Discrete time crystals are periodically driven systems that display spontaneous symmetry breaking of time translation invariance in the form of indefinite subharmonic oscillations. We introduce a thermodynamically consistent model for a discrete time crystal and analyze it using the framework of stochastic thermodynamics. In particular, we evaluate the rate of energy dissipation of this many-body system of interacting noisy subharmonic oscillators in contact with a heat bath. The mean-field model displays the phenomenon of subharmonic synchronization, which corresponds to collective subharmonic oscillations of the individual units. The 2D model does not display synchronization but it does show a time-crystalline phase, which is characterized by a power-law behavior of the number of coherent subharmonic oscillations with system size. This result demonstrates that the emergence of coherent oscillations is possible even in the absence of synchronization.

中文翻译：

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随机离散时间晶体：熵产生和次谐波同步

离散时间晶体是周期性驱动的系统，以不确定的次谐波振荡的形式自发对称地打破时间平移不变性。我们介绍了一个离散时间晶体的热力学一致性模型，并使用随机热力学框架对其进行了分析。特别是，我们评估了与热浴接触的相互作用多噪声次谐波振荡器的多体系统的能量耗散率。平均场模型显示了次谐波同步现象，该现象对应于各个单元的集体次谐波振荡。2D模型没有显示同步，但确实显示了时晶相，其特征是相干次谐波振荡的数量随系统大小的幂律行为。