Thermodynamic uncertainty relation for underdamped dynamics driven by time-dependent protocols

Abstract

The thermodynamic uncertainty relation (TUR) for overdamped dynamics has recently been derived for general time-dependent protocols. However, its precedent TUR for underdamped dynamics has not been known yet. Even for the case of steady states, a proper way to match underdamped and overdamped TURs has not been found. We derive the TUR for underdamped systems subject to general time-dependent protocols, that covers steady states, by using the celebrating Cramér-Rao inequality. We show the resultant TUR to give rise to the inequality for the product of the variance and entropy production. We prove it to approach to the known overdamped result for large viscosity limit. It is the eventual step to pursue the TUR for Langevin systems. We present three examples to confirm our rigorous result.