  We presented a methodology to approximate the entropy production for Brownian
motion in a tilted periodic potential. The approximation stems from the well
known thermodynamic uncertainty relation. By applying a virial-like expansion,
we provided a tighter lower limit solely in terms of the drift velocity and
diffusion. The approach presented is systematically analysed in the
tight-binding regime. We also provide a relatively simple rule to validate
using the tight-binding approach based on drift and diffusion relations rather
than energy barriers and forces. We also discuss the implications of our
results outside the tight-binding regime.
