Diffusion in a Random Potential: Hopping as a Dynamical Consequence of Localization

A model of diffusion $\psi (\mathrm{x},t)=\Delta \psi (\mathrm{x},t)+V\left(\mathrm{x}\right)\psi (\mathrm{x},t)$ is studied, where $V\left(\mathrm{x}\right)$ is the Gaussian random potential. It is found that the probability distribution function is concentrated (localized) at some metastable potential attractor, while the localization center hops discontinuously in search of a better metastable attractor. The sample-averaged localization-center displacement is found as ${\chi }_c\sim \frac{t}{2\ln t}$, i.e., it is sub-ballistic.