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Clocks are inherently out-of-equilibrium because, due to friction, theyconstantly consume free energy to keep track of time. The ThermodynamicUncertainty Relation (TUR) quantifies the trade-off between the precision ofany time-antisymmetric observable and entropy production. In the context ofclocks, the TUR implies that a minimum entropy production is needed in order toachieve a certain level of precision in timekeeping. But the TUR has only beenproven for overdamped systems. Recently, a toy model of a classical underdampedpendulum clock was proposed that violated these relations (Phys. Rev. Lett.128, 130606), thus demonstrating that the TUR do not hold for underdampeddynamics. We propose an electronic implementation of such a clock, using aresistor-inductor-capacitor (RLC) circuit and a biased CMOS inverter (NOTgate), which can work at different scales. We find that in the nanoscopicsingle-electron regime of the circuit, we essentially recover the toy modelviolating the TUR bound. However, in different macroscopic regimes of thecircuit, we show that the TUR bound is restored and analyze the thermodynamicefficiency of timekeeping.

Thermodynamic cost of precise timekeeping in an electronic underdamped clock