Classical pendulum clocks break the thermodynamic uncertainty relation

Abstract

A mechanical clock, such as a pendulum clock, uses an escapement to couple the motion of an oscillator to regulate the motion of another degree of freedom (a "hand") driven by an external force. Clocks built after this principle can yield high precision at low energetic cost. They provide a counterexample to the thermodynamic uncertainty relation, which is thus shown not to hold for systems undergoing driven Brownian diffusion with inertia. 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.

Bounds on the precision of currents in underdamped Langevin dynamics

“…This agrees with the arguments made in Ref. [24], where the coupling between different degrees of freedom-here, the motion along different spatial directions-was identified as the main ingredient for constructing counter-examples to the TUR. The bound Eq.…”

“…As discussed in Ref. [24], a particle, whose motion in the longitudinal direction is too fast compared to the transverse oscillations will find itself climbing the potential and thus be slowed down. Similarly, if the particle is too slow, it will be accelerated by sliding down the potential.…”

“…( 14). However, more recently, it has been shown [24] that it is indeed possible to violate the TUR by exploiting the coherent oscillating motion of the particle. We note that bounds on the precision of a current of the type…”

“…( 14) are known in the one-dimensional case [20], it was recently shown in Ref. [24] that in two-and higherdimensional systems the TUR can be violated in a suitably designed potential landscape. The mechanism for violating the TUR rests on exploiting the coherent oscillations of an underdamped particle to reduce the fluctuations of the measured current.…”

“…[20]. Very recently, however, Pietzonka [24] showed that, by constructing a potential landscape that exploits the coherent oscillations of a finite-mass particle, the TUR can indeed be violated in underdamped systems.…”