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 $ΔU$ and period $\ell$, and fine-scale `roughness' of magnitude $ΔU'\lesssim ΔU$ and small period $\ell'\ll \ell$. Numerical solution of the Smoluchowski equation and analytical predictions from Kramers theory document distinct regimes at different distances $|Δq|=|q-q_E|$ from stable equilibrium at $q=q_E$. The physical diffusivity $D$ prescribed by dissipative effects can be observed farther than a distance $|Δq'| \propto (ΔU'/\ell' + ΔU/\ell)$. Rescaling the physical diffusivity to account for the fine-scale `roughness' is strictly valid when $|Δq| < Δq_I \propto (ΔU'/\ell' - ΔU/\ell)$. Farther than a critical distance $Δq_{II}\propto ΔU/\ell$ 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 $Δq_{II} > |Δq| > Δq_{III}$ and exponential decay over time for $|Δq| < Δq_{III}\propto 1/\ell$. 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 $ΔU'/k_B T \gg 1$ agrees closely with mean first-passage time estimates currently employed, which depend solely on the fine-scale energy barrier.