All clocks, in some form or another, use the evolution of nature toward higher entropy states to quantify the passage of time. Because of 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 toward 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 nonthermal 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.

中文翻译：

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测量计时的热力学成本

所有时钟都以某种形式或其他形式使用自然界向更高熵态的演化来量化时间的流逝。由于第二定律的统计性质和相应的熵流，波动从根本上限制了任何时钟的性能。这表明熵的增加与时钟滴答的质量之间存在深层的关系。的确，量子领域中自主时钟的最小模型揭示了可以得出线性关系，其中在有限的范围内，熵的每一位都会线性地增加量子时钟的准确性。但是，当我们朝着更经典的系统发展时，这样的线性关系会持续吗？我们通过在纳米级时钟中对此热力学关系进行首次实验研究来肯定地回答这一问题。我们随机地驱动一个纳米厚的膜，并用一个射频腔读出它的位移，从而使我们能够识别出一个时钟的滴答声。我们从理论上证明，此经典时钟的最大可能精度与每个滴答声所产生的熵成正比，类似于弱耦合量子时钟的已知极限，但比例常数不同。我们测量准确性和熵。一旦考虑了非热噪声，我们就会发现精度和熵之间存在线性关系，并且时钟在理论界限的数量级内工作。我们从理论上证明，此经典时钟的最大可能精度与每个滴答声所产生的熵成正比，类似于弱耦合量子时钟的已知极限，但比例常数不同。我们测量准确性和熵。一旦考虑了非热噪声，我们就会发现精度和熵之间存在线性关系，并且时钟在理论界限的数量级内工作。我们从理论上证明，此经典时钟的最大可能精度与每个滴答声所产生的熵成正比，类似于弱耦合量子时钟的已知极限，但比例常数不同。我们测量准确性和熵。一旦考虑了非热噪声，我们就会发现精度和熵之间存在线性关系，并且时钟在理论界限的数量级内工作。