of fact, they were rst studied by Benoît Mandelbrot in the framework of random walks over fractals.13 Mandelbrot named this type of motion a Lévy ›ight and used it himself to model the ƒuctuations of cotton price.14 Lévy ƒights are now commonly used in the study of economy indices.15
