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A quantum Otto engine using two-interacting spins as its working medium is analyzed within framework of stochastic thermodynamics. The time-dependent power fluctuations and average power are explicitly derived for a complete cycle of engine operation. We find that the efficiency and power fluctuations are affected significantly by interparticle interactions, but both of them become interaction-independent under maximal power via optimizing the external control parameter. The behavior of the efficiency at maximum power is further explained by analyzing the optimal