Energetics and performance of a microscopic heat engine based on exact calculations ofwork and heat distributions

Abstract

We investigate a microscopic motor based on an externally controlled two-level system. Onecycle of the motor operation consists of two strokes. Within each stroke, the two-levelsystem is in contact with a given thermal bath and its energy levels are driven at aconstant rate. The time evolutions of the occupation probabilities of the two states arecontrolled by one rate equation and represent the system’s response with respect to theexternal driving. We give the exact solution of the rate equation for the limit cycle anddiscuss the emerging thermodynamics: the work done on the environment, the heatexchanged with the baths, the entropy production, the motor’s efficiency, and the poweroutput. Furthermore we introduce an augmented stochastic process which reflects, at agiven time, both the occupation probabilities for the two states and the time spent inthe individual states during the previous evolution. The exact calculation of theevolution operator for the augmented process allows us to discuss in detail theprobability density for the work performed during the limit cycle. In the stronglyirreversible regime, the density exhibits important qualitative differences withrespect to the more common Gaussian shape in the regime of weak irreversibility.