Efficiency at and near maximum power of low-dissipation heat engines

Temperature fluctuations in mesoscopic systems

Temperature is a fundamental concept in thermodynamics. In macroscopic thermodynamics, systems possess their own intrinsic temperature which equals the reservoir temperature when they equilibrate. In stochastic thermodynamics for simple systems at the microscopic level, thermodynamic quantities other than temperature (a deterministic parameter of the reservoir) are stochastic. To bridge the disparity in the perspectives about temperature between the micro- and macroregimes, we assign a generic mesoscopic N-body system an intrinsic fluctuating temperature T in this work. We simplify the complicated dynamics of numerous particles to one stochastic differential equation with respect to T, where the noise term accounts for finite-size effects arising from random energy transfer between the system and the reservoir. Our analysis reveals that these fluctuations make the extensive quantities (in the thermodynamic limit) deviate from being extensive. Moreover, we derive finite-size corrections, characterized by heat capacity of the system, to the Jarzynski equality. A possible violation of the principle of maximum work that scales with N^{-1} is also discussed. Additionally, we examine the impact of temperature fluctuations in a finite-size Carnot engine. We show that irreversible entropy production resulting from the temperature fluctuations of the working substance diminishes the average efficiency of the cycle as η_{C}-〈η〉∼N^{-1}, highlighting the unattainability of the Carnot efficiency η_{C} for mesoscopic heat engines even under the quasistatic limit. Our general framework paves the way for further exploration of nonequilibrium thermodynamics and the corresponding finite-size effects in a mesoscopic regime.

The effects of asymmetry in active noises on the efficiency of single colloidal Stirling engines with active noises

Molecular machines, which operate in highly fluctuating environments far from equilibrium, may benefit from their non-equilibrium environments. It is, however, a topic of controversy how the efficiency of the microscopic engines can be enhanced. Recent experiments showed that microscopic Stirling engines in bacterial reservoirs could show high performance beyond the equilibrium thermodynamics. In this work, we perform overdamped Langevin dynamics simulations for microscopic Stirling heat engines in bacterial reservoirs and show that the temperature dependence of the magnitude of active noises should be responsible for such high efficiency. Only when we introduce temperature-dependent active noises, the efficiency of the microscopic Stirling engines is enhanced significantly as in experiments.

Carnot, Stirling, and Ericsson stochastic heat engines: Efficiency at maximum power

This work uses the low-dissipation strategy to obtain efficiency at maximum power from a stochastic heat engine performing Carnot-, Stirling- and Ericsson-like cycles at finite time. The heat engine consists of a colloidal particle trapped by optical tweezers, in contact with two thermal baths at different temperatures, namely hot (T_{h}) and cold (T_{c}). The particle dynamics is characterized by a Langevin equation with time-dependent control parameters bounded to a harmonic potential trap. In a low-dissipation approach, the equilibrium properties of the system are required, which in our case, can be calculated through a statelike equation for the mean value 〈x^{2}〉_{eq} coming from a macroscopic expression associated with the Langevin equation.

Generalizing the isothermal efficiency by using Gaussian distributions

Unlike the Carnot heat engine efficiency published in 1824, an isothermal efficiency derived from thermodynamics and information theory can be applied to biological systems. The original approach by Pierce and Cutler in 1959 to derive the isothermal efficiency equation came from Shannon's channel capacity of 1949 and from Felker's 1952 determination of the minimum energy dissipation needed to gain a bit. In 1991 and 2010 Schneider showed how the isothermal efficiency equation can be applied to molecular machines and that this can be used to explain why several molecular machines are 70% efficient. Surprisingly, some macroscopic biological systems, such as whole ecosystems, are also 70% efficient but it is hard to see how this could be explained by a thermodynamic and molecular theory. The thesis of this paper is that the isothermal efficiency can be derived without using thermodynamics by starting from a set of independent Gaussian distributions. This novel derivation generalizes the isothermal efficiency equation for use at all levels of biology, from molecules to ecosystems.

