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In this work, we study the dynamic robustness of an endoreversible Carnot cycle working at the maximum per-unit-time performance regime, based on the linearization technique for dynamical systems and the local stability analysis. Our analysis is focused on the endoreversible Carnot refrigerator model, which works in the maximum per-unit-time coefficient of performance. At the steady-state of the maximum performance, the expressions of the relaxation times describing the stability of the system are derived. It is found that the relaxation times in the cycle condition are the function of thermal conductances σh and σc, the temperatures of the heat reservoirs Th and Tc, and the heat capacity C. The influence of the temperature ratio τ = Tc/Th and the thermal conductance ratio σr = σh/σc on the relaxation times is discussed in detail. The results obtained here are useful and provide a potential guidance for the design of an endoreversible Carnot refrigerator working in the maximum performance per cycle time optimization condition.

In this paper, the model of a three-terminal quantum dot heat engine consists of two quantum dots with a single energy level, a cavity and two electron reservoirs is established. According to Landauer formula the expressions for the heat current, the output power and the efficiency are derived analytically. The performance characteristic curves of the output power versus the efficiency are plotted by numerical calculation. Moreover, the optimal performance parameters are determined. Then we optimize the output power and the efficiency respectively, the influence of the width of energy level and the heat leak on performance of the three-terminal thermoelectric heat engine is discussed. Lastly, the variation of the corresponding efficiency at the maximum power output with the Carnot efficiency between two reservoirs is discussed and the corresponding efficiency is compared with Carnot efficiency and CA efficiency. It is shown that this three-terminal heat engine is irreversible due to the existence of the width of energy level and the heat leak. Thus the curve of the power output versus the efficiency is a loop-shaped one. And due to the heat leak, the characteristics of the efficiency and the width of energy level is a non-monotone curve. The efficiency at the maximum power will be greater than the CA efficiency when the heat leak and the width of energy level are not taken into account.

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.

We consider the finite-time performance of a quantum Otto engine working between a hot squeezed and a cold thermal bath at inverse temperatures β_{h} and β_{c}(>β_{h}) with (k_{B}≡1)β=1/T. We derive the analytical expressions for work, efficiency, power, and power fluctuations, in which the squeezing parameter is involved. By optimizing the power output with respect to two frequencies, we derive the efficiency at maximum power as η_{mp}=(η_{C}^{gen})^{2}/[η_{C}^{gen}-(1-η_{C}^{gen})ln(1-η_{C}^{gen})], where the generalized Carnot efficiency η_{C}^{gen} in the high-temperature or small squeezing limit simplifies to an analytic function of squeezing parameter γ: η_{C}^{gen}=1-β_{h}/[β_{c}cosh(2γ)]. Within the context of irreversible thermodynamics, we demonstrate that the expression of efficiency at maximum power satisfies a general form derived from nonlinear steady state heat engines. We show that, the power fluctuations are considerably increased, although the engine efficiency is enhanced by squeezing.

We study the minimally nonlinear irreversible heat engines in which the time-reversal symmetry for the systems may be broken. The expressions for the power and the efficiency are derived, in which the effects of the nonlinear terms due to dissipations are included. We show that, as within the linear responses, the minimally nonlinear irreversible heat engines can enable attainment of Carnot efficiency at positive power. We also find that the Curzon-Ahlborn limit imposed on the efficiency at maximum power can be overcome if the time-reversal symmetry is broken.

We study the performance of a cyclic heat engine which uses a small system with a finite number of ultracold atoms as its working substance and works between two heat reservoirs at constant temperatures T h and T c (

The Diesel refrigeration cycle using an ideal quantum gas as the working substance is called quantum Diesel refrigeration cycle, which is different from Carnot, Ericsson, Brayton, Otto and Stirling refrigeration cycles. For ideal quantum gases, a corrected equation of state, which considers the quantum behavior of gas particles, is used instead of the classical one. The purpose of this paper is to investigate the effect of quantum gas as the working substance on the performance of a quantum Diesel refrigeration cycle. It is found that coefficients of performance of the cycle are not affected by the quantum degeneracy of the working substance, which is the same as that of the classical Diesel refrigeration cycle. However, the refrigeration load is different from those of the classical Diesel refrigeration cycle. Lastly, the influence of the quantum degeneracy on the performance characteristics of the quantum Diesel refrigeration cycle operated in different temperature regions is discussed.

