Abstract
We discuss the combined effects of overdamped motion in a quenched random potential and diffusion, in one dimension, in the limit where the diffusion coefficient is small. Our analysis considers the statistics of the mean first-passage time T(x) to reach position x, arising from different realisations of the random potential. Specifically, we contrast the median {bar{T}}(x), which is an informative description of the typical course of the motion, with the expectation value langle T(x)rangle , which is dominated by rare events where there is an exceptionally high barrier to diffusion. We show that at relatively short times the median {bar{T}}(x) is explained by a ‘flooding’ model, where T(x) is predominantly determined by the highest barriers which are encountered before reaching position x. These highest barriers are quantified using methods of extreme value statistics.
Highlights
There are many situations where particles move under the combined influence of thermal diffusion and a static random potential [1]
V (x) is a random potential, D is the diffusion coefficient, and η(t) is a white noise signal with statistics defined by η(t) = 0, η(t)η(t ) = δ(t − t )
We note that the central limit theorem is applicable to the quantity T (x) obtained by equation (9) in the limit as x → ∞, so that x2/2T (x) does approach a limit, which we identify as the effective diffusion coefficient
Summary
There are many situations where particles move under the combined influence of thermal diffusion and a static (or quenched) random potential [1]. Page 3 of 18 54 has been extended to consider a periodised random potential, so that the long-time dynamics is diffusive: in this case it is found [12] that the diffusion coefficient has a very broad probability distribution, and these studies have inspired other works on dispersion in the presence of traps, of which [13] is a recent example Another related problem is the motion of a quantum particle in a statistically homogeneous random potential which exhibits even slower spatial dispersion: it may be completely localised [14], and in one or two dimensions localisation is almost certain [15,16]: see [17] for a review. After a sufficiently long time, the dynamics becomes diffusive, with a diffusion coefficient given by (5)
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