Lecture 1: Introduction to Random Walks and Diffusion

The term “random walk” was originally proposed by Karl Pearson in 19051. In a letter to Nature, he gave a simple model to describe a mosquito infestation in a forest. At each time step, a single mosquito moves a fixed length a, at a randomly chosen angle. Pearson wanted to know the distribution of the mosquitos after many steps had been taken. The letter was answered by Lord Rayleigh, who had already solved a more general form of this problem in 1880, in the context of sound waves in heterogeneous materials. Modeling a sound wave traveling through the material can be thought of as summing up a sequence of random wavevectors k of constant amplitude but random phase: sound waves in the material have roughly constant wavelength, but their directions are altered at scattering sites within the material. We wish to find the probability density function of the sound waves after many steps have been taken. We let PN (R)dR be the probability of traveling a distance between R and R + dR in N steps. For steps of unit length, Lord Rayleigh showed that as N →∞,