Low-dissipation engines: Microscopic construction via shortcuts to adiabaticity and isothermality, the optimal relation between power and efficiency

We construct a microscopic model of low-dissipation engines by driving a Brownian particle in a time-dependent harmonic potential. Shortcuts to adiabaticity and shortcuts to isothermality are introduced to realize the adiabatic and isothermal branches in a thermodynamic cycle, respectively. We derive an analytical formula of the efficiency at maximum power with explicit expressions of dissipation coefficients under the optimized protocols. When the relative temperature difference between the two baths in the cycle is insignificant, this expression satisfies the universal law of efficiency at maximum power up to the quadratic term of the Carnot efficiency. For large relative temperature differences, the efficiency at maximum power tends to be 1/2. Furthermore, we analyze the issue of power at any given efficiency for general low-dissipation engines and then obtain the supremum of the power in three limiting cases, respectively. These expressions of maximum power at given efficiency provide the optimal relations between power and efficiency which are tighter than the results in previous references.

Thermodynamics and efficiency of sequentially collisional Brownian particles: The role of drivings

Brownian particles placed sequentially in contact with distinct thermal reservoirs and subjected to external driving forces are promising candidates for the construction of reliable engine setups. In this contribution, we address the role of driving forces for enhancing the collisional machine performance. Analytical expressions for thermodynamic quantities such as power output and efficiency are obtained for general driving schemes. A proper choice of these driving schemes substantially increases both power output and efficiency and extends the working regime. Maximizations of power and efficiency, whether with respect to the strength of the force, driving scheme, or both have been considered and exemplified for two kind of drivings: generic power-law and harmonic (sinusoidal) drivings.

Perturbative expansion of irreversible work in symmetric and asymmetric processes

The systematic expansion method of the solution of the Fokker-Planck equation is developed by generalizing the formulation proposed in [J. Phys. A: Math. Theor. 50, 325001 (2017)10.1088/1751-8121/aa7af4]. Using this method, we obtain an alternative formula to calculate the mean work perturbatively which is applicable to systems with degeneracy in the eigenvalues of the Fokker-Planck operator. This method enables us to study how the geometrical symmetry affects thermodynamic description of a Brownian particle. To illustrate the application of the derived theory, we consider the Fokker-Planck equation with a two-dimensional harmonic potential. To investigate the effect of symmetry of the potential, we study thermodynamic properties in symmetric and asymmetric deformation processes of the potential: the rotational symmetry of the harmonic potential is held in the former, but it is broken in the latter. Optimized deformations in these processes are defined by minimizing mean work. Comparing these optimized processes, we find that the difference between the symmetric and asymmetric processes is maximized when the deformation time of the potential is given by a critical time which is characterized by the relaxation time of the Fokker-Planck equation. This critical time in the mean work is smaller than that of the change of the mean energy because of the hysteresis effect in the irreversible processes.

Maximum efficiency of low-dissipation heat pumps at given heating load

We derive an analytical expression for maximum efficiency at fixed power of heat pumps operating along a finite-time reverse Carnot cycle under the low-dissipation assumption. The result is cumbersome, but it implies simple formulas for tight upper and lower bounds on the maximum efficiency and various analytically tractable approximations. In general, our results qualitatively agree with those obtained earlier for endoreversible heat pumps. In fact, we identify a special parameter regime when the performance of the low-dissipation and endoreversible devices is the same. At maximum power, heat pumps operate as work to heat converters with efficiency 1. Expressions for maximum efficiency at given power can be helpful in the identification of more practical operation regimes.

Optimizing thermodynamic cycles with two finite-sized reservoirs

We study the nonequilibrium thermodynamics of a heat engine operating between two finite-sized reservoirs with well-defined temperatures. Within the linear response regime, it is found that the uniform temperature of the two reservoirs at final time τ is bounded from below by the entropy production σ_{min}∝1/τ. We discover a general power-efficiency tradeoff depending on the ratio of heat capacities (γ) of the reservoirs for the engine, and a universal efficiency at maximum average power of the engine for arbitrary γ is obtained. For practical purposes, the operation protocol of an ideal gas heat engine to achieve the optimal performance associated with σ_{min} is demonstrated. Our findings can be used to develop a general optimization scenario for thermodynamic cycles with finite-sized reservoirs in real-world circumstances.

