Anomalous Features of Diffusion in Corrugated Potentials with Spatial Correlations: Faster than Normal, and Other Surprises

Normal diffusion in corrugated potentials with spatially uncorrelated Gaussian energy disorder famously explains the origin of non-Arrhenius exp[-sigma(2)/(k(B)T(2))] temperature dependence in disordered systems. Here we show that unbiased diffusion remains asymptotically normal also in the presence of spatial correlations decaying to zero. However, because of a temporal lack of self-averaging, transient subdiffusion emerges on the mesoscale, and it can readily reach macroscale even for moderately strong disorder fluctuations of sigma similar to 4 - 5k(B)T. Because of its nonergodic origin, such subdiffusion exhibits a large scatter in single-trajectory averages. However, at odds with intuition, it occurs essentially faster than one expects from the normal diffusion in the absence of correlations. We apply these results to diffusion of regulatory proteins on DNA molecules and predict that such diffusion should be anomalous, but much faster than earlier expected on a typical length of genes for a realistic energy disorder of several room k(B)T, or merely 0.05-0.075 eV.