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.

我们讨论了在阻尼系数小的极限中，在一维中，超阻尼运动在淬灭的随机势和扩散中的综合作用。我们的分析考虑了由于随机势的不同实现而导致的到达位置x的平均首次通过时间T（x）的统计量。具体来说，我们将中位数\（{\ bar {T}}（x）\）与期望值\（\ langle T（x）\ rangle \）进行对比，后者是典型运动过程的有益描述。，这是由罕见的事件所主导，这些事件的传播壁垒非常高。我们显示，在相对较短的时间内，中位数\（{\ bar {T}}（x）\）用“溢流”模型来解释，其中T（x）主要由到达位置x之前遇到的最高障碍确定。这些最高壁垒是使用极值统计方法量化的。