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

Diffusion in a "rough" potential parameterized by a reaction coordinate q is analyzed for systems that exhibit a coarse-scale structure with barrier height ΔU and period ℓ, superimposed with fine-scale "roughness" of barrier height ΔU′ and small period ℓ′ ≪ ℓ. Numerical solution of the Smoluchowski equation and analytical predictions identify distinct dynamical regimes as a function of the distance |Δq| = |q − q_E| from the stable equilibrium at q = q_E. The physical diffusivity D, set by dissipative effects, can be rescaled to an effective diffusivity D′ that accounts for the fine-scale roughness. This rescaling is strictly valid when |Δq| < Δq_I ∝ (ΔU′/ℓ′ − ΔU/ℓ). Farther than a critical distance Δq_II ∝ ΔU/ℓ, the coarse-scale barriers control activated transport and the dynamics follow Kramers-like escape with an effective diffusivity D′ inferred from the reaction-coordinate trajectory. Closer to equilibrium, the coarse-scale structure induces two diffusive regimes separated by a crossover at Δq_III ∝ 1/ℓ: an initial regime with approximately constant D′ and a near-equilibrium regime where D′ decreases over time for |Δq| < Δq_III. The effective diffusivity obtained here is sensitive to the estimation method used from time series, which depends solely on the fine-scale energy barrier. These results have implications for activated transport in complex media, including protein folding landscapes, colloidal stabilization, and self-assembly.