Brownian motion in disordered media

The Brownian motion in quenched disordered media is studied from a stochastic point of view using random walk models and to a lesser extent a Langevin approach. Focus is placed on highly ramified media that exhibits self-similarity over a wide range of length scales. Accurate estimates are obtained for the dynamical exponents for the blind and myopic ant random walks on percolation clusters in two and three dimensions at criticality and DLA (diffusion limited aggregation) clusters in two dimensions. The method employed exactly enumerates all Brownian paths from all starting points in a cluster. The scaling properties found in the probability density for the position of the random walker is exploited to develop an approximation scheme for the mean squared displacement. Moreover, a simple connection between the anisotropic properties of the diffusion process to the geometric structure of the clusters is established. Regular oscillations are found to persist in the step correlation function for the myopic ant on bipartite clusters. In the regime of self-similarity, these oscillations decay as a power-law with an exponent given by half the spectral dimension. A formal mapping from a discrete time random walk to a continuous time master equation applicable to quenched disordered media is developed. Moreover, the frequency dependent diffusion coefficient is expressed in terms of the step correlation function. A kinematic analysis reveals that a generalized Langevin equation with a power-law friction kernel adequately describes the anomalous diffusion. A random walk model with persistence is introduced for the analysis of the electrical conductivity of disordered composite materials. Features of the model include; a phenomenological parameterization with a high frequency cut off determined by the coarse grain length scale, a Drude-like mobility in homogeneous media, and the relative carrier concentrations and mobilities of the constituents are treated independently. The interplay between the mean free paths of each constituent, the disorder length scale, and the correlation length of the disorder is investigated. Results are presented which indicate that scaling theories utilizing only conductivities are generally inadequate