Universality of Efficiency at Maximum Power

Abstract

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...

Stochastically driven single-level quantum dot: A nanoscale finite-time thermodynamic machine and its various operational modes

“…This latter result thus again supports the universality of this value (for systems with a left-right symmetry) [3]. Finally, we turn to the efficiency at maximum power for the refrigerator in the tight coupling regime.…”

“…We investigate in analytic detail various operational regimes. When operating under tight coupling conditions, familiar features are recovered in appropriate limits: Carnot efficiency for reversible operation when the reservoirs are at different temperatures, universal features of efficiency at maximum power [3,7], and efficiency at maximum power close to the Curzon-Ahlborn efficiency [8]. When the reservoirs are at the same temperature, the work done on the dot by the switching can reverse the "normal" direction (from high to low chemical potential) of the current.…”

“…Parallel to spectacular developments in bio-and nanotechnology, there has been great theoretical interest in the study of small-scale machines. A well-documented case is the small-scale Carnot engine, in which the operational unit is subject to thermal fluctuations [1][2][3][4]. Of greater biological relevance are machines that convert one form of work to another, and yet these have received far less attention [5].…”

Stochastic thermodynamics under coarse graining

“…They coincided with an unprecedented development in the experimental techniques used to manipulate small systems and triggered a great deal of experimental studies in a variety of contexts, such as single molecule stretching experiments [35,[38][39][40], nanomechanical oscillator work measurements [41], spectroscopic measurement of trajectory entropies [42,43], and electronic current fluctuations in full counting statistics experiments [44]. Since stochastic thermodynamics combines kinetics and thermodynamics, it has also proved extremely useful to describe the finite-time thermodynamics (e.g., efficiency at finite power) of various nanodevices operating as thermodynamic machines [45][46][47][48][49][50][51][52][53][54]. Overall, this theory is becoming a fundamental tool for the study of nanosciences.…”

Ensemble and trajectory thermodynamics: A brief introduction

“…Another question that has received a lot of attention is the universal features of the efficiency of such machines when operating at maximum power [20][21][22][23][24][25][26][27][28][29]. We finally mention the simplification in the absence of particle transport, achieved by setting in the above formulas all the chemical potentials equal to zero, µ ν = 0, ∀ν.…”