The local stability of an irreversible Carnot heat engine has been studied based on the linearization technique for dynamical systems and local stability analysis. At two steady-states of the maximum power output and the maximum efficiency the expressions of the relaxation time of an irreversible Carnot heat engine are derived. It is found that the relaxation time is a function of the heat-transfer coefficient α and β, heat capacity C, temperatures of the heat reservoirs TH and TL, the degree of internal irreversibility ϕ and the internal heat conductance k. The influence of heat resistance, internal irreversibility and heat leak on the relaxation time is discussed. Phase portraits for the trajectories are presented in some representative cases. The results obtained here are more general and useful for the realistic irreversible heat engine than endoreversible heat engine.

The new model of a quantum refrigeration cycle composed of two adiabatic and two isomagnetic field processes is established. The working substance in the cycle consists of many non-interacting spin-1/2 systems. The performance of the cycle is investigated, based on the quantum master equation and semi-group approach. The general expressions of several important performance parameters, such as the coefficient of performance, cooling rate and power input, are given. It is found that the coefficient of performance of this cycle is a close analogue of the classical Carnot-cycle. Some performance characteristic curves relating the cooling rate, the coefficient of performance and power input are plotted. Further, for high temperatures, the optimal relations between the cooling rate and the coefficient of performance are analyzed in detail.

We propose and theoretically investigate a system of two coupled harmonic oscillators as a heat engine. We show how these two coupled oscillators within undamped regime can be controlled to realize an Otto cycle that consists of two adiabatic and two isochoric processes. During the two isochores the harmonic system is embedded in two heat reservoirs at constant temperatures T(h) and T(c)(

The irreversible model of a quantum refrigeration cycle composed of two adiabatic and two isomagnetic field processes is established. The working substance in the cycle consists of many noninteracting spin-1/2 systems. The performance of the cycle is investigated, based on the quantum master equation and semi-group approach. The general expressions of several important performance parameters, such as the coefficient of performance, cooling rate, and power input, are given. It is found that the coefficient of performance of this cycle is close analogues of that of classical Carnot cycle. Some performance characteristics curves between the cooling rate and the maximum “temperature” ratio of the working substances are plotted. Further, at high temperatures the optimal relations of the cooling rate and the maximum cooling rate are analyzed in detail. The results obtained are further generalized and discussed, so that they may be directly used to describe the performance of the quantum refrigerator using spin- J systems as the working substance. Finally, the optimum characteristics of the quantum Carnot and Ericsson refrigeration cycles are derived analogously.

We propose an Otto heat engine that undergoes processes involving a special class of feedback and analyze theoretically its response. We use stochastic thermodynamics to determine the performance characteristics of the heat engine and indicate the possibility that its maximum efficiency can surpass the Carnot value. The analytical expression for efficiency at maximum power, including the effects resulting from feedback, reduces to that previously derived based on an engine without feedback.

We study the coefficient of performance (COP) and its bounds for a Carnot-like refrigerator working between two heat reservoirs at constant temperatures ${T}_{h}$ and ${T}_{c}$, under two optimization criteria $\ensuremath{\chi}$ and $\ensuremath{\Omega}$. In view of the fact that an ``adiabatic'' process usually takes finite time and is nonisentropic, the nonadiabatic dissipation and the finite time required for the adiabatic processes are taken into account by assuming low dissipation. For given optimization criteria, we find that the lower and upper bounds of the COP are the same as the corresponding ones obtained from the previous idealized models where any adiabatic process is undergone instantaneously with constant entropy. To describe some particular models with very fast adiabatic transitions, we also consider the influence of the nonadiabatic dissipation on the bounds of the COP, under the assumption that the irreversible entropy production in the adiabatic process is constant and independent of time. Our theoretical predictions match the observed COPs of real refrigerators more closely than the ones derived in the previous models, providing a strong argument in favor of our approach.

We consider the finite-time operation of a quantum heat engine whose working substance is composed of a two-level atomic system. The engine cycle, consisting of two quantum adiabatic and two quantum isochoric (constant-frequency) processes and working between two heat reservoirs at temperatures T(h) and T(c)(

A Brownian refrigerator with the cold and hot reservoirs alternating along a space coordinate is established. The heat flux couples with the movement of the Brownian particles due to an external force in the spatially asymmetric but periodic potential. After using the Arrhenius factor to describe the behaviors of the forward and backward jumps of the particles, the expressions for coefficient of performance (COP) and cooling rate are derived analytically. Then, through maximizing the product of conversion efficiency and heat flux flowing out, a new upper bound only depending on the temperature ratio of the cold and hot reservoirs is found numerically in the reversible situation, and it is a little larger than the so-called Curzon and Ahlborn COP ε(CA)=(1/√[1-τ])-1. After considering the irreversible factor owing to the kinetic energy change of the moving particles, we find the optimized COP is smaller than ε(CA) and the external force even does negative work on the Brownian particles when they jump from a cold to hot reservoir.

