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