Transport properties of 1D disordered models: a novel approach

A new method is developed for the study of transport properties of 1D models with random potentials. It is based on an exact transformation that reduces discrete Schr\"odinger equation in the tight-binding model to a two-dimensional Hamiltonian map. This map describes the behavior of a classical linear oscillator under random parametric delta-kicks. We are interested in the statistical properties of the transmission coefficient $T_L$ of a disordered sample of length $L$. In the ballistic regime we derive expressions for the mean value of the transmission coefficient $T_L$, its second moment and variance, that are more accurate than the existing ones. In the localized regime we analyze the global characteristics of $\ln T_L$, and demonstrate that its distribution function approaches the Gaussian form if $L\to \infty$. For any finite $L$ there are deviations from the Gaussian law that originate from the subtle correlation effects between different trajectories of the Hamiltonian map.