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

TL;DR

This paper derives a thermodynamic uncertainty relation for underdamped systems with time-dependent protocols, bridging the gap with overdamped systems and confirming the relation through examples.

Contribution

The paper introduces a TUR for underdamped dynamics under general protocols, extending the known overdamped TUR and using the Cramér-Rao inequality.

Findings

Derived TUR for underdamped systems with time-dependent protocols.

Shows the TUR approaches the overdamped limit at large viscosity.

Confirmed the theoretical results with three example systems.

Abstract

The thermodynamic uncertainty relation (TUR) for underdamped dynamics has intriguing problems while its counterpart for overdamped dynamics has recently been derived. 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 Cram\'{e}r-Rao inequality. We show the resultant TUR to give rise to the inequality of the product of the variance and entropy production. We prove it to approach to the known overdamped result for large viscosity limit. We present three examples to confirm our rigorous result.