Fluctuation relations for heat engines in time-periodic steady states

A fluctuation relation for heat engines has been derived recently. In the beginning, the system is in contact with the cooler bath. The system is then coupled to the hotter bath and external parameters are changed cyclically, eventually bringing the system back to its initial state, once the coupling with the hot bath is switched off. In this work, we lift the condition of initial thermal equilibrium and derive a new fluctuation relation for the central system (heat engine) being in a time-periodic steady state (TPSS). Carnot's inequality for classical thermodynamics follows as a direct consequence of this fluctuation theorem even in the TPSS. For the special cases of the absence of hot bath and no extraction of work, we obtain the integral fluctuation theorem for total entropy and the generalized exchange fluctuation theorem, respectively. Recently, microsized heat engines have been realized experimentally in the TPSS. We numerically simulate the same model and verify our proposed theorems.