We study the efficiency at maximum power, η * , of engines performing finite-time Carnot cycles between a hot and a cold reservoir at temperatures T h and Tc, respectively. For engines reaching Carnot efficiency ηC = 1 − Tc/T h in the reversible limit (long cycle time, zero dissipation), we find in the limit of low dissipation that η * is bounded from above by ηC /(2 − ηC ) and from below by ηC /2. These bounds are reached when the ratio of the dissipation during the cold and hot isothermal phases tend respectively to zero or infinity. For symmetric dissipation (ratio one) the Curzon-Ahlborn efficiency ηCA = 1 − Tc/T h is recovered.

We derive an exact (classical and quantum) expression for the entropy production of a finite system placed in contact with one or several finite reservoirs each of which is initially described by a canonical equilibrium distribution. Whereas the total entropy of system plus reservoirs is conserved, we show that the system entropy production is always positive and is a direct measure of the system-reservoir correlations and/or entanglements. Using an exactly solvable quantum model, we illustrate our novel interpretation of the Second Law in a microscopically reversible finite-size setting, with strong coupling between system and reservoirs. With this model, we also explicitly show the approach of our exact formulation to the standard description of irreversibility in the limit of a large reservoir.PACS numbers: 05.70. Ln, Starting with the groundbreaking work of Boltzmann, there have been numerous attempts to construct a microscopic derivation of the Second Law. The main difficulty is that the prime microscopic candidate for the entropy, namely, the von Neumann entropy S = −Trρ ln ρ with ρ the density matrix of the total or compound system, is a constant in time by virtue of Liouville's theorem. Related difficulties are the time-reversibility of the microscopic laws and the recurrences of the micro-states. A common way to bypass these difficulties is to introduce irreversibility in an ad hoc way, for example by reasoning that the system is in contact with idealized infinitely large heat reservoirs. Nevertheless, as was realized early on by Onsager, a consistent description of the resulting irreversible behavior still carries the undiluted imprint of the underlying time-reversibility and Liouville's theorem for the system. Examples are the symmetry of the Onsager coefficients and the fluctuation dissipation theorem. As examples of more recent discussions we cite results on work theorems and fluctuation theorems [1,2,3]. Even more relevant to the question pursued here, we cite the microscopic expression for the entropy production as the breaking, in a statistical sense, of the arrow of time [4,5,6,7,8,9,10]. We also mention that significant effort has been devoted to a detailed description and understanding of the interaction with the heat reservoirs, in particular the difficulties of dealing with the case of strong coupling [11,12].In this letter we show that the problem of entropy production can be addressed within a microscopically exact description of a finite system, without resorting to infinitely large heat reservoirs and without any assumption of weak coupling. Whereas the von Neumann entropy of system plus reservoirs is conserved, the entropy production of the system is always positive, even though it displays oscillations and recurrences typical of the finite total system. Interestingly, this entropy production is expressed in terms of the correlations and/or entanglement between system and reservoirs, so that its positivity can be explained by a corresponding negative entropy contribution contained in the...

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...

We identify the operational conditions for maximum power of a nanothermoelectric engine consisting of a single quantum level embedded between two leads at different temperatures and chemical potentials. The corresponding thermodynamic efficiency agrees with the Curzon-Ahlborn expression up to quadratic terms in the gradients, supporting the thesis of universality beyond linear response.PACS numbers:

We evaluate the efficiency at maximum power of a quantum-dot Carnot heat engine. The universal values of the coefficients at the linear and quadratic order in the temperature gradient are reproduced. Curzon-Ahlborn efficiency is recovered in the limit of weak dissipation.