Random Walks in One Dimension: New Results and Applications.

Although the first explicit formulation of the random walk problem was given by Pearson as early as 1905, random walk theory continues to be, even today, a field of very active and exciting research with a wide range of very distinct applications. Our purpose here is to add a few new technical results, in particular for the case of one-dimensional random walks, and to add some further applications to the long list of existing ones. We start with a review of the techniques that can be used to calculate the Green's function of a random walk. We present a few "tricks", and derive some new explicit results for the Green's function of a one-dimensional random walk. We then proceed with two applications of random walks in the field of polymer physics. In the first model, a polymer chain is described as a random walk with persistence. We give a complete analytic treatment of the model, including expressions for quantities of interest such as the moments of the end-to-end distance of the polymer. Next, we investigate the orientational relaxation in the so-called reptation model, introduced to describe the dynamics of entangled polymers. Compact, analytic results are given for the variables that are observed in experiments, notably the stationary anisotropy. Random walks have successfully been used to describe one of the basic transport processes in nature: Fickian diffusion. The combination of diffusion with convection, for example when Brownian particles are suspended in a fluid in non-uniform flow, leads to the interesting phenomenon known as Taylor diffusion. We study here another variant of the Taylor dispersion problem: the dispersion of particles in spatially periodic flows. Our main contribution is to show that the effective dispersion coefficient can be expressed in terms of the Green's function of the random walk inside the unit cell. Finally, we discuss a new, interesting phenomenon of stochastic resonance for particles suspended in an oscillating flow. We find that the interplay between the frequencies that characterize the stochastic process, e.g. a random walk, and the frequency of the oscillating flow, can lead to a resonance phenomenon, characterized by a strong increase of the effective diffusion coefficient.