Stochastic control of thermodynamic heat engines

Thermodynamics historically developed out of a desire to quantify the maximal efficiency of early thermodynamic heat engines, especially through the work of French physicist Sadi Carnot. However, the more practical problem about quantifying the limits of power output that can be delivered from the system remained unclear due to the fact that quasistatic process requires infinite operation time, resulting in a vanishing power output. Recent advances in the field of stochastic thermodynamics appear to link the theory and practice, which enables us to mathematically analyze the maximal power and also control design of a thermodynamic heat engine on the microscopic scale. This review aims at summarizing and categorizing previous research on the optimal performance of two kinds of finite-time stochastic thermodynamic engines (a Carnot-like heat engine and the heat engine with a single heat bath) both in the linear and nonlinear response regimes. Thus, this is to be expected, estimated bounds for maximal power output and optimal control can provide physical insights and guidelines for engineering design. We start by reviewing the optimal performance for the Carnot-like engine that alternates between two heat baths of different constant temperatures. Then we discuss the fundamental bounds of the power output for the heat engine with a single periodic heat bath. In each setting, we provide a comprehensive analysis of the maximal power and efficiency both in the linear and nonlinear regimes. Finally, several challenges and future research directions are concluded.