  Through multiscale analysis of the adjoint Fokker-Planck equation, strict
bounds are derived for the center of mass diffusivity of an overdamped harmonic
chain in a periodic potential, often known as the discrete Frenkel-Kontorova
model. Significantly, it is shown that the free energy barrier is a lower bound
to the true finite temperature migration barrier for this general and popular
system. Numerical simulation confirms the analysis, whilst effective migration
potentials implied by the bounds are employed to give a surprisingly accurate
prediction of the non-linear response.
