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arXiv:1002.4941 · Physical Review E

Universal efficiency at optimal work with Bayesian statistics

Ramandeep S. Johal

Published Dec 7, 2010• 20 citations
arXivHTMLPDF

Abstract

If the work per cycle of a quantum heat engine is averaged over an appropriate prior distribution for an external parameter a , the work becomes optimal at Curzon-Ahlborn (CA) efficiency. More general priors of the form Π(a) ∝ 1/a(γ) yield optimal work at an efficiency which stays close to CA value, in particular near equilibrium the efficiency scales as one-half of the Carnot value. This feature is analogous to the one recently observed in literature for certain models of finite-time thermodynamics. Further, the use of Bayes' theorem implies that the work estimated with posterior probabilities also bears close analogy with the classical formula. These findings suggest that the notion of prior information can be used to reveal thermodynamic features in quantum systems, thus pointing to a connection between thermodynamic behavior and the concept of information.

DOI
10.1103/physreve.82.061113

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If the work per cycle of a quantum heat engine is averaged over an appropriate prior distribution for an external parameter a , the work becomes optimal at Curzon-Ahlborn (CA) efficiency. More general priors of the form Π(a) ∝ 1/a(γ) yield optimal work at an efficiency which stays close to CA value, in particular near equilibrium the efficiency scales as one-half of the Carnot value. This feature is analogous to the one recently observed in literature for certain models of finite-time thermodynamics. Further, the use of Bayes' theorem implies that the work estimated with posterior probabilities also bears close analogy with the classical formula. These findings suggest that the notion of prior information can be used to reveal thermodynamic features in quantum systems, thus pointing to a connection between thermodynamic behavior and the concept of information.
When incomplete information about the control parameters is quantified as a prior distribution, a subtle connection emerges between quantum heat engines and their classical analogues. We study the quantum model where the uncertain parameters are the intrinsic energy scales and compare it with the classical models where the intermediate temperature is the uncertain parameter. The prior distribution quantifying the incomplete information has the form π ( x )∝1/ x in both the quantum and the classical models. The expected efficiency calculated in the near-equilibrium limit approaches the value of one third of Carnot efficiency.
We consider a class of quantum heat engines consisting of two subsystems interacting with a work-source and coupled to two separate baths at different temperatures Th>Tc. The purpose of the engine is to extract work due to the temperature difference. Its dynamics is not restricted to the near equilibrium regime. The engine structure is determined by maximizing the extracted work under various constraints. When this maximization is carried out at finite power, the engine dynamics is described by well-defined temperatures and satisfies the local version of the second law. In addition, its efficiency is bounded from below by the Curzon-Ahlborn value 1-radical Tc/Th and from above by the Carnot value 1-(Tc/Th). The latter is reached-at finite power--for a macroscopic engine, while the former is achieved in the equilibrium limit Th-->Tc . The efficiency that maximizes the power is strictly larger than the Curzon-Ahloborn value. When the work is maximized at a zero power, even a small (few-level) engine extracts work right at the Carnot efficiency.
We estimate the expected behavior of the quantum model of a heat engine when we have incomplete information about external macroscopic parameters such as the magnetic field controlling the intrinsic energy scales of the working medium. We explicitly derive the prior probability distribution for these unknown parameters ai (i=1,2). Based on a few simple assumptions, the prior probability distribution is found to be of the form Π(ai)∝1/ai. By calculating the expected values of various physical quantities related to this engine, we find that the expected behavior of the quantum model exhibits thermodynamiclike features. This leads us to a surprising proposal that incomplete information quantified as an appropriate prior distribution can lead us to expect classical thermodynamic behavior in quantum models.
The area of quantum thermodynamics is a relatively new field and there are great efforts in research to study the effects of microscopic work, heat, efficiency and other associated properties due to its vast potential applications in micro engineering and biophysics. The transition from large scale classical macro systems to small scale micro systems is not straightforward and the effects of different properties pose different responses in the system. As the systems miniaturise large scale fluctuations that in classical thermodynamics were once neglected, become of great importance in micro systems. In this thesis, a review of the current understandings of quantum thermodynamics is given with particular interest in Carnot engines and the maximum efficiency proposed by Curzon-Ahlborn. From a quantum analog of the Carnot engine, found in previous studies, the effects of thermal variations were investigated and the resulting response of the systems shows correlation to classical Carnot engines. It was found that increasing thermal fluctuations result in a decrease in maximum efficiency. Furthermore, to optimise these engines decreasing thermal fluctuations is necessary in order to build quantum engines that can trend towards or potentially reach the maximum Carnot efficiency.
Optimizing the performance of thermal machines is an essential task of thermodynamics. We here consider the optimization of information engines that convert information about the state of a system into work. We concretely introduce a generalized finite-time Carnot cycle for a quantum information engine and optimize its power output in the regime of low dissipation. We derive a general formula for its efficiency at maximum power valid for arbitrary working media. We further investigate the optimal performance of a qubit information engine subjected to weak energy measurements.
The quantum analog of Carnot cycles in few-particle systems consists of two quantum adiabatic steps and two isothermal steps. This construction is formally justified by use of a minimum work principle. It is then shown, using minimal assumptions of work or heat in nanoscale systems, that the heat-to-work efficiency of such quantum heat engine cycles can be further optimized via two conditions regarding the expectation value of some generalized force operators evaluated at equilibrium states. In general the optimized efficiency is system specific, lower than the Carnot efficiency, and dependent upon both temperatures of the cold and hot reservoirs. Simple computational examples are used to illustrate our theory. The results should be an important guide towards the design of favorable working conditions of a realistic quantum heat engine.