Heat leakage in equilibrium processes

The difference between the zero-mass limit of the heat exchanged with a thermal reservoir, and its value as determined from overdamped dynamics, is termed `heat leakage' or `hidden heat' in the Smoluchowski limit. If present, heat leakages are the sign of the unsuitability of the overdamped approximation for addressing thermodynamics. It is accepted that no hidden heat arises in an isothermal process driven by conservative forces. Here, we challenge that conclusion. The heat exchanged with a reservoir in any isothermal and quasistatic process connecting two equilibrium states, indeed exhibits hidden contributions. Our results imply that the overdamped dynamics misrepresents thermodynamics quite generally. Surprisingly, the hidden heat is described by an universal distribution in slow processes, easing the correction of the heat statistics in that context.