Thermodynamic cost of precise timekeeping in an electronic underdamped clock

Clocks are inherently out-of-equilibrium because, due to friction, they constantly consume free energy to keep track of time. The thermodynamic uncertainty relation (TUR) quantifies the trade-off between the precision of any time-antisymmetric observable and entropy production. In the context of clocks, the TUR implies that a minimum entropy production is needed in order to achieve a certain level of precision in timekeeping. But the TUR has only been proven for overdamped systems. Recently, a toy model of a classical underdamped pendulum clock was proposed that violated this relation Phys. Rev. Lett. 128, 130606 (2022), thus demonstrating that the TUR does not hold for underdamped dynamics. We propose an electronic implementation of such a clock, using a resistor-inductor-capacitor circuit and a biased CMOS inverter (not gate), which can work at different scales. We find that in the nanoscopic single-electron regime of the circuit, we essentially recover the toy model violating the TUR bound. However, in different macroscopic regimes of the circuit, we show that the TUR bound is restored and analyze the thermodynamic efficiency of timekeeping.