Finite-Time Thermodynamics of Fluctuations in Microscopic Heat Engines

Fluctuations of thermodynamic quantities become nonnegligible and play an important role when the system size is small. We develop finite-time thermodynamics of fluctuations in microscopic heat engines whose environmental temperature and mechanical parameter are driven periodically in time. This formalism universally holds in coarse-grained time scale whose resolution is much longer than the correlation time of the fluctuations, and is shown to be consistent with the relation analogous to the fluctuation-dissipation relation. Employing a geometric argument, a scenario to simultaneously minimize both the average and fluctuation of the dissipation in the Carnot cycle is identified. Furthermore, we demonstrate our optimized protocol can improve the dissipation and its fluctuation over the current experiment.