Statistical scattering of waves in disordered waveguides: From microscopic potentials to limiting macroscopic statistics.
We study the statistical properties of wave scattering in a disordered waveguide. The statistical properties of a "building block" of length deltaL are derived from a potential model and used to find the evolution with length of the expectation value of physical quantities. In the potential model the scattering units consist of thin potential slices, idealized as delta slices, perpendicular to the longitudinal direction of the waveguide; the variation of the potential in the transverse direction may be arbitrary. The sets of parameters defining a given slice are taken to be statistically independent from those of any other slice and identically distributed. In the dense-weak-scattering limit, in which the potential slices are very weak and their linear density is very large, so that the resulting mean free paths are fixed, the corresponding statistical properties of the full waveguide depend only on the mean free paths and on no other property of the slice distribution. The universality that arises demonstrates the existence of a generalized central-limit theorem. Our final result is a diffusion equation in the space of transfer matrices of our system, which describes the evolution with the length L of the disordered waveguide of the transport properties of interest. In contrast to earlier publications, in the present analysis the energy of the incident particle is fully taken into account. For one propagating mode, N=1 , we have been able to solve the diffusion equation for a number of particular observables, and the solution is in excellent agreement with the results of microscopic calculations. In general, we have not succeeded in finding a solution of the diffusion equation. We have thus developed a numerical simulation, to be called "random walk in the transfer matrix space," in which the universal statistical properties of a "building block" are first implemented numerically, and then the various building blocks are combined to find the statistical properties of the full waveguide. The reported results thus obtained (in which use was made of a "short-wavelength approximation") are in very good agreement with those arising from truly microscopic calculations, for both bulk and surface disorder. Since the paper has a clear pedagogical aim, we have included, for the benefit of experts and non-experts, a number of appendixes that contain the more involved calculations.