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

TL;DR

The paper investigates heat dissipation in linear time-delayed Langevin systems by decomposing the dissipation spectrum through the Harada-Sasa equality, which links FRR violation to energy loss. It analytically solves single-delay dynamics and extends to multi-delay cases, revealing a characteristic oscillatory spectrum in the FRR-violation across all frequencies with a period set by the delay $\tau$ and an envelope decaying as $1/\omega$, while the low-frequency portion dictates the sign and magnitude of dissipation. The results show that delayed forces induce non-Markovian features that can be probed experimentally via velocity fluctuations and linear response, even with limited data. This work provides a practical framework for detecting and characterizing time-delayed dissipation in diverse systems, including colloidal particles under delayed feedback and other non-Markovian or engineered-delay processes, and motivates extensions to nonlinear or embedded models to generalize the findings.

Abstract

Time-delayed effects are widely present in nature, often accompanied by distinctive nonequilibrium features, such as negative apparent heat dissipation. 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.

Figures (2)

• Figure 1: The heat dissipation rate $\langle J\rangle_0$ for different values of $a$, $b$, and $\tau$ (here, $a<0$ and $|b|<|a|$). We fix $T=1$ and $\gamma=1$.
• Figure 2: The upper row: frequency-resolved spectrum of heat dissipation rate of a linear Langevin system with (a) a single delay and (b) two delays. Blue solid lines denote the frequency-resolved spectrum $\tilde{C}_v(\omega) -2T\tilde{R}'_v(\omega)$, green dashed lines show the asymptotic forms: (a) $-2Tb\sin(\omega\tau)/(\gamma^2\omega)$ and (b) $-(2T/\gamma^2\omega)\cdot[A_1\sin(\omega\tau_1) +$$A_2\sin(\omega\tau_2)]$, and purple crosses denote results of numerical simulations (red error bars show the standard error of the mean). Parameters: (a) $a=-2$, $b=1$, $\tau=1$, $T=0.1$, $\gamma=1$; (b) $A_0 = -3$, $A_1=A_2=1$, $\tau_2 = 2\tau_1 = 2$, $\gamma=1$, $T=0.1$. The middle row: how the single delay time $\tau$ and delay strength $b$ affect the resolved spectrum of heat dissipation rate. We fix $a=-1$, $T=1$, and $\gamma=1$. (c) The spectrum with different delay times $\tau$. We fix $b=0.5$. (d) The spectrum with different strengths $b$ of the time-delayed force. Here $\tau=1$. The lower row: how the two delay times $\tau_1, \tau_2$ and corresponding delay strengths $A_1, A_2$ affect the resolved spectrum of heat dissipation rate. We fix $A_0=-1$, $T=1$, and $\gamma=1$. (e) The spectrum with different delay times $\tau_1, \tau_2$. Here $A_1=A_2=0.2$. (f) The spectrum with different delay strengths $A_1,A_2$ of the same sign. Here $\tau_1=1$ and $\tau_2=2$. (g) The spectrum with the same delay strengths $A_1,A_2$ of different signs. Here $\tau_1=1$ and $\tau_2=2$.