The efficiency of engines operating around a steady state at finite frequencies

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Abstract
Linear response theory is applied to a study of the performance of engines which operate harmonically around a stable reference state which is either an equilibrium or a nonequilibrium steady state. Two kinds of quantities are relevant for efficiency and power output: those which are integrals over a semicycle (e.g., the heat input in a heat engine or the heat uptake in a refrigeration machine), and net quantities of the whole cycle, like the work exchanged with the mechanical surroundings, the net dissipative losses, and certain refractive quantities which, via Kramers–Kronig relations, correspond to the absorptive ones. We discuss their dependence on the frequency of operation and on the amplitudes of the driving forces. The general analysis is applied to three examples: The first is completely dissipative and corresponds to chemical reactions near equilibrium; the second has an inertial term in the mechanical operation; and the third is designed to have similar properties as the second example but with first‐order time derivatives, to correspond to a chemical system far from equilibrium. In the last two examples oscillatory behavior may occur; if the frequency of oscillation is slow compared to the rate of heat conduction, then, depending on the amplitudes of the driving forces, there may be one or two power maxima. In case of two maxima, optimization from the point of view of power generation, or of efficiency, leaves a choice between fast and slow operation. This may have a bearing on our understanding of metabolic systems which often do exhibit oscillatory behavior.