On the classical limit of quantum thermodynamics in finite time

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Abstract
The finite time performance of quantum heat engines has been examined with emphasis on the classical, high temperature, limit. Two basic engine models were studied, differing by their consistency of working fluid: the harmonic engine, consisting of noninteracting harmonic oscillators, and the spin‐j engine, consisting of noninteracting spin‐j subsystems. The two models represent two distinct types of engines, with bounded vs unbounded Hamiltonians, and with creation and annihilation operators of the Fermionic vs the Bosonic type. The analysis is based on the time derivatives of the first and second laws of thermodynamics. Explicit relations linking quantum observables to thermodynamic quantities are utilized. The dynamics of the engines was modeled by the semigroup approach. The engines were optimized with respect to various target functions: power, entropy production, and efficiency, while subject to finite cycle duration. The main strategy of optimization was based on the Euler–Lagrange equation, and is similar to that previously applied by Salamon and Nitzan for the investigation of Newtonian engines [J. Chem. Phys. 74, 3546 (1981)]. The optimal cycles obtained at the classical limit are not of the Curzon–Ahlborn type, i.e., the internal temperature along the thermal branches is not constant. This result is in conflict with Newtonian thermodynamics, where the optimal cycles are of the Curzon–Ahlborn type. Nonetheless, some of the main features of Newtonian thermodynamics, such as the Curzon–Ahlborn efficiency at maximum power, are reproduced at the classical limit. This makes it possible to establish a ‘‘thermodynamic correspondence principle.’’ This principle asserts that the phenomenological Newtonian thermodynamic approach provides an asymptote of a theory based upon the more fundamental semigroup quantum approach. It is argued that the asymptotic nature of Newtonian thermodynamics is twofold since its validity is restricted by two demands: that of high temperatures and that of proximity to equilibrium.