Optimal work associated with off-centered harmonic Brownian motion at any friction damping

There is extensive literature on how to determine the work involving a Brownian particle interacting with an external field and submerged in a thermal reservoir. However, the information supplied is essentially theoretical without specific calculations to show how this property changes with the system parameters and initial conditions. In this article, we provide explicit calculations of the optimal work considering the particle is under the influence of a time-dependent off-centered moving harmonic potential. It is done for all physical values of the friction coefficient. The system is modeled through a more general version of the Langevin equation which encompasses its classical and quasiclassical version. From the equation that defines the work, the external protocol is found through a fairly current extended version of the Euler-Lagrange equation that unifies the local and nonlocal contributions in a simple expression. The protocol is linear and, unlike previous work, not only changes the initial velocity of the particle but also its acceleration. Calculations were done for friction constants γ spanning all possible values. The periodic γ=1 shows discontinuities in the optimal work of the interplay of concentration and diffusion processes acting periodically in the dynamics. For higher values work appears to be as a smooth function of time, while the truly overdamped, where the inertial effect can be discarded, agrees with the analytical result up to a time where the numerical overdamped algorithm provides a different solution due to its inability to discard entirely the inertial effect.

Maximum efficiency of absorption refrigerators at arbitrary cooling power

We consider absorption refrigerators consisting of simultaneously operating Carnot-type heat engine and refrigerator. Their maximum efficiency at given power (MEGP) is given by the product of MEGPs for the internal engine and refrigerator. The only subtlety of the derivation lies in the fact that the maximum cooling power of the absorption refrigerator is not limited just by the maximum power of the internal refrigerator, but, due to the first law, also by that of the internal engine. As a specific example, we consider the simultaneous absorption refrigerators composed of low-dissipation (LD) heat engines and refrigerators, for which the expressions for MEGPs are known. The derived expression for maximum efficiency implies bounds on the MEGP of LD absorption refrigerators. It also implies that a slight decrease in power of the absorption refrigerator from its maximum value results in a large nonlinear increase in efficiency, observed in heat engines, whenever the ratio of maximum powers of the internal engine and the refrigerator does not diverge. Otherwise, the increase in efficiency is linear as observed in LD refrigerators. Thus, in all practical situations, the efficiency of LD absorption refrigerators significantly increases when their cooling power is slightly decreased from its maximum.

Shift a laser beam back and forth to exchange heat and work in thermodynamics

Although the equivalence of heat and work has been unveiled since Joule's ingenious experiment in 1845, they rarely originate from the same source in experiments. In this study, we theoretically and experimentally demonstrated how to use a high-precision optical feedback trap to combine the generation of virtual temperature and potential to simultaneously manipulate the heat and work of a small system. This idea was applied to a microscopic Stirling engine consisting of a Brownian particle under a time-varying confining potential and temperature. The experimental results justified the position and the velocity equipartition theorem, confirmed several theoretically predicted energetics, and revealed the engine efficiency as well as its trade-off relation with the output power. The small theory-experiment discrepancy and high flexibility of the swift change of the particle condition highlight the advantage of this optical technique and prove it to be an efficient way for exploring heat and work-related issues in the modern thermodynamics for small systems.

Geometric Optimisation of Quantum Thermodynamic Processes

Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We review and connect different frameworks where it emerges in the quantum regime: adiabatically driven closed systems, time-dependent Lindblad master equations, and discrete processes. A geometric lower bound on entropy production in finite-time is then presented, which represents a quantum generalisation of the original classical bound. Following this, we review and develop some general principles for the optimisation of thermodynamic processes in the linear-response regime. These include constant speed of control variation according to the thermodynamic metric, absence of quantum coherence, and optimality of small cycles around the point of maximal ratio between heat capacity and relaxation time for Carnot engines.

Thermodynamic optimization subsumed in stability phenomena

In the present paper the possibility of an energetic self-optimization as a consequence of thermodynamic stability is addressed. This feature is analyzed in a low dissipation refrigerator working in an optimized trade-off regime (the so-called Omega function). The relaxation after a perturbation around the stable point indicates that stability is linked to trajectories in which the thermodynamic performance is improved. Furthermore, a limited control over the system is analyzed through consecutive external random perturbations. The statistics over many cycles corroborates the preference for a better thermodynamic performance. Endoreversible and irreversible behaviors play a relevant role in the relaxation trajectories (as well as in the statistical performance of many cycles experiencing random perturbations). A multi-objective optimization reveals that the well-known endoreversible limit works as an attractor of the system evolution coinciding with the Pareto front, which represents the best energetic compromise among efficiency, entropy generation, cooling power, input power and the Omega function. Meanwhile, near the stable state, performance and stability are dominated by an irreversible behavior.

