  All clocks, in some form or another, use the evolution of nature towards
higher entropy states to quantify the passage of time. Due to the statistical
nature of the second law and corresponding entropy flows, fluctuations
fundamentally limit the performance of any clock. This suggests a deep relation
between the increase in entropy and the quality of clock ticks. Indeed, minimal
models for autonomous clocks in the quantum realm revealed that a linear
relation can be derived, where for a limited regime every bit of entropy
linearly increases the accuracy of quantum clocks. But can such a linear
relation persist as we move towards a more classical system? We answer this in
the affirmative by presenting the first experimental investigation of this
thermodynamic relation in a nanoscale clock. We stochastically drive a
nanometer-thick membrane and read out its displacement with a radio-frequency
cavity, allowing us to identify the ticks of a clock. We show theoretically
that the maximum possible accuracy for this classical clock is proportional to
the entropy created per tick, similar to the known limit for a weakly coupled
quantum clock but with a different proportionality constant. We measure both
the accuracy and the entropy. Once non-thermal noise is accounted for, we find
that there is a linear relation between accuracy and entropy and that the clock
operates within an order of magnitude of the theoretical bound.
