The thermodynamic uncertainty relation expresses a seemingly universal trade-off between the cost for driving an autonomous system and precision in any output observable. It has so far been proven for discrete systems and for overdamped Brownian motion. Its validity for the more general class of underdamped Brownian motion, where inertia is relevant, was conjectured based on numerical evidence. We now disprove this conjecture by constructing a counterexample. Its design is inspired by a classical pendulum clock, which uses an escapement to couple the motion of an oscillator to another degree of freedom (a “hand”) driven by an external force. Considering a thermodynamically consistent, discrete model for an escapement mechanism, we first show that the oscillations of an underdamped harmonic oscillator in thermal equilibrium are sufficient to break the thermodynamic uncertainty relation. We then show that this is also the case in simulations of a fully continuous underdamped system with a potential landscape that mimics an escaped pendulum.

• Received 7 October 2021
• Revised 22 December 2021
• Accepted 1 March 2022
• Corrected 19 April 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.130606

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society