Thermodynamic anomalies of overdamped systems with time-dependent temperature

The overdamped approximation is commonly used to analyze mesoscopic experiments in fluids with high viscosity or negligible inertia effect. However, it fails to fully capture the thermodynamics of mesoscopic systems when the temperature varies over time. Specifically, heat dissipation and entropy production calculated under the overdamped approximation deviate from their true values, leading to what are termed thermodynamic anomalies. For a general time-varying temperature, we analytically derive expressions for these anomalies. One notable feature is that high viscosity and small mass, though both leading to the same overdamped dynamics, result in different thermodynamic anomaly relations. Our results have significant implications, particularly for the correct calculation of efficiency in heat engines operating in overdamped environments with time-varying temperatures. Additionally, our findings offer an improved method for estimating the kinetic energy of an overdamped system, surpassing traditional experimental approaches.