Heat dissipation in marginally stable linear time-delayed Langevin systems

TL;DR

This work analyzes heat dissipation in two marginally stable classes of linear time-delayed Langevin systems: diffusive criticality and oscillatory criticality. By deriving the fundamental solution and employing Stratonovich heat definitions, the authors show that the long-time average heat dissipation rate is constant for diffusive criticality but diverges linearly in time for oscillatory criticality, even though both exhibit linear variance growth. They further characterize the approach to these criticalities from the stable regime, revealing distinct asymptotics and spectral features, including a stationary Harada–Sasa spectrum for the diffusive case and a resonant peak that diverges as the oscillatory boundary is approached. Numerical simulations support the analytic results, and the distributions of heat dissipation exhibit qualitatively different aging behavior between the two classes. The findings provide a concrete thermodynamic framework for non-stationary time-delayed systems and motivate extensions to nonlinear and many-body delayed dynamics.

Abstract

The thermodynamic properties of time-delayed dynamics remain largely unexplored, especially for systems that exhibit asymptotically non-stationary behavior. Here, we investigate heat dissipation in two classes of marginally stable linear time-delayed Langevin dynamics: (i) diffusive criticality, which asymptotically manifests as scaled Brownian diffusion, and (ii) oscillatory criticality, which shows oscillation with diffusive amplitude. By analytical derivations, we find fundamentally different thermodynamic signatures: the average heat dissipation rate asymptotically approaches a constant for diffusive criticality but diverges linearly with oscillations for oscillatory criticality, despite both showing linearly growing variance over time. We discuss in detail how the heat dissipation rate behaves differently as the dynamics asymptotically approaches these two criticality classes from the stable regime. We also numerically study the probability distributions of heat dissipation rates for both types of critical dynamics. Our results demonstrate that non-stationary time-delayed dynamics with similar scaling of variance can yield qualitatively distinct heat dissipation behaviors, depending on the underlying dynamical details. This work provides a concrete foundation for future investigations into thermodynamic properties of general nonlinear time-delayed systems.

Figures (8)

• Figure 1: (a) A single trajectory simulation of the dynamics with diffusive criticality. Here $a=-5$, $b=5$, $\tau =1$, $T = 1$, $\textcolor{black}{\gamma=1}$, and $\phi(t)=0 \ \ (t\leq0)$. (b) 100 trajectories of the dynamics with diffusive criticality for each set of parameters. Blue: $a=0.5$ and $b=-0.5$; red: $a=b=0$ (Brownian motion) ; green: $a=-1$ and $b=1$. Here $T=1$, $\textcolor{black}{\gamma=1}$, $\tau=1$, and $\phi(t)=0 \ \ (t\leq0)$.
• Figure 2: (a) A single trajectory simulation of the dynamics with oscillatory criticality. Here $a=-2$, $b=-3$, $\tau \approx1.029$, $T = 1$, $\textcolor{black}{\gamma=1}$, and $\phi(t)=0 \ \ (t\leq0)$. (b) 100 trajectories of the dynamics with oscillatory criticality for each set of parameters. Blue: $a=-2$, $b=-10$, and $\tau \approx 0.181$; red: $a=-2$, $b = -5$, and $\tau\approx0.433$; green: $a=-2$, $b=-2.5$, and $\tau\approx1.665$. Here $T=1$, $\textcolor{black}{\gamma=1}$, and $\phi(t)=0 \ \ (t\leq0)$.
• Figure 3: (a) Time evolution of the average heat dissipation rate in diffusive critical dynamics. Solid line: analytical results [calculated by Eqs. \ref{['var']}, \ref{['cri_con1']} and \ref{['dq0']}, where $x_0(t)$ is calculated through the numerical integral of Eq. \ref{['det_eq']}]; dashed line: the asymptotic behavior (Eq. \ref{['dq1']}); crosses: numerical simulations with $50000$ trajectories of precision $dt=10^{-4}$, where $\langle \Delta q/\Delta t\rangle$ is obtained by averaging the values of $\langle dq/dt\rangle$ within a interval $\Delta t=0.1$ centered at each specified time. Red error bars indicate the standard error of the mean. Here $a = -5$, $b=5$, $\tau = 1$, $T = 0.1$, $\textcolor{black}{\gamma=1}$, and $\phi(t) = 0 \ (t\leq0)$. (b) Average heat dissipation rate of the diffusive critical dynamics in the long-time limit. Here $T=1$ and $\textcolor{black}{\gamma=1}$.
• Figure 4: (a) Time evolution of the average heat dissipation rate in oscillatory critical dynamics. Solid line: the analytical result (calculated by Eqs. \ref{['var']}, \ref{['cri_con2']}, and \ref{['dq0']}, where $x_0(t)$ is calculated through the numerical integral of Eq. \ref{['det_eq']} ); dashed line: the asymptotic behavior (Eq. \ref{['dq2']}); crosses: numerical simulations with $50000$ trajectories of precision $dt=10^{-4}$, where $\langle\Delta q/\Delta t\rangle$ is obtained by averaging the values of $\langle dq/dt\rangle$ within a interval $\Delta t=0.1$ centered at each specified time. Red error bars indicate the standard error of the mean. Here $a = -2$, $b=-2.1$, $\tau \approx 4.422$, $T = 0.1$, $\textcolor{black}{\gamma=1}$, and $\phi(t) = 0 \ (t\leq0)$. (b) Divergence speed of the average heat dissipation rate in oscillatory critical dynamics (Eq. \ref{['lim_q']}). Here $b<-|a|$, $T=1$, and $\textcolor{black}{\gamma=1}$.
• Figure 5: Probability distribution of the heat dissipation rate $\Delta q/\Delta t$ for diffusive critical dynamics after a sufficiently long time (at $t=70$), where each sample of $\Delta q /\Delta t$ is averaged over $\Delta t=\textcolor{black}{1}$ . We take $10^6$ samples with simulation precision $dt=10^{-3}$, and set $T=0.1$ and $\textcolor{black}{\gamma=1}$. (a) Varying $a$ with $a=-b<0$ and $\tau=1$. (b) Varying $a$ with $a=-b>0$ and $\tau=1$. (c) Varying $\tau$ with $a=-2$ and $b=2$. (d) Varying $\tau$ with $a=0.5$ and $b=-0.5$.