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arXiv:1205.0667 · Physica Scripta

Informative priors and the analogy between quantum and classical heat engines

George Thomas, Preety Aneja, Ramandeep S Johal

Published Nov 1, 2012• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0031-8949/2012/t151/014031

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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 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.
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.
It is possible to extract work from a quantum–mechanical system whose dynamics is governed by a time–dependent cyclic Hamiltonian. An energy bath is required to operate such a quantum engine in place of the heat bath used to run a conventional classical thermodynamic heat engine. The effect of the energy bath is to maintain the expectation value of the system Hamiltonian during an isoenergetic process. It is shown that the existence of such a bath leads to equilibrium quantum states that maximize the von Neumann entropy. Quantum analogues of certain thermodynamic relations are obtained that allow one to define the temperature of the energy bath.
In the present study we have developed an alternative formulation for the quantum stochastic thermodynamics based on the c-number Langevin equation for the system-reservoir model. This is analogous to the classical one. Here we have considered both Markovian and non-Markovian dynamics (NMD). Consideration of the NMD is an important issue at the current state of the stochastic thermodynamics. Applying the present formalism, we have carried out a comparative study on the heat absorbed and the change of entropy with time for a linear quantum system and its classical analog for both Markovian and NMD. Here the strength of the thermal noise and its correlation time for the respective cases are the leading quantities to explain any distinguishable feature which may appear with the equilibration kinetics. For another application, we have proposed a formulation with classical look for a quantum stochastic heat engine. Using it we have presented a comparative study on the efficiency and its value at maximum power for a quantum stochastic heat engine and its classical analog. The engines are Carnot like which are coupled with their respective Markovian thermal baths. Here also the noise strength as well as the diffusion constant are the leading quantities to explain any noticeable feature.
t is possible to extract work from a quantum-mechanical system whose dynamics is governed by a time-dependent cyclic Hamiltonian. An energy bath is required to operate such a quantum engine in place of the heat bath used to run a conventional classical thermodynamic heat engine. The effect of the energy bath is to maintain the expectation value of the system Hamiltonian during an isoenergetic process. It is shown that the existence of such a bath leads to equilibrium quantum states that maximise the von Neumann entropy. Quantum analogues of certain thermodynamic relations are obtained that allow one to define the temperature of the energy bath.
The concept of a quantum heat pump is proposed as a convenient model in the thermodynamic interpretation of certain multilevel processes. The ideal quantum heat engine is defined as an idealization of realistic pumped multilevel systems in much the same way that the well-known Carnot cycle is an idealization of physically realizable, classical processes or engines. There is evidence that the conventional Carnot cycle can be operated only between reservoirs at absolute temperatures of identical sign. No such restriction applies, however, to the quantum heat engine. Thus it may be used to calibrate negative absolute temperatures by relating them directly to positive temperatures. Negative efficiencies or efficiencies greater than unity have particularly simple interpretations in the quantum-heat-engine model. An important application of these concepts is in the calculation of optical maser parameters.