Thermodynamic anomalies in overdamped systems with time-dependent temperature

One of the key objectives in investigating small stochastic systems is the development of micrometer-sized engines and the understanding of their thermodynamics. However, the primary mathematical tool used for this purpose, the overdamped approximation, has a critical limitation: it fails to fully capture the thermodynamics when the temperature varies over time, as the velocity is not considered in the approximation. Specifically, we show that heat dissipation and entropy production calculated under the overdamped approximation deviate from their true values. These discrepancies are termed thermodynamic anomalies. To overcome this limitation, we analytically derive expressions for these anomalies in the presence of a general time-varying temperature. One notable feature of the result is that high viscosity and small mass, though both leading to the same overdamped dynamic equations, result in different thermodynamic anomaly relations. Our results have significant implications, particularly for accurately calculating the efficiency of heat engines operating in overdamped environments with time-varying temperatures, without requiring velocity measurements. Additionally, our findings offer a simple method for estimating the kinetic energy of an overdamped system.