Abstract
We introduce a general framework for analyzing the thermodynamics of small systems that are driven by both a periodic temperature variation and some external parameter modulating their energy. This setup covers, in particular, periodic micro- and nano-heat engines. In a first step, we show how to express total entropy production by properly identified time-independent affinities and currents without making a linear response assumption. In linear response, kinetic coefficients akin to Onsager coefficients can be identified. Specializing to a Fokker-Planck-type dynamics, we show that these coefficients can be expressed as a sum of an adiabatic contribution and one reminiscent of a Green-Kubo expression that contains deviations from adiabaticity. Furthermore, we show that the generalized kinetic coefficients fulfill an Onsager-Casimir-type symmetry tracing back to microscopic reversibility. This symmetry allows for nonidentical off-diagonal coefficients if the driving protocols are not symmetric under time reversal. We then derive a novel constraint on the kinetic coefficients that is sharper than the second law and provides an efficiency-dependent bound on power. As one consequence, we can prove that the power vanishes at least linearly when approaching Carnot efficiency. We illustrate our general framework by explicitly working out the paradigmatic case of a Brownian heat engine realized by a colloidal particle in a time-dependent harmonic trap subject to a periodic temperature profile. This case study reveals inter alia that our new general bound on power is asymptotically tight.
- Received 21 May 2015
DOI:https://doi.org/10.1103/PhysRevX.5.031019
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Published by the American Physical Society
Popular Summary
Reciprocating heat engines like James Watt’s famous steam engine convert thermal energy into useful mechanical work by cyclically compressing and expanding a working fluid whose temperature is altered periodically. In recent decades, much work has focused on the miniaturization of such devices. A comprehensive understanding of the fundamental principles governing the performance of these miniaturized devices is accordingly of paramount importance. Here, we develop a general framework for the thermodynamic description of small periodically driven systems. We also derive a universal constraint on the power output of mesoscopic heat engines.
Our approach is inspired by concepts of irreversible thermodynamics, a powerful theory originally designed for the characterization of nonequilibrium steady states. We show that the central notions of irreversible thermodynamics can be applied consistently to periodically driven systems on the level of cycle averages. In particular, we recover the generic bilinear expression for the total rate of entropy production in terms of fluxes and affinities. Assuming linear response conditions, we obtain a generalization of Onsager’s celebrated reciprocal relations and, using a novel method, prove a general bound on the power of stochastic micro-heat-engines. We show that the power and efficiency of such systems are quadratically related and that the power vanishes as the efficiency approaches the Carnot value.
Since our analysis is entirely classical, our results might have to be reassessed on even smaller scales where genuine quantum effects such as coherence and entanglement become important. Exploring this issue constitutes a challenging and exciting subject for future research.