Heat dissipation in marginally stable linear time-delayed Langevin systems

The thermodynamic properties of time-delayed dynamics remain largely unexplored, especially for systems that exhibit asymptotically nonstationary 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 nonstationary time-delayed dynamics with similar time-dependence 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.