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We investigate a microscopic motor based on an externally controlledtwo-level system. One cycle of the motor operation consists of two strokes.Within each stroke, the two-level system is in contact with a given thermalbath and its energy levels are driven with a constant rate. The time evolutionof the occupation probabilities of the two states are controlled by one rateequation and represent the system's response with respect to the externaldriving. We give the exact solution of the rate equation for the limit cycleand discuss the emerging thermodynamics: the work done on the environment, theheat exchanged with the baths, the entropy production, the motor's efficiency,and the power output. Furthermore we introduce an augmented stochastic processwhich reflects, at a given time, both the occupation probabilities for the twostates and the time spent in the individual states during the previousevolution. The exact calculation of the evolution operator for the augmentedprocess allows us to discuss in detail the probability density for theperformed work during the limit cycle. In the strongly irreversible regime, thedensity exhibits important qualitative differences with respect to the morecommon Gaussian shape in the regime of weak irreversibility.

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