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Normal diffusion in corrugated potentials with spatially uncorrelatedGaussian energy disorder famously explains the origin of non-Arrhenius$\exp[-\sigma^2/(k_BT^2)]$ temperature-dependence in disordered systems. Herewe show that unbiased diffusion remains asymptotically normal also in thepresence of spatial correlations decaying to zero. However, due to a temporallack of self-averaging transient subdiffusion emerges on mesoscale, and it canreadily reach macroscale even for moderately strong disorder fluctuations of$\sigma\sim 4-5\, k_BT$. Due to its nonergodic origin such subdiffusionexhibits a large scatter in single trajectory averages. However, at odds withintuition, it occurs essentially faster than one expects from the normaldiffusion in the absence of correlations. We apply these results to diffusionof regulatory proteins on DNA molecules and predict that such diffusion shouldbe anomalous, but much faster than earlier expected on a typical length ofgenes for a realistic energy disorder of several room $k_BT$, or merely$0.05-0.075$ eV.

Anomalous features of diffusion in corrugated potentials with spatial correlations: faster than normal, and other surprises