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The Carnot theorem, one expression of the second law of thermodynamics, places a fundamental upper bound on the efficiency of heat engines operating between two heat baths. The Carnot theorem can be stated in a more generalized form for heat engines operating with multiple heat baths, where the maximum efficiency is achieved for reversible heat engines operating quasistatically between two heat baths. In this study, we determine the irreversible efficiency of heat engines operating with multiple heat baths in a linear response regime, i.e., under small temperature differences and a slow variation of the control parameters, by quantifying the impact of the dissipation by irreversible operations. The Carnot theorem is derived as a natural consequence of it. Because the result obtained is based on the linear response relation and fluctuation-dissipation theorem in the universal framework of linear response theory, it has wide applicability to irreversible heat engines operating in the linear response regime., Comment: 6 pages, 1 figure

In existing linear response theories for adiabatically driven cyclic heat engines, Onsager symmetry is identified only phenomenologically, and a relation between global and local Onsager coefficients, defined over one cycle and at any instant of a cycle, respectively, is not derived. To address this limitation, we develop a linear response theory for the speed of adiabatically changing parameters and temperature differences in generic Gaussian heat engines obeying Fokker--Planck dynamics. We establish a hierarchical relationship between the global linear response relations, defined over one cycle of the heat engines, and the local ones, defined at any instant of the cycle. This yields a detailed expression for the global Onsager coefficients in terms of the local Onsager coefficients. Moreover, we derive an efficiency bound, which is tighter than the Carnot bound, for adiabatically driven linear irreversible heat engines based on the detailed global Onsager coefficients. Finally, we demonstrate the application of the theory using the simplest stochastic Brownian heat engine model., Comment: 8 pages (main text and supplemental material)

Achieving the Carnot efficiency at finite power is a challenging problem in heat engines due to the trade-off relation between efficiency and power that holds for general heat engines. It is pointed out that the Carnot efficiency at finite power may be achievable in the vanishing limit of the relaxation times of a system without breaking the trade-off relation. However, any explicit model of heat engines that realizes this scenario for arbitrary temperature difference has not been proposed. Here, we investigate an underdamped Brownian Carnot cycle where the finite-time adiabatic processes connecting the isothermal processes are tactically adopted. We show that in the vanishing limit of the relaxation times in the above cycle, the compatibility of the Carnot efficiency and finite power is achievable for arbitrary temperature difference. This is theoretically explained based on the trade-off relation derived for our cycle, which is also confirmed by numerical simulations., Comment: 16pages, 8 figures

The autonomous heat engine is a model system of autonomous nonequilibrium systems like biological cells, exploiting nonequilibrium flow for operations. As the Carnot engine has essentially contributed to the equilibrium thermodynamics, autonomous heat engine is expected to play a critical role in the challenge of constructing nonequilibrium thermodynamics. However, the high complexity of the engine involving an intricate coupling among heat, gas flow, and mechanics has prevented simple modeling. Here, we experimentally characterized the nonequilibrium dynamics and thermodynamics of a low-temperature-differential Stirling engine, which is a model autonomous heat engine. Our experiments demonstrated that the core engine dynamics are quantitatively described by a minimal dynamical model with only two degrees of freedom. The model proposes a novel concept that illustrates the engine as a thermodynamic pendulum driven by a thermodynamic force. This work will open a new approach to explore the nonequilibrium thermodynamics of autonomous systems based on a simple dynamical system., Comment: 6 pages, 7 figures

Low-temperature-differential (LTD) Stirling heat engines are able to operate with a small temperature difference between low-temperature heat reservoirs that exist in our daily lives, and thus they are considered to be an important sustainable energy technology. The author recently proposed a nonlinear dynamics model of an LTD kinematic Stirling heat engine to study the rotational mechanism of the engine [Y. Izumida, EPL \textbf{121}, 50004 (2018)]. This paper presents our study of the nonequilibrium thermodynamics analysis of this engine model, where a load torque against which the engine does work is introduced. We demonstrate that the engine's rotational state is in a quasi-linear response regime where the thermodynamic fluxes show a linear dependency on the thermodynamic forces. Significantly, it is found that the response coefficients of the quasi-linear relations are symmetric, which is similar to Onsager symmetry in linear irreversible thermodynamics. Based on these relations, we formulate the maximum efficiency of the engine. We also elucidate that the symmetry of the quasi-linear response coefficients emerges by reflecting the (anti-)reciprocity of the Onsager kinetic coefficients identified for the relaxation dynamics of the engine in the vicinity of an equilibrium state. We expect that the present study paves the way for developing nonequilibrium thermodynamics of autonomous heat engines described as a nonlinear dynamical system., Comment: 17 pages, 8 figures, accepted by Phys. Rev. E

