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 $\sigma \sim 4-5k_{B}T$. 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 $0.05–0.075\mathrm{eV}$.

Figure 1

Ensemble-averaged diffusion for different values of disorder strength $\sigma$ in units of $k_{B}T$ for exponentially decaying correlations. In doing numerics, we fixed $\sigma ={\sigma }_0$ and varied temperature. Distance is measured in units of correlation length $\lambda$ and time in units of ${\tau }_0={\lambda }^2\eta /{\sigma }_0$. For ${\sigma }_0=2k_B{T}_{\text{room}}=0.05\mathrm{eV}$, $D_0=k_B{T}_{\text{room}}/\eta =3\mu {\mathrm{m}}^2/\mathrm{s}$, and $\lambda =5.25\mathrm{nm}$ (15 bp), ${\tau }_0\approx 4.6\mu \mathrm{s}$. Initially, diffusion is normal, $⟨\delta x^2(t)⟩=2D_{0}t$. The dashed lines present asymptotically normal behavior $⟨\delta x^2(t)⟩=2D_{\text{ren}}t$, for $\sigma /(k_{B}T)=1/2,1,3/2,2$. Transient subdiffusion is much faster than this limit. Averaging over ${10}^4$ particles is done in 10 different realizations of random potential replicated with period $L={10}^4$ (${10}^3$ particles per a potential realization). Particles are initially uniformly distributed over the length $L$.

Figure 2

(a) Single-trajectory averages are scattered between the free and disorder-renormalized diffusion limits (depicted with dash-dotted lines). Time window $T_w=2\times {10}^4$ for averaging is chosen 100 times larger than the maximal time $t$. Each trajectory is characterized by individual subdiffusion index $\alpha$ distributed as shown in the lower inset. Arrow indicates the value of $\alpha =0.67$ which corresponds to the ensemble average depicted with dashed line. The ensemble average of time averages with $\alpha =0.78$ is depicted as full green line. It is also depicted in (b) as function of time $t$ for $T_w={10}^4$, and also in (c) as a function of $T_w$ for two fixed values of $t$. The latter decays as $⟨\overline{\delta x^2(t)}⟩\sim {T_w}^{-0.07}$. Moreover, the corresponding ergodicity breaking parameter in the upper inset of (a) gradually decays as a function of $T_w$. This indicates that no ergodicity breaking takes place asymptotically, $T_w\to \infty$.

Figure 3

(a)–(c) Probability density function (PDF) of waiting times derived from numerics (symbols), its fit with a log-normal distribution (full lines), and the result expected from normal diffusion with $D_{\text{ren}}$ (dashed lines) for three different values of $\sigma /k_{B}T$. (d) Mean first passage time (MPT) and its standard deviation (STD) derived from numerics (symbols) and their exponential fits (full lines), as well as MPT and STD expected from normal diffusion characterized by $D_{\text{ren}}$, which obey a $\text{const}\times \exp [-{\sigma }^2/(k_{B}T{)}^2]$ dependence.