Boffi Tarcisio, De Gregorio Paolo
Independent Researcher, 00179 Rome, Italy.
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
Entropy (Basel). 2024 Dec 12;26(12):1087. doi: 10.3390/e26121087.
We study two harmonic oscillators with high quality factors, driven by equilibrium and off equilibrium thermal noise, the latter mimicked by establishing a temperature gradient. The two oscillators are coupled via a third reciprocal harmonic interaction. We deepen the case of a weak coupling between the two oscillators, and show the emergence of a "spike" in the displacement variance of the colder oscillator, when the respective elastic constants approach each other. Away from the peak, the displacement variance of each oscillator only reflects the value of the local temperature. We name this phenomenon the variance resonance, or alternatively covariance resonance, in the sense that it comes about as one element of the covariance matrix describing both oscillators. In fact, all of the elements of the covariance matrix show some distinctive behavior. The oscillator at the lower temperature, therefore, oscillates as if driven by a higher temperature, resonating with the other one. By converse, the variance of the hotter oscillator develops a deep dent, or depression, around the same region. We could not reproduce this behavior if either the coupling constant is not small compared to those of the two oscillators, or if the quality factors are not large enough. In fact, in such instances the system tends to resemble one which is in equilibrium at the average temperature, regardless of the relative strengths of the elastic constants of the two oscillators. Our results could have various applications including for example precision measurement systems, when not all parts of the apparatuses are at the same temperature.
我们研究了两个具有高品质因数的谐振子，它们由平衡热噪声和非平衡热噪声驱动，后者通过建立温度梯度来模拟。这两个谐振子通过第三种互易谐波相互作用耦合。我们深入研究了两个谐振子之间弱耦合的情况，并表明当各自的弹性常数彼此接近时，较冷的谐振子的位移方差会出现一个“尖峰”。在远离峰值的情况下，每个谐振子的位移方差仅反映局部温度的值。我们将这种现象称为方差共振，或者也称为协方差共振，因为它是作为描述两个谐振子的协方差矩阵的一个元素出现的。实际上，协方差矩阵的所有元素都表现出一些独特的行为。因此，温度较低的谐振子振荡起来就好像是由较高温度驱动的，与另一个谐振子发生共振。相反，温度较高的谐振子的方差在同一区域周围会出现一个深深的凹陷。如果耦合常数与两个谐振子的耦合常数相比不小，或者如果品质因数不够大，我们就无法重现这种行为。实际上，在这种情况下，无论两个谐振子弹性常数的相对强度如何，系统往往类似于处于平均温度下的平衡系统。我们的结果可能有各种应用，例如在精密测量系统中，当仪器的并非所有部分都处于相同温度时。
