Characteristic oscillations in frequency-resolved heat dissipation of linear time-delayed Langevin systems: Approach from the violation of the fluctuation response relation

Time-delayed effects are widely present in nature, often accompanied by distinctive nonequilibrium features, such as net heat flow from a single thermal bath to the system. To elucidate detailed structures of the dissipation, we study the frequency decomposition of the heat dissipation in linear time-delayed Langevin systems. We decompose the heat dissipation into frequency spectrum using the Harada-Sasa equality, which relates the heat dissipation to the violation of the fluctuation response relation (FRR). We find a characteristic oscillatory behavior in the spectrum, and the oscillation asymptotically decays with an envelope inversely proportional to the frequency in the high-frequency region. Furthermore, the oscillation over the low-frequency region reflects the magnitude and sign of the heat dissipation. We confirm the generality of the results by extending our analysis to systems with multiple delay times. Since the violation of FRR is experimentally accessible, our results suggest an experimental direction for detecting and analyzing detailed characteristics of dissipation in time-delayed systems.