Thermodynamic constraints on the power spectral density in and out of equilibrium

The power spectral density of an observable quantifies the amount of fluctuations at a given frequency and can reveal the influence of different timescales on the observable's dynamics. Here, we show that the spectral density in a continuous-time Markov process can be both lower and upper bounded by an expression involving two constants that depend on the observable and the properties of the system. In equilibrium, we identify these constants with the low- and high-frequency limit of the spectral density, respectively; thus, the spectrum at arbitrary frequency is bounded by the short- and long-time behavior of the observable. Out of equilibrium, on the other hand, the constants can no longer be identified with the limiting behavior of the spectrum, allowing for peaks that correspond to oscillations in the dynamics. We show that the height of these peaks is related to dissipation, allowing to infer the degree to which the system is out of equilibrium from the measured spectrum.