Variance Resonance in Weakly Coupled Harmonic Oscillators Driven by Thermal Gradients

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.