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 $\mathrm{\Delta }{U}^{\mathrm{\prime }}\lesssim \mathrm{\Delta }U$ and small period ${\mathrm{\ell }}^{\mathrm{\prime }}\ll \mathrm{\ell }$. Numerical solution of the Smoluchowski equation and analytical predictions from Kramers theory document distinct regimes at different distances $\mathrm{\mid }\mathrm{\Delta }q\mathrm{\mid }=\mathrm{\mid }q-q_E\mathrm{\mid }$ from stable equilibrium at $q=q_E$. The physical diffusivity $D$ prescribed by dissipative effects can be observed farther than a distance $\mathrm{\mid }\mathrm{\Delta }{q}^{\mathrm{\prime }}\mathrm{\mid }\propto (\mathrm{\Delta }{U}^{\mathrm{\prime }}\mathrm{/}{\mathrm{\ell }}^{\mathrm{\prime }}+\mathrm{\Delta }U\mathrm{/}\mathrm{\ell })$. Rescaling the physical diffusivity to account for the fine-scale `roughness’ is strictly valid when $\mathrm{\mid }\mathrm{\Delta }q\mathrm{\mid }<\mathrm{\Delta }{q}_I\propto (\mathrm{\Delta }{U}^{\mathrm{\prime }}\mathrm{/}{\mathrm{\ell }}^{\mathrm{\prime }}-\mathrm{\Delta }U\mathrm{/}\mathrm{\ell })$. Farther than a critical distance $\mathrm{\Delta }{q}_{II}\propto \mathrm{\Delta }U\mathrm{/}\mathrm{\ell }$ the diffusion process is free of coarse-scale metastable states, which facilitates determining the effective diffusivity $D^{\mathrm{\prime }}$ from the reaction coordinate trajectory. Closer to equilibrium the coarse-scale structure induces two diffusive regimes: nearly logarithmic evolution for $\mathrm{\Delta }{q}_{II}>\mathrm{\mid }\mathrm{\Delta }q\mathrm{\mid }>\mathrm{\Delta }{q}_{III}$ and exponential decay over time for $\mathrm{\mid }\mathrm{\Delta }q\mathrm{\mid }<\mathrm{\Delta }{q}_{III}\propto 1\mathrm{/}\mathrm{\ell }$. The effective diffusivity derived in this work is sensitive to the coarse- and fine-scale energy barriers and periods, and for ${\mathrm{\ell }}^{\mathrm{\prime }}\mathrm{/}\mathrm{\ell }\to 0$ and $\mathrm{\Delta }{U}^{\mathrm{\prime }}\mathrm{/}{k}_{B}T\gg 1$ agrees closely with mean first-passage time estimates currently employed, which depend solely on the fine-scale energy barrier.