We generalize the quantum heat engine (QHE) model which was first proposed by Bender et al. [J. Phys. A 33, 4427 (2000)] to the case in which an ideal Fermi gas with an arbitrary number N of particles in a box trap is used as the working substance. Besides two quantum adiabatic processes, the engine model contains two isoenergetic processes, during which the particles are coupled to energy baths at a high constant energy E(h) and a low constant energy E(c), respectively. Directly employing the finite-time thermodynamics, we find that the power output is enhanced by increasing particle number N (or decreasing minimum trap size L(A)) for given L(A) (or N), without reduction in the efficiency. By use of global optimization, the efficiency at possible maximum power output (EPMP) is found to be universal and independent of any parameter contained in the engine model. For an engine model with any particle-number N, the efficiency at maximum power output (EMP) can be determined under the condition that it should be closest to the EPMP. Moreover, we extend the heat engine to a more general multilevel engine model with an arbitrary 1D power-law potential. Comparison between our engine model and the Carnot cycle shows that, under the same conditions, the efficiency η = 1 - E(c)/E(h) of the engine cycle is bounded from above the Carnot value η(c) =1 - T(c)/T(h).

We study the efficiency at maximum power, $\eta_m$, of irreversible quantum Carnot engines (QCEs) that perform finite-time cycles between a hot and a cold reservoir at temperatures $T_h$ and $T_c$, respectively. For QCEs in the reversible limit (long cycle period, zero dissipation), $\eta_m$ becomes identical to Carnot efficiency $\eta_{_C}=1-\frac{T_c}{T_h}$. For QCE cycles in which nonadiabatic dissipation and time spent on two adiabats are included, the efficiency $\eta_m$ at maximum power output is bounded from above by $\frac{\eta_{_C}}{2-\eta_{_C}}$ and from below by $\frac{\eta_{_C}}2$. In the case of symmetric dissipation, the Curzon-Ahlborn efficiency $\eta_{_{CA}}=1-\sqrt{\frac{T_c}{T_h}}$ is recovered under the condition that the time allocation between the adiabats and the contact time with the reservoir satisfy a certain relation., Comment: to be published in Phys. Rev. E (2012)

We present a performance analysis of a two-state heat engine model working with a single-mode radiation field in a cavity. The heat engine cycle consists of two adiabatic and two isoenergetic processes. Assuming the wall of the potential moves at a very slow speed, we determine the optimization region and the positive work condition of the heat engine model. Furthermore, we generalize the results to the performance optimization for a two-state heat engine with a one-dimensional power-law potential. Based on the generalized model with an arbitrary one-dimensional potential, we obtain the expression of efficiency as $\ensuremath{\eta}=1\ensuremath{-}\frac{{E}_{C}}{{E}_{H}}$, with ${E}_{H}$ (${E}_{C}$) denoting the expectation value of the system Hamiltonian along the isoenergetic process at high (low) energy. This expression is an analog of the classical thermodynamical result of Carnot, ${\ensuremath{\eta}}_{c}=1\ensuremath{-}\frac{{T}_{C}}{{T}_{H}}$, with ${T}_{H}$ (${T}_{C}$) being the temperature along the isothermal process at high (low) temperature. We prove that under the same conditions, the efficiency $\ensuremath{\eta}=1\ensuremath{-}\frac{{E}_{C}}{{E}_{H}}$ is bounded from above the Carnot efficiency, ${\ensuremath{\eta}}_{c}=1\ensuremath{-}\frac{{T}_{C}}{{T}_{H}}$, and even quantum dynamics is reversible.

We present an approach to the energetics of a diode based on the Langevin equation for an ideal $RC$ circuit and establish a simple model consisting of two diodes switched in the same directions located in two heat reservoirs with different temperatures. We calculate the thermoelectric voltage and the efficiency of the diode system as a fluctuation rectifier and show the efficiency is always smaller than the Carnot value, thus, indicating the irreversible mode of operation. The Carnot efficiency is achieved only when the diode works as the normal resistor.

