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 analyzed in the tight-binding regime. We also provide a relative 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.
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