Statistical Properties of Waves in a Random Medium

Abstract

Averages of a number of properties of waves or quantum particles in random media are determined. Problems are formulated in terms of functional integrals. Techniques of the theory of Markov processes are employed to express the averaged properties in terms of the solutions of Fokker-Planck-Kolmogorov equations. Explicitly, the average density of states and Green's function of a particle in a one-dimensional (white noise) random medium are re-examined. A three-dimensional model system with a spherically symmetric, random potential $V\left(\right|\mathrm{}\overrightarrow{\mathrm{r}}\mathrm{}\left|\right)\mathrm{}$ is also considered. This model is relevant to several physical problems. Finally, statistics of the reflectivity of a slab with random, complex dielectric constant $\epsilon \mathrm{}\left(x\right)\mathrm{}$, are determined. A discussion is included of mathematical aspects of the type of functional integral involved in the wave-random-medium problem.