Maximum efficiency of low-dissipation refrigerators at arbitrary cooling power

We analytically derive maximum efficiency at given cooling power for Carnot-type low-dissipation refrigerators. The corresponding optimal cycle duration depends on a single parameter, which is a specific combination of irreversibility parameters and bath temperatures. For a slight decrease in power with respect to its maximum value, the maximum efficiency exhibits an infinitely fast nonlinear increase, which is standard in heat engines, only for a limited range of parameters. Otherwise, it increases only linearly with the slope given by ratio of irreversibility parameters. This behavior can be traced to the fact that maximum power is attained for vanishing duration of the hot isotherm. Due to the lengthiness of the full solution for the maximum efficiency, we discuss and demonstrate these results using simple approximations valid for parameters yielding the two different qualitative behaviors. We also discuss relation of our findings to those obtained for minimally nonlinear irreversible refrigerators.

Optimal Cycles for Low-Dissipation Heat Engines

We consider the optimization of a finite-time Carnot engine characterized by small dissipations. We bound the power with a simple inequality and show that the optimal strategy is to perform small cycles around a given working point, which can be, thus, chosen optimally. Remarkably, this optimal point is independent of the figure of merit combining power and efficiency that is being maximized. Furthermore, for a general class of dissipative dynamics the maximal power output becomes proportional to the heat capacity of the working substance. Since the heat capacity can scale supraextensively with the number of constituents of the engine, this enables us to design optimal many-body Carnot engines reaching maximum efficiency at finite power per constituent in the thermodynamic limit.

Energetic Self-Optimization Induced by Stability in Low-Dissipation Heat Engines

The local stability of a weakly dissipative heat engine is analyzed and linked to an energetic multi-objective optimization perspective. This constitutes a novel issue in the unified study of cyclic energy converters, opening the perspective to the possibility that stability favors self-optimization of thermodynamic quantities including efficiency, power and entropy generation. To this end, a dynamics simulating the restitution forces, which mimics a harmonic potential, bringing the system back to the steady state is analyzed. It is shown that relaxation trajectories are not arbitrary but driven by the improvement of several energetic functions. Insights provided by the statistical behavior of consecutive random perturbations show that the irreversible behavior works as an attractor for the energetics of the system, while the endoreversible limit acts as an upper bound and the Pareto front as a global attractor. Fluctuations around the operation regime reveal a difference between the behavior coming from fast and slow relaxation trajectories: while the former are associated to an energetic self-optimization evolution, the latter are ascribed to better performances. The self-optimization induced by stability and the possible use of instabilities in the operation regime to improve the energetic performance might usher into new useful perspectives in the control of variables for real engines.

Thermodynamic Geometry of Microscopic Heat Engines

We develop a general framework to describe the thermodynamics of microscopic heat engines driven by arbitrary periodic temperature variations and modulations of a mechanical control parameter. Within the slow-driving regime, our approach leads to a universal trade-off relation between efficiency and power, which follows solely from geometric arguments and holds for any thermodynamically consistent microdynamics. Focusing on Lindblad dynamics, we derive a second bound showing that coherence as a genuine quantum effect inevitably reduces the performance of slow engine cycles regardless of the driving amplitudes. To show how our theory can be applied in practice, we work out a specific example, which lies within the range of current solid-state technologies.

Performance optimization of low-dissipation thermal machines revisited

We revisit the optimization of performance of finite-time Carnot machines satisfying the low-dissipation assumption. The standard procedure seeks to optimize an objective function, such as power output of the engine, over the durations of contacts between the working medium and the heat reservoirs. This procedure may lead to unwieldy equations at the optimum of some objective functions. We propose an alternate scheme in which the output or input work is first optimized for a given cycle time, followed by an optimization of another objective function over the cycle time. This optimization problem is solved in a much simplified manner, with closed-form expressions for figures of merit. The approach is demonstrated for various objective functions, both for engines as well as refrigerators.

Optimization induced by stability and the role of limited control near a steady state

A relationship between stability and self-optimization is found for weakly dissipative heat devices. The effect of limited control on operation variables around an steady state is such that, after instabilities, the paths toward relaxation are given by trajectories stemming from restitution forces which improve the system thermodynamic performance (power output, efficiency, and entropy generation). Statistics over random trajectories for many cycles shows this behavior as well. Two types of dynamics are analyzed, one where an stability basin appears and another one where the system is globally stable. Under both dynamics there is an induced trend in the control variables space due to stability. In the energetic space this behavior translates into a preference for better thermodynamic states, and thus stability could favor self-optimization under limited control. This is analyzed from the multiobjective optimization perspective. As a result, the statistical behavior of the system is strongly influenced by the Pareto front (the set of points with the best compromise between several objective functions) and the stability basin. Additionally, endoreversible and irreversible behaviors appear as very relevant limits: The first one is an upper bound in energetic performance, connected with the Pareto front, and the second one represents an attractor for the stochastic trajectories.

