Thermodynamic anomalies in overdamped systems with time-dependent temperature

One of the objectives in investigating small stochastic systems is the development of micrometer-sized engines and the understanding of their thermodynamics. However, the main mathematical tool used for this purpose, the overdamped approximation, has a critical limitation: it fails to capture the thermodynamics when the temperature varies over time. We show that heat dissipation and entropy production computed under this approximation deviate from their true values, and these discrepancies are termed thermodynamic anomalies. To address this, we analytically derive expressions for these anomalies in the presence of a general time-varying temperature. A key feature of the result is that high viscosity and small mass, though both leading to the same overdamped equations, yield different anomaly relations. Our results have broad implications, particularly for accurately evaluating engine efficiency in overdamped environments with time-varying temperatures and provide a method for estimating the kinetic energy of an overdamped system. The overdamped approximation, widely used to study micrometer-sized engines, fails to capture their thermodynamics when temperature changes over time, leading to discrepancies in heat dissipation and entropy production that we identify as thermodynamic anomalies. Here, the authors derive analytical expressions for these anomalies and show that viscosity and mass produce distinct behaviors, improving the evaluation of engine efficiency and enabling simple estimation of kinetic energy in overdamped systems.