Based on quantum thermodynamic processes, we make a quantum-mechanical (QM) extension of the typical heat engine cycles, such as the Carnot, Brayton, Otto, Diesel cycles, etc., with no introduction of the concept of temperature. When these QM engine cycles are implemented by an ideal gas confined in an arbitrary power-law trap, a relation between the quantum adiabatic exponent and trap exponent is found. The differences and similarities between the efficiency of a given QM engine cycle and its classical counterpart are revealed and discussed.

The cycle model of a quantum refrigerator composed of two isothermal and two isomagnetic field processes is established. The working substance in the cycle consists of many noninteracting spin-1/2 systems. The performance of the cycle is investigated, based on the quantum master equation and semigroup approach. The general expressions of several important performance parameters, such as the coefficient of performance, cooling rate, and power input, are given. Especially, the case at high temperatures is analyzed in detail. The results obtained are further generalized and discussed, so that they may be directly used to describe the performance of the quantum refrigerator using spin-J systems as the working substance. Finally, the optimum characteristics of the quantum Carnot refrigerator are derived simply.

We setup a three-level heat engine model that works with two noninteracting fermions in a one-dimensional box trap. Besides two quantum adiabatic processes, the quantum heat engine cycle consists of two isoenergetic processes, along which the particles are coupled to energy baths at a high constant energy EH and a low constant energy EC, respectively. Based on the assumption that the potential wall moves at a very slow speed and there exists a heat leakage between two energy baths, we derive the expressions of the power output and the efficiency, and then obtain the optimization region for the heat engine cycle. Finally, we present a brief performance analysis of a Carnot engine between a hot and a cold bath at temperatures TH and TC, respectively. We demonstrate that under the same conditions, the efficiency η=1-(EC/EH) of the engine cycle is bounded from above the Carnot efficiency ηc=1-(TC/TH).

Particle hopping on a one-dimensional lattice driven by an external force in a periodic sawtooth potential and temperature field may act as a micro-Brownian refrigerator. In order to clarify the underlying physical pictures of the refrigerator, heat flows via both the potential energy and the kinetic energy of the particle are considered simultaneously. Based on the master equation describing the jump of the particle among the three states, expressions for the cooling rate and the coefficient of performance of the refrigerator are derived analytically. The general performance characteristic curves are plotted by numerical calculation. It is found that the characteristic curve between the cooling rate and the coefficient of performance is a loop-shaped one; the Brownian refrigerator is irreversible and its coefficient of performance is always less than the Carnot value. The influence of the temperature ratio of the heat reservoirs and the height of the sawtooth potential on the optimal performance characteristic parameters is analyzed. When heat flow via the kinetic energy of the particle is neglected, the characteristic curve between the cooling rate and the coefficient of performance is an open-shaped one. In this case, the Brownian refrigerator is reversible and its coefficient of performance reaches the Carnot value in the quasistatic limit.

A local stability analysis of an endoreversible Carnot refrigerator, working at the maximum objective function of the product of the cooling rate R and the coefficient of performance e, is presented. The endoreversible Carnot refrigerator consists of a reversible Carnot refrigerator that exchanges heat with the heat reservoirs TH through the thermal conductance α and with the cold reservoirs TL through the thermal conductance β. In addition, the working fluid has the same heat capacity C in the two isothermal branches of the cycle. By linearization and stability analysis, we find that the relaxation times are a function of α, β, the heat capacity C and τ=TL /TH; that the endoreversible Carnot refrigerator is stable for every value of α, β, C and τ; that after a perturbation, the system state exponentially decays to the steady state with either of two different relaxation times; that both relaxation times are proportional to α/2C; and that one of them is a monotonically increasing function τ and the other is almost independent of τ. Finally, the phase portraits for the trajectories after a small perturbation over the steady-state values of internal temperatures are presented.

In this paper, the work output (W) and efficiency (η) of a Carnot cycle in micro-/nanoscale are calculated, in which the quantum boundary effect of gas particles as a working substance of cycle is considered. It is found that under the quantum boundary effect conditions, the work output of cycle depends sensitively on the surface area of the system (boundary of cycle) and cannot be predicted by the classical approximate result (WC), while the efficiency of cycle is independent of the surface area of the system and equals the Carnot efficiency (ηC). Further, the difference of work output of the micro-/nanoscaled Carnot cycle to that of the classical one, ΔW=W−WC, is introduced and can be expressed as the function of the temperature ratio τ=TL/TH of the two heat reservoirs, the volume ratio rV=V3/V1, and the surface area ratio rA=A3/A1 of the two isochoric processes. Variations of difference of work output, ΔW, with the temperature ratio τ and volume ratio rV for the different surface ratio rA are examined,...