Finite-time thermodynamics of fluctuations in microscopic heat engines

Fluctuations of thermodynamic quantities become non-negligible and play an important role when the system size is small. We develop finite-time thermodynamics fluctuations in microscopic heat engines whose environmental temperature mechanical parameter are driven periodically time. Within slow-driving regime, this formalism universally holds a coarse-grained time scale resolution much longer than correlation fluctuations, shown to be consistent with relation analogous fluctuation-dissipation relation. Employing geometric argument, scenario simultaneously minimize both average fluctuation dissipation Carnot cycle identified. For simultaneous optimization, existence zero eigenvalue singular metric for invariant equilibrium state found essential. Furthermore, we demonstrate that our optimized protocol can improve its over current experiment.