Enhanced diffusion and the eigenvalue band structure of Brownian motion in tilted periodic potentials

We consider enhanced diffusion for Brownian motion on a tilted periodic potential. Expressing the effective diffusion in terms of the eigenvalue band structure, we establish a connection between band gaps in the eigenspectrum and enhanced diffusion. We explain this connection for a simple cosine potential with a linear force and then generalize to more complicated potentials including one-dimensional potentials with multiple frequency components and nonseparable multidimensional potentials. We find that potentials with multiple band gaps in the eigenspectrum can lead to multiple maxima or broadening of the force-diffusion curve. These features are likely to be observable in experiments.