Finite-power performance of quantum heat engines in linear response

We investigate the finite-power performance of quantum heat engines working in the linear response regime where the temperature gradient is small. The engine cycles with working substances of ideal harmonic systems consist of two heat transfer and two adiabatic processes, such as the Carnot cycle, Otto cycle, and Brayton cycle. By analyzing the optimal protocol under maximum power we derive the explicitly analytic expression for the irreversible entropy production, which becomes the low dissipation form in the long duration limit. Assuming the engine to be endoreversible, we derive the universal expression for the efficiency at maximum power, which agrees well with that obtained from the phenomenological heat transfer laws holding in the classical thermodynamics. Through appropriate identification of the thermodynamic fluxes and forces that a linear relation connects, we find that the quantum engines under consideration are tightly coupled, and the universality of efficiency at maximum power is confirmed at the linear order in the temperature gradient.

Cycling Tames Power Fluctuations near Optimum Efficiency

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 quasistatic 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 quasistatic ones are different stochastic processes.

Entropy generation and unified optimization of Carnot-like and low-dissipation refrigerators

The connection between Carnot-like and low-dissipation refrigerators is proposed by means of their entropy generation and the optimization of two unified, compromise-based figures of merit. Their optimization shows that only a limited set of heat transfer laws in the Carnot-like model are compatible with the results stemming from the low-dissipation approximation, even though there is an agreement of the related physical spaces of variables. A comparison between two operation regimes and relations among entropy generation, efficiency, cooling power. and power input are obtained, with emphasis on the role of dissipation symmetries. The results extend previous findings for heat engines at maximum power conditions.

General relations between the power, efficiency, and dissipation for the irreversible heat engines in the nonlinear response regime

We derive the general relations between the maximum power, maximum efficiency, and minimum dissipation for the irreversible heat engine in a nonlinear response regime. In this context, we use the minimally nonlinear irreversible model and obtain the lower and upper bounds of the above relations for the asymmetric dissipation limits. These relations can be simplified further when the system possesses the time-reversal symmetry or antisymmetry. We find that our results are the generalization of various such relations obtained earlier for different heat engines.

Diverging, but negligible power at Carnot efficiency: Theory and experiment

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, overdamped stochastic HEs and an underdamped 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.

Local-stability analysis of a low-dissipation heat engine working at maximum power output

In this paper we address the stability of a low-dissipation (LD) heat engine (HE) under maximum power conditions. The LD system dynamics are analyzed in terms of the contact times between the engine and the external heat reservoirs, which determine the amount of heat exchanged by the system. We study two different scenarios that secure the existence of a single stable steady state. In these scenarios, contact times dynamics are governed by restitutive forces that are linear functions of either the heat amounts exchanged per cycle, or the corresponding heat fluxes. In the first case, according to our results, preferably locating the system irreversibility sources at the hot-reservoir coupling improves the system stability and increases its efficiency. On the other hand, reducing the thermal gradient increases the system efficiency but deteriorates its stability properties, because the restitutive forces are smaller. Additionally, it is possible to compare the relaxation times with the total cycle time and obtain some constraints upon the system dynamics. In the second case, where the restitutive forces are assumed to be linear functions of the heat fluxes, we find that although the partial contact time presents a locally stable stationary value, the total cycle time does not; instead, there exists an infinite collection of steady values located in the neighborhood of the fixed point, along a one-dimensional manifold. Finally, the role of dissipation asymmetries on the efficiency, the stability, and the ratio of the total cycle time to the relaxation time is emphasized.

Route towards the optimization at given power of thermoelectric heat engines with broken time-reversal symmetry

We investigate the performance at a given power of a thermoelectric heat engine with broken time-reversal symmetry, and derive analytically the efficiency at a given power of a thermoelectric generator within linear irreversible thermodynamics. A universal bound on the efficiency of the thermoelectric heat engine is achieved under a strong constraint on the Onsager coefficients, and some interesting features are further revealed. Our results demonstrate that there exists a trade-off between efficiency and power output, and the efficiency at a given power may surpass the Curzon-Ahlborn limit due to broken time-reversal symmetry. Moreover, optimal efficiency at a given power can be achieved, which indicates that broken time-reversal symmetry offers physically allowed ways to optimize the performance of heat engines. Our study may contribute to the interesting guidelines for optimizing actual engines.

