Transport and localization of waves in one-dimensional disordered media: Random phase approximation and beyond

We report a systematic and detailed numerical study of statistics of the reflection coefficient $(|R(L)|^2)$ and its associated phase ($\theta$) for a plane wave reflected from a one-dimensional (1D) disordered medium beyond the random phase approximation (RPA) for Gaussian white-noise disorder. We solve numerically the full Fokker-Planck (FP) equation for the probability distribution in the ($|R(L)|^2,\theta(L)$)-space for different lengths of the sample with different "disorder strengths". The statistical electronic transport properties of 1D disordered conductors are calculated using the Landauer four-probe resistance formula and the FP equation. This constitutes a complete solution for the reflection statistics and many aspects of electron transport in a 1D Gaussian white-noise potential. Our calculation shows the contribution of the phase distribution to the different averages and its effects on the one-parameter scaling theory of localization.