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

Abstract

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 $\Delta U^{\prime }\lesssim \Delta U$ and small period ${\ell }^{\prime }\ll \ell$. 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 $|\Delta q^{\prime }|\propto (\Delta U^{\prime }/{\ell }^{\prime }+\Delta U/\ell )$. Rescaling the physical diffusivity to account for the fine-scale `roughness' is strictly valid when $|\Delta q|<\Delta q_I\propto (\Delta U^{\prime }/{\ell }^{\prime }-\Delta U/\ell )$. 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 $D^{\prime }$ 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 ${\ell }^{\prime }/\ell \to 0$ and $\Delta U^{\prime }/k_{B}T\gg 1$ agrees closely with mean first-passage time estimates currently employed, which depend solely on the fine-scale energy barrier.

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