Van den Broeck, Parrondo, and Toral Reply:

Broeck, C. Van den; Parrondo, Juan M. R.; Toral, Raúl

doi:10.1103/physrevlett.75.4787

Cited by 120 publications

(288 citation statements)

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“…The upper limit is reached for a specific class of models, namely, those for which the heat flux is strongly coupled to the work-generating flux. Interestingly, such strong coupling is also a prerequisite for open systems to achieve Carnot efficiency [3,4]. In the nonlinear regime, no general result is known.…”

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confidence: 99%

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Universality of Efficiency at Maximum Power

Esposito

Lindenberg²,

Broeck³

2009

Phys. Rev. Lett.

403

447

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We investigate the efficiency of power generation by thermo-chemical engines. For strong coupling between the particle and heat flows and in the presence of a left-right symmetry in the system, we demonstrate that the efficiency at maximum power displays universality up to quadratic order in the deviation from equilibrium. A maser model is presented to illustrate our argument. The concept of Carnot efficiency is a cornerstone of thermodynamics. It states that the efficiency of a cyclic ("Carnot") thermal engine that transforms an amount Q r of energy extracted from a heat reservoir at temperature T r into an amount of work W is at most η = W/Q r ≤ η c = 1 − T l /T r , where T l is the temperature of a second, colder reservoir. The theoretical implications of this result are momentous, as they lie at the basis of the introduction by Clausius of the entropy as a state function. The practical implications are more limited, since the upper limit η c ("Carnot efficiency") is only reached for engines that operate reversibly. As a result, when the efficiency is maximal, the output power is zero. By optimizing the Carnot cycle with respect to power rather than efficiency, Curzon and Ahlborn found that the corresponding efficiency is given by η CA = 1 − T l /T r [1]. They obtained this result for a specific model, using in addition the so-called endo-reversible approximation (i.e., neglecting the dissipation in the auxiliary, work producing entity). Subsequently, the validity of this result as an upper bound, as well as its universal character, were the subject of a longstanding debate. In the regime of linear response, more precisely to linear order in η c , it was proven that the efficiency at maximum power is indeed limited by the Curzon-Ahlborn efficiency, which in this regime is exactly half of the Carnot efficiency,. The upper limit is reached for a specific class of models, namely, those for which the heat flux is strongly coupled to the work-generating flux. Interestingly, such strong coupling is also a prerequisite for open systems to achieve Carnot efficiency [3,4]. In the nonlinear regime, no general result is known. Efficiencies at maximum power, not only below but also above Curzon-Ahlborn efficiency, have been reported [5,6,7,8]. However, it was also found, again in several strong coupling models [7,8,9], that the efficiency at maximum power agrees with η CA up to quadratic order in η c , i.e., η = η c /2+η 2 c /8+O(η 3 c ), again raising the question of universality at least to this order. In this letter we prove that the coefficient 1/8 is indeed universal for strong coupling models that possess a left-right symmetry. Such a universality is remarkable in view of the fact that most explicit macroscopic relationships, for example the symmetry of Onsager coefficients, are limited to the regime of linear response. The interest in strong coupling is further motivated by the observation that it can naturally be achieved in nano-devices [10,11,12]. To complement our theoretical discussion, we also present a deta...

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confidence: 99%

“…This does not require that the forces F E and F M be zero separately. In fact in the vicinity of F = 0 the device can operate at Carnot efficiency, see for example [3,4,11]. Using Eq.…”

mentioning

confidence: 99%

Universality of Efficiency at Maximum Power

Esposito

Lindenberg²,

Broeck³

2009

Phys. Rev. Lett.

403

447

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“…Then we can confirm that q = 1 in our finite-time Carnot cycle from Eqs. (19), (23), (25) and (35). This condition gives a proof that our finitetime Carnot cycle shows the CA efficiency in the limit of ∆T → 0 as suggested in [4,5] from the viewpoint of the linear-response theory.…”

Section: The Efficiency At the Maximal Powermentioning

confidence: 99%

Onsager coefficients of a finite-time Carnot cycle

2009

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We study a finite-time Carnot cycle of a weakly interacting gas which we can regard as a nearly ideal gas in the limit of T h − Tc → 0 where T h and Tc are the temperatures of the hot and cold heat reservoirs, respectively. In this limit, we can assume that the cycle is working in the linearresponse regime and can calculate the Onsager coefficients of this cycle analytically using the elementary molecular kinetic theory. We reveal that these Onsager coefficients satisfy the so-called tight-coupling condition and this fact explains why the efficiency at the maximal power ηmax of this cycle can attain the Curzon-Ahlborn efficiency from the viewpoint of the linear-response theory.

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“…Since all variables of the engine inevitably fluctuate, the attainability of the CE with autonomous engines is a non-trivial problem. In the linear response regime (i.e., T H − T L ≃ 0), it is well known that the tight-coupling condition is necessary for autonomous engines to attain the CE [4]. On the other hand, for the case of finite difference of temperatures or chemical potentials, most of studies have paid attention to specific and elaborated models .…”

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Attainability of Carnot efficiency with autonomous engines

Shiraishi

2015

Phys. Rev. E

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The maximum efficiency of autonomous engines with finite chemical potential difference is investigated. We show that without a particular type of singularity autonomous engines cannot attain the Carnot efficiency. This singularity is realized in two ways: single particle transports and the thermodynamic limit. We demonstrate that both of two ways really lead to the Carnot efficiency in concrete setups. Our results clearly illustrate that the singularity plays a crucial role for the maximum efficiency of autonomous engines.

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Van den Broeck, Parrondo, and Toral Reply:

Cited by 120 publications

References 1 publication

Universality of Efficiency at Maximum Power

Universality of Efficiency at Maximum Power

Onsager coefficients of a finite-time Carnot cycle

Attainability of Carnot efficiency with autonomous engines

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