Generalized Lyapunov Exponent and Transmission Statistics in One-dimensional Gaussian Correlated Potentials

TL;DR

This paper investigates how correlations in Gaussian disorder affect transmission statistics in one-dimensional systems, revealing that disorder correlations significantly enhance transmission fluctuations and deviation from log-normal distribution.

Contribution

It introduces analytical and numerical methods to study the impact of disorder correlations on the generalized Lyapunov exponent and transmission cumulants in 1D systems.

Findings

Disorder correlations significantly alter transmission statistics.

Transmission fluctuations are greatly enhanced beyond weak disorder.

Relations between cumulants and generalized Lyapunov exponent are established.

Abstract

Distribution of the transmission coefficient T of a long system with a correlated Gaussian disorder is studied analytically and numerically in terms of the generalized Lyapunov exponent (LE) and the cumulants of lnT. The effect of the disorder correlations on these quantities is considered in weak, moderate and strong disorder for different models of correlation. Scaling relations between the cumulants of lnT are obtained. The cumulants are treated analytically within the semiclassical approximation in strong disorder, and numerically for an arbitrary strength of the disorder. A small correlation scale approximation is developed for calculation of the generalized LE in a general correlated disorder. An essential effect of the disorder correlations on the transmission statistics is found. In particular, obtained relations between the cumulants and between them and the generalized LE show that, beyond weak disorder, transmission fluctuations and deviation of their distribution from the log-normal form (in a long but finite system) are greatly enhanced due to the disorder correlations. Parametric dependence of these effects upon the correlation scale is presented.