According to the laws of thermodynamics, no heat engine can beat the efficiency of a Carnot cycle. This efficiency traditionally comes with vanishing power output and practical designs, optimized for power, generally achieve far less. Recently, various strategies to obtain Carnot's efficiency at large power were proposed. However, a thermodynamic uncertainty relation implies that steady-state heat engines can operate in this regime only at the cost of large fluctuations that render them immensely unreliable. Here, we demonstrate that this unfortunate trade-off can be overcome by designs operating cyclically under quasi-static conditions. The experimentally relevant yet exactly solvable model of an overdamped Brownian heat engine is used to illustrate the formal result. Our study highlights that work in cyclic heat engines and that in quasi-static ones are different stochastic processes.

We investigate maximum efficiency at a given power for low-dissipation heat engines. Close to maximum power, the maximum gain in efficiency scales as a square root of relative loss in power and this scaling is universal for a broad class of systems. For the low-dissipation engines, we calculate the maximum gain in efficiency for an arbitrary fixed power. We show that the engines working close to maximum power can operate at considerably larger efficiency compared to the efficiency at maximum power. Furthermore, we introduce universal bounds on maximum efficiency at a given power for low-dissipation heat engines. These bounds represent direct generalization of the bounds on efficiency at maximum power obtained by Esposito et al. Phys. Rev. Lett. 105, 150603 (2010). We derive the bounds analytically in the regime close to maximum power and for small power values. For the intermediate regime we present strong numerical evidence for the validity of the bounds.

A new universality in optimization of trade-off between power and efficiency for low-dissipation Carnot cycles is presented. It is shown that any trade-off measure expressible in terms of efficiency and the ratio of power to its maximum value can be optimized independently of most details of the dynamics and of the coupling to thermal reservoirs. The result is demonstrated on two specific trade-off measures. The first one is designed for finding optimal efficiency for a given output power and clearly reveals diseconomy of engines working at maximum power. As the second example we derive universal lower and upper bounds on the efficiency at maximum trade-off given by the product of power and efficiency. The results are illustrated on a model of a diffusion-based heat engine. Such engines operate in the low-dissipation regime given that the used driving minimizes the work dissipated during the isothermal branches. The peculiarities of the corresponding optimization procedure are reviewed and thoroughly discussed.

We investigate fluctuations of output work for a class of Stirling heat engines with working fluid composed of interacting units and compare these fluctuations to an average work output. In particular, we focus on engine performance close to a critical point where Carnot's efficiency may be attained at a finite power as reported by M. Campisi and R. Fazio [Nat. Commun. 7, 11895 (2016)2041-172310.1038/ncomms11895]. We show that the variance of work output per cycle scales with the same critical exponent as the heat capacity of the working fluid. As a consequence, the relative work fluctuation diverges unless the output work obeys a rather strict scaling condition, which would be very hard to fulfill in practice. Even under this condition, the fluctuations of work and power do not vanish in the infinite system size limit. Large fluctuations of output work thus constitute inseparable and dominant element in performance of the macroscopic heat engines close to a critical point.

We discuss the possibility of reaching the Carnot efficiency by heat engines (HEs) out of quasistatic conditions at nonzero power output. We focus on several models widely used to describe the performance of actual HEs. These models comprise quantum thermoelectric devices, linear irreversible HEs, minimally nonlinear irreversible HEs, HEs working in the regime of low dissipation, over-damped stochastic HEs and an under-damped stochastic HE. Although some of these HEs can reach the Carnot efficiency at nonzero and even diverging power, the magnitude of this power is always negligible compared to the maximum power attainable in these systems. We provide conditions for attaining the Carnot efficiency in the individual models and explain practical aspects connected with reaching the Carnot efficiency at large power output. Furthermore, we show how our findings can be tested in practice using a standard Brownian HE realizable with available micromanipulation techniques.

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We discuss the efficiency of a heat engine operating in a nonequilibrium steady state maintained by two heat reservoirs. Within the general framework of linear irreversible thermodynamics we derive a universal upper bound on the efficiency of the engine operating at arbitrary fixed power. Furthermore, we show that a slight decrease of the power below its maximal value can lead to a significant gain in efficiency. The presented analysis yields the exact expression for this gain and the corresponding upper bound.PACS numbers: 05.70.Ln, 07.20.Pe The Carnot efficiency η C = 1 − T c /T h [1, 2] provides the upper bound on efficiency of heat engines working between two reservoirs at temperatures T h and T c , T h > T c . Though crucial from the theoretical point of view [3], practical applications of η C are rather limited, since the Carnot efficiency can be reached only when the heat engine operates reversibly. Reversible operation implies extremely long duration of the working cycle. As a result, when the engine efficiency reaches the upper bound η C , the output power is zero. Appealing universality of the upper bound η C , which depends solely on the two temperatures, and the needs of engineering solutions stimulated an intensive search for a more practical upper bound on the efficiency of heat engines operating at finite power. A promising candidate for which at least some universal properties can be derived was introduced about half century ago [4][5][6], it is the efficiency at maximum power η ⋆ .The upper bound on the efficiency at maximum power (EMP) in the linear response regime (linear in η C ) is equal to the famous Curzon-Ahlborn [7] formula η CA = 1 − T c /T h , which is to the linear order in η C equal to the half of the Carnot efficiency,The upper bound η ⋆ = η C /2 is achieved by a particular class of heat engines with strongly coupled thermodynamic fluxes. The assumption of strong coupling (see discussion below Eq. (5)) means that the heat flux is proportional to the flux, which generates work on the surrounding [9][10][11].In the present study we stay in the linear response regime (linear in η C ), however, we go beyond the regime of maximum power and study the engine efficiency at an arbitrary power P , 0 ≤ P ≤ P ⋆ (P ⋆ stands for the maximum power). One of the main messages is that the universal bounds on efficiency can be derived for an arbitrary P and not only at the point of maximum power which was considered in several recent studies [8][9][10][11][12][13][14], see however [15][16][17][18][19][20][21][22] for optimal regimes other than that with maximum power. To this end we introduce relative deviations from the regime of maximum power, the relative gain in efficiency δη and power δP :where −1 ≤ δP ≤ 0. Such normalization of the two principal engine characteristics allows us to derive several explicit results. One of them is that it is possible to provide a universal upper bound for the efficiency at an arbitrary power P . The bound depends explicitly on δP and it readsAt the maximum powe...

Abstract. We investigate the distribution of work performed on a Brownian particle in a time-dependent asymmetric potential well. The potential has a harmonic component with time-dependent force constant and a time-independent logarithmic barrier at the origin. For arbitrary driving protocol, the problem of solving the FokkerPlanck equation for the joint probability density of work and particle position is reduced to the solution of the Riccati differential equation. For a particular choice of the driving protocol, an exact solution of the Riccati equation is presented. Asymptotic analysis of the resulting expression yields the tail behavior of the work distribution for small and large work values. In the limit of vanishing logarithmic barrier, the work distribution for the breathing parabola model is obtained.