Diffusion in a rough potential: Dual-scale structure and regime crossovers

Diffusion in a `rough' potential parameterized by a reaction coordinate $q$ is relevant to a wide spectrum of problems ranging from protein folding and charge transport in complex media to colloidal stabilization and self-assembly. This work studies the case of a potential having coarse-scale structure with characteristic energy barrier $\Delta U$ and period $\ell$, and fine-scale `roughness' of magnitude $\Delta U'\lesssim \Delta U$ and small period $\ell'\ll \ell$. Numerical solution of the Smoluchowski equation and analytical predictions from Kramers theory document distinct regimes at different distances $|\Delta q|=|q-q_E|$ from stable equilibrium at $q=q_E$. The physical diffusivity $D$ prescribed by dissipative effects can be observed farther than a distance $|\Delta q'| \propto (\Delta U'/\ell' + \Delta U/\ell)$. Rescaling the physical diffusivity to account for the fine-scale `roughness' is strictly valid when $|\Delta q| < \Delta q_I \propto (\Delta U'/\ell' - \Delta U/\ell)$. Farther than a critical distance $\Delta q_{II}\propto \Delta U/\ell$ the diffusion process is free of coarse-scale metastable states, which facilitates determining the effective diffusivity $D'$ from the reaction coordinate trajectory. Closer to equilibrium the coarse-scale structure induces two diffusive regimes: nearly logarithmic evolution for $\Delta q_{II} > |\Delta q| > \Delta q_{III}$ and exponential decay over time for $|\Delta q| < \Delta q_{III}\propto 1/\ell$. The effective diffusivity derived in this work is sensitive to the coarse- and fine-scale energy barriers and periods, and for $\ell'/\ell \to 0$ and $\Delta U'/k_B T \gg 1$ agrees closely with mean first-passage time estimates currently employed, which depend solely on the fine-scale energy barrier.