Heat engines at optimal power: Low-dissipation versus endoreversible model

The low-dissipation model and the endoreversible model of heat engines are two of the most commonly studied models of machines in finite-time thermodynamics. In this paper we compare the performance characteristics of these two models under optimal power output. We point out a basic equivalence between them, in the linear response regime.

From maximum power to a trade-off optimization of low-dissipation heat engines: Influence of control parameters and the role of entropy generation

For a low-dissipation heat engine model we present the role of the partial contact times and the total operational time as control parameters to switch from maximum power state to maximum Ω trade-off state. The symmetry of the dissipation coefficients may be used in the design of the heat engine to offer, in such switching, a suitable compromise between efficiency gain, power losses, and entropy change. Bounds for entropy production, efficiency, and power output are presented for transitions between both regimes. In the maximum power and maximum Ω trade-off cases the relevant space of parameters are analyzed together with the configuration of minimum entropy production. A detailed analysis of the parameter's space shows physically prohibited regions in which there is no longer a heat engine and another region that is physically well behaved but is not suitable for possible optimization criteria.

Efficiency and its bounds of minimally nonlinear irreversible heat engines at arbitrary power

The efficiency for minimally nonlinear irreversible heat engines at any arbitrary power has been systematically evaluated, and general lower and upper efficiency bounds under the tight coupling condition for different operating regions have been proposed, which can be seen as the generalization of the bounds [η_{C}/2<η_{maxP}<η_{C}/(2-η_{C})] on efficiency at maximum power (η_{maxP}), where η_{C} means the Carnot efficiency. We have also calculated the universal bounds of the maximum gain in efficiency in different operating regions to give further insight into the efficiency gain with the power away from the maximum power. In the region of higher loads (higher than the load which corresponds to the maximum power), a small power loss away from the maximum power induces a much larger gain in efficiency. As actual heat engines may not work at the maximum power condition, this paper may contribute to operating actual heat engines more efficiently.

Power-Efficiency-Dissipation Relations in Linear Thermodynamics

We derive general relations between the maximum power, maximum efficiency, and minimum dissipation regimes from linear irreversible thermodynamics. The relations simplify further in the presence of a particular symmetry of the Onsager matrix, which can be derived from detailed balance. The results are illustrated on a periodically driven system and a three-terminal device subject to an external magnetic field.

Maximum efficiency of steady-state heat engines at arbitrary power

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.

Optimal performance of periodically driven, stochastic heat engines under limited control

We consider the performance of periodically driven stochastic heat engines in the linear response regime. Reaching the theoretical bounds for efficiency and efficiency at maximum power typically requires full control over the design and the driving of the system. We develop a framework which allows us to quantify the role that limited control over the system has on the performance. Specifically, we show that optimizing the driving entering the work extraction for a given temperature protocol leads to a universal, one-parameter dependence for both maximum efficiency and maximum power as a function of efficiency. In particular, we show that reaching Carnot efficiency (and, hence, Curzon-Ahlborn efficiency at maximum power) requires to have control over the amplitude of the full Hamiltonian of the system. Since the kinetic energy cannot be controlled by an external parameter, heat engines based on underdamped dynamics can typically not reach Carnot efficiency. We illustrate our general theory with a paradigmatic case study of a heat engine consisting of an underdamped charged particle in a modulated two-dimensional harmonic trap in the presence of a magnetic field.

Optimizing work output for finite-sized heat reservoirs: Beyond linear response

We uncover an optimization principle for the finite-time heat-work conversion process performed between two finite-sized heat reservoirs in the nonlinear response regime that is characterized by rather generic flux-force relations. We solve the problem of maximizing work output in a given time interval by means of the variational method. Moreover, in the limiting case that the cold reservoir is infinite, we find the corresponding optimized process can be determined by a single quantity, which plays the role similar to that of the Hamiltonian in classical mechanics. Some theoretical implications are discussed consequently, under the generalized tight-coupling condition which applies to both linear and nonlinear response cases. Our results can hopefully help design and control realistic thermodynamical processes.