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

Abstract

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.

A quantum Otto engine with finite heat baths: energy, correlations, and degradation

“…m c m m m h m m so that the WM's state remains unchanged. The fact that the WM is detached from the baths during the swap ensures that the process is adiabatic, i.e., thermally isolated, in the thermodynamic sense 6 . It is important to note that the change in equation (22) is not equivalent to simply quenching the frequency of the oscillator.…”

“…The ideal engine converts the internal energy of the hot bath into work with an efficiency given by Carnot's formula, h = -T T 1 c h C . The idealizations needed for the machine to operate at such an efficiency are that (i) the baths interact with the WM weakly [1,2], (ii) the cycle is a quasiequilibrium process and hence it takes infinite time to complete [1,3,4], and (iii) the baths are infinitely large [1,[5][6][7][8]. It has to be noted, however, that the size of the WM itself is of no relevance-it can be anything from a two-level quantum system [9][10][11] to a giant steam engine [12].…”

Surpassing the Carnot efficiency by extracting imperfect work

“…Nevertheless, the study of using such non-thermal reservoirs can still be of interest, since they potentially may boost other features of the heat engine, such as the rate of extracting work. However, in this manuscript we are focusing on the standard setting of a QHE, in which the baths are thermal, where in classical thermodynamics, it is proven that although CE can be approached, it can never be surpassed [14].Even without additional resources such as those in EQHEs, QHEs are already radically different from classical engines, since energy fluctuations are much more prominent due to the small number of particles involved. The laws of thermodynamics for small quantum systems are more restrictive due to finite-size effects [14][15][16][17][18][19].…”

“…The laws of thermodynamics for small quantum systems are more restrictive due to finite-size effects [14][15][16][17][18][19]. It is known that such second laws introduce additional restrictions on the performance of QHEs [14]. Specifically, not all QHEs can even achieve the CE.…”

“…Nevertheless, the study of using such non-thermal reservoirs can still be of interest, since they potentially may boost other features of the heat engine, such as the rate of extracting work. However, in this manuscript we are focusing on the standard setting of a QHE, in which the baths are thermal, where in classical thermodynamics, it is proven that although CE can be approached, it can never be surpassed [14].…”