The maximum efficiency of nano heat engines depends on more than temperature

Mischa P. Woods1,2, Nelly Huei Ying Ng2,3, and Stephanie Wehner2

1Institute for Theoretical Physics, ETH Zurich, Switzerland
2QuTech, Delft University of Technology, Lorentzweg 1, 2611 CJ Delft, Netherlands
3Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

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Sadi Carnot's theorem regarding the maximum efficiency of heat engines is considered to be of fundamental importance in thermodynamics. This theorem famously states that the maximum efficiency depends only on the temperature of the heat baths used by the engine, but not on the specific structure of baths. Here, we show that when the heat baths are finite in size, and when the engine operates in the quantum nanoregime, a revision to this statement is required. We show that one may still achieve the Carnot efficiency, when certain conditions on the bath structure are satisfied; however if that is not the case, then the maximum achievable efficiency can reduce to a value which is strictly less than Carnot. We derive the maximum efficiency for the case when one of the baths is composed of qubits. Furthermore, we show that the maximum efficiency is determined by either the standard second law of thermodynamics, analogously to the macroscopic case, or by the non increase of the max relative entropy, which is a quantity previously associated with the single shot regime in many quantum protocols. This relative entropic quantity emerges as a consequence of additional constraints, called generalized free energies, that govern thermodynamical transitions in the nanoregime. Our findings imply that in order to maximize efficiency, further considerations in choosing bath Hamiltonians should be made, when explicitly constructing quantum heat engines in the future. This understanding of thermodynamics has implications for nanoscale engineering aiming to construct small thermal machines.

Nicolas Carnot famously showed that all heat engines, regardless of their heat source or working substance, could, in principle, achieve a theoretical maximum efficiency for converting heat into work; which only depended on the temperature of its thermal baths. This principle has underpinned a transport revolution from steam engines to jet planes. However, the principle was derived for macroscopic machines, and with the advent of quantum nano-devices, the question of whether this universal principle still holds is of great importance. Here, we show that in the realm of the very small where particles obey the laws of quantum physics, Carnot’s theoretical maximum still holds — but with a catch; that the quantized baths must have sufficiently small energy gaps. Baths with energy gaps that are too large will still allow for functioning heat engines, but with a reduced maximum efficiency. This imposes an important fundamental limitation on the future design of nanoscale heat engines.
When the system of interest is characterized by only a few quantum particles, the statistical arguments based on the law of large numbers break down. Therefore, in addition to the usual second law of thermodynamics, additional restrictions arise to govern the possibility of a thermodynamical state transition. It is due to these additional constraints that Carnot’s efficiency ceases to be universally achievable. This work is an important milestone in a growing body of work aiming to revolutionize our understanding of the nature of work and heat at the nanoscale.

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