  The spreading of a particle along a chain, and its relaxation, are central
themes in statistical and quantum mechanics. One wonders what are the
consequences of the interplay between coherent and stochastic transitions. This
fundamental puzzle has not been addressed in the literature, though closely
related themes were in the focus of the Physics literature throughout the last
century, highlighting quantum versions of Brownian motion. Most recently this
question has surfaced again in the context of photo-synthesis. Here we consider
both an infinite tight-binding chain and a finite ring within the framework of
an Ohmic master equation. With added disorder it becomes the quantum version of
the Sinai-Derrida-Hatano-Nelson model, which features sliding and
delocalization transitions. We highlight non-monotonic dependence of the
current on the bias, and a counter-intuitive enhancement of the effective
disorder due to coherent hopping.
