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 a coarse-scale structure with characteristic energy barrier ΔU and period ℓ and fine-scale "roughness" of magnitude ΔU′ ≤ ΔU and small period ℓ′ << ℓ. The 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′| (ΔU′/ℓ′ + ΔU/ℓ). Rescaling the physical diffusivity to account for the fine-scale "roughness" is strictly valid when |Δq| < Δq_I (ΔU′/ℓ′-ΔU/ℓ). Farther than a critical distance Δq_II ΔU/ℓ, 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 1/ℓ. The effective diffusivity derived in this work is sensitive to the coarse-and fine-scale energy barriers and periods and for ℓ′/ℓ → 0 and ΔU′/k_BT ≫ 1 agrees closely with mean first-passage time estimates currently employed, which depend solely on the fine-scale energy barrier.