The unlikely Carnot efficiency
Christian Van den Broeck, Université Hasselt, Belgique
The efficiency of a 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 in the case of a symmetric driving protocol. More generally, the long-time probability for observing a reversible efficiency in a given engine is identical to that for the same engine working under the time-reversed driving. 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 several model systems, including the work-to-work conversion via a Brownian particle, effusion as a thermal engine, and an asymmetrically driven quantum dot.