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

TL;DR

This paper reveals that in corrugated potentials with spatial correlations, diffusion can be anomalous yet faster than normal, with transient subdiffusion and large trajectory variability, challenging previous assumptions about diffusion slowing down in disordered systems.

Contribution

It demonstrates that spatial correlations do not prevent asymptotic normal diffusion but induce transient subdiffusion and faster-than-expected diffusion in correlated disordered potentials.

Findings

Transient subdiffusion occurs on mesoscale due to lack of self-averaging.

Diffusion can be faster than normal despite correlations.

Predicted rapid diffusion of proteins on DNA with realistic disorder levels.

Abstract

Normal diffusion in corrugated potentials with spatially uncorrelated Gaussian energy disorder famously explains the origin of non-Arrhenius $\exp[-\sigma^2/(k_BT^2)]$ 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, due to a temporal lack of self-averaging transient subdiffusion emerges on mesoscale, and it can readily reach macroscale even for moderately strong disorder fluctuations of $\sigma\sim 4-5\, k_BT$. Due to 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 $k_BT$, or merely $0.05-0.075$ eV.