The unlikely Carnot efficiency

The efficiency of an heat engine is traditionally defined as the ratio of its average output work
over its average input heat. Its highest possible value was discovered by Carnot in 1824 and is
a cornerstone concept in thermodynamics. It led to the discovery of the second law and to the
definition of the Kelvin temperature scale. Small-scale engines operate in the presence of
highly fluctuating input and output energy fluxes. They are therefore much better
characterized by fluctuating efficiencies. In this study, using the fluctuation theorem, we
identify universal features of efficiency fluctuations. While the standard thermodynamic
efficiency is, as expected, the most likely value, we find that the Carnot efficiency is,
surprisingly, the least likely in the long time limit. Furthermore, the probability distribution for
the efficiency assumes a universal scaling form when operating close-to-equilibrium. We
illustrate our results analytically and numerically on two model systems.