The fast-forward (FF) scheme proposed by Masuda and Nakamura (\textit{Proc. R. Soc. A} \textbf{466}, 1135 (2010)) in the context of conservative quantum dynamics can reproduce a quasi-static dynamics in an arbitrarily short time. We apply the FF scheme to the classical stochastic Carnot-like heat engine which is driven by a Brownian particle coupled with a time-dependent harmonic potential and working between the high ($T_h$)- and low ($T_c$)-temperature heat reservoirs. Concentrating on the underdamped case where momentum degree of freedom is included, we find the explicit expressions for the FF protocols necessary to accelerate both the isothermal and thermally-adiabatic processes, and obtain the reversible and irreversible works. The irreversible work is shown to consist of two terms with one proportional to and the other inversely proportional to the friction coefficient. The optimal value of efficiency $\eta$ at the maximum power of this engine is found to be $\eta^*=\frac{1}{2} \left( 1+\frac{1}{2}\left(\frac{T_c}{T_h}\right)^{\frac{1}{2}} - \frac{5}{4}\frac{T_c}{T_h} +O\left(\left(\frac{T_c}{T_h}\right)^{\frac{3}{2}}\right)\right)$ and $\eta^*= 1- \left(\frac{T_c}{T_h}\right)^{\frac{1}{2}}$, respectively in the cases of strong and weak dissipation. The result is justified for a wide family of time scaling functions, making the FF protocols very flexible. We also revealed that the accelerated full cycle of the Carnot-like stochastic heat engine cannot be conceivable within the framework of the overdamped case, and the power and efficiency can be evaluated only when the momentum degree of freedom is taken into consideration., Comment: 14 pages, 3 figures

The low-temperature-differential (LTD) Stirling heat engine technology constitutes one of the important sustainable energy technologies. The basic question of how the rotational motion of the LTD Stirling heat engine is maintained or lost based on the temperature difference is thus a practically and physically important problem that needs to be clearly understood. Here, we approach this problem by proposing and investigating a minimal nonlinear dynamic model of an LTD kinematic Stirling heat engine. Our model is described as a driven nonlinear pendulum where the motive force is the temperature difference. The rotational state and the stationary state of the engine are described as a stable limit cycle and a stable fixed point of the dynamical equations, respectively. These two states coexist under a sufficient temperature difference, whereas the stable limit cycle does not exist under a temperature difference that is too small. Using a nonlinear bifurcation analysis, we show that the disappearance of the stable limit cycle occurs via a homoclinic bifurcation, with the temperature difference being the bifurcation parameter., Comment: 7 pages, 7 figures

Shortage of upward/downward operating reserve is concerned by integrated large amounts of variable renewable generations (VRGs). We analyzed characteristics of prediction errors of VRGs output by using historical weather data to calculate forecasted and actual output. We defined required operating reserve which considers prediction errors of VRGs, and evaluated shortage of operating reserve by unit commitment and economic load dispatch in a power system with large integration of VRGs. As a result, shortage of reserve is decreased by operating reserve which considers prediction errors of VRGs. Shortage of downward reserve can be avoided by decreasing units of gas-fired combined cycle when forecasted output of VRGs is lower.

We identify a velocity distribution function of ideal gas particles that is compatible with the local equilibrium assumption and the fundamental thermodynamic relation satisfying the endoreversibility. We find that this distribution is a Maxwell--Boltzmann distribution with a spatially uniform temperature and a spatially varying local center-of-mass velocity. We construct the local equilibrium Carnot cycle of an ideal gas, based on this distribution, and show that the efficiency of the present cycle is given by the endoreversible Carnot efficiency using the molecular kinetic temperatures of the gas. We also obtain an analytic expression of the efficiency at maximum power of our cycle under a small temperature difference. Our theory is also confirmed by a molecular dynamics simulation., 9 pages, 5 figures

Heat engines are ubiquitous in our industrial society and are also crucially important to the basis of thermodynamics. They allow us to extract useful work from the heat flowing from a hot heat reservoir into a cold heat reservoir by expanding and compressing the working substance. We have already known that the upper bound of the efficiencies of all the existing heat engines is limited by the Carnot efficiency ηC ≡ 1− Tc/Th where Th and Tc are the temperatures of the hot and the cold reservoirs, respectively. ηC can be realized only when heat engines are working infinitely slowly (quasistatic limit). This implies that the heat engine working in the quasistatic limit produces zero power because it takes infinite time to produce a finite amount of work. Though the Carnot efficiency is a fundamental result which manifests the natural law of the heat energy conversion and supplies certain guidance for designing efficient heat engines, it is not useful for describing the realistic heat engines which should produce finite power. Therefore, it is important to question: Are there any physical laws which can be applied to finite-time heat engines? Motivated by the above considerations, Curzon and Ahlborn studied the efficiency of a finite-time Carnot cycle and derived the result that the efficiency at the maximal power output ηmax becomes

