Classical pendulum clocks break the thermodynamic uncertainty relation

TL;DR

This paper demonstrates that classical pendulum clocks can violate the thermodynamic uncertainty relation by constructing a counterexample involving underdamped oscillators, challenging previous assumptions of its universality.

Contribution

The authors disprove the conjecture that the thermodynamic uncertainty relation holds for underdamped Brownian motion by providing explicit counterexamples inspired by pendulum clocks.

Findings

Underdamped harmonic oscillators can break the thermodynamic uncertainty relation.

A classical pendulum clock mechanism serves as a counterexample.

The violation occurs both in discrete models and continuous simulations.

Abstract

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.