We formulate an endoreversible finite-time Carnot cycle model based on the assumptions of local equilibrium and constant energy flux, where the efficiency and the power are expressed in terms of the thermodynamic variables of the working substance. By analyzing the entropy production rate caused by the heat transfer in each isothermal process during the cycle, and using an endoreversible condition applied to the linear response regime, we identify the thermodynamic flux and force of the present system and obtain a linear relation that connects them. We calculate the efficiency at maximum power in the linear response regime by using the linear relation, which agrees with the Curzon-Ahlborn efficiency known as the upper bound in this regime. This reason is also elucidated by rewriting our model into the form of the Onsager relations, where our model turns out to satisfy the tight-coupling condition leading to the Curzon-Ahlborn efficiency., Comment: 12 pages, 1 figure

We formulate the work output and efficiency for linear irreversible heat engines working between a finite-sized hot heat source and an infinite-sized cold heat reservoir until the total system reaches the final thermal equilibrium state with a uniform temperature. We prove that when the heat engines operate at the maximum power under the tight-coupling condition without heat leakage the work output is just half of the exergy, which is known as the maximum available work extracted from a heat source. As a consequence, the corresponding efficiency is also half of its quasistatic counterpart., Comment: 5 pages, 1 figure

We present results obtained by using nonlinear irreversible models for heat devices. In particular, we focus on the global performance characteristics, the maximum efficiency and the efficiency at maximum power regimes for heat engines, and the maximum coefficient of performance (COP) and the COP at maximum cooling power regimes for refrigerators. We analyze the key role played by the interplay between irreversibilities coming from heat leaks and internal dissipations. We also discuss the relationship between these results and those obtained by different models., Comment: 14 pages, 7 figures

We perform a molecular dynamics computer simulation of a heat engine model to study how the engine size difference affects its performance. Upon tactically increasing the size of the model anisotropically, we determine that there exists an optimum size at which the model attains the maximum power for the shortest working period. This optimum size locates between the ballistic heat transport region and the diffusive heat transport one. We also study the size dependence of the efficiency at the maximum power. Interestingly, we find that the efficiency at the maximum power around the optimum size attains a value that has been proposed as a universal upper bound, and it even begins to exceed the bound as the size further increases. We explain this behavior of the efficiency at maximum power by using a linear response theory for the heat engine operating under a finite working period, which naturally extends the low-dissipation Carnot cycle model [M. Esposito, R. Kawai, K. Lindenberg, C. Van den Broeck, Phys. Rev. Lett. 105, 150603 (2010)]. The theory also shows that the efficiency at the maximum power under an extreme condition may reach the Carnot efficiency in principle.

We apply the model of minimally nonlinear irreversible heat engines developed by Izumida and Okuda [EPL {\bf 97}, 10004 (2012)] to refrigerators. The model assumes extended Onsager relations including a new nonlinear term accounting for dissipation effects. The bounds for the optimized regime under an appropriate figure of merit and the tight-coupling condition are analyzed and successfully compared with those obtained previously for low-dissipation Carnot refrigerators in the finite-time thermodynamics framework. Besides, we study the bounds for the nontight-coupling case numerically. We also introduce a leaky low-dissipation Carnot refrigerator and show that it serves as an example of the minimally nonlinear irreversible refrigerator, by calculating its Onsager coefficients explicitly., Comment: Typo in eq.(34) is fixed

We study a Brownian Carnot cycle introduced by T. Schmiedl and U. Seifert [Europhys. Lett. \textbf{81}, 20003 (2008)] from a viewpoint of the linear irreversible thermodynamics. By considering the entropy production rate of this cycle, we can determine thermodynamic forces and fluxes of the cycle and calculate the Onsager coefficients for general protocols, that is, arbitrary schedules to change the potential confining the Brownian particle. We show that these Onsager coefficients contain the information of the protocol shape and they satisfy the tight-coupling condition irrespective of whatever protocol shape we choose. These properties may give an explanation why the Curzon-Ahlborn efficiency often appears in the finite-time heat engines.

We study the efficiency at the maximal power $\eta_\mathrm{max}$ of a finite-time Carnot cycle of a weakly interacting gas which we can reagard as a nearly ideal gas. In several systems interacting with the hot and cold reservoirs of the temperatures $T_\mathrm{h}$ and $T_\mathrm{c}$, respectively, it is known that $\eta_\mathrm{max}=1-\sqrt{T_\mathrm{c}/T_\mathrm{h}}$ which is often called the Curzon-Ahlborn (CA) efficiency $\eta_\mathrm{CA}$. For the first time numerical experiments to verify the validity of $\eta_\mathrm{CA}$ are performed by means of molecular dynamics simulations and reveal that our $\eta_\mathrm{max}$ does not always agree with $\eta_\mathrm{CA}$, but approaches $\eta_\mathrm{CA}$ in the limit of $T_\mathrm{c} \to T_\mathrm{h}$. Our molecular kinetic analysis explains the above facts theoretically by using only elementary arithmetic., Comment: 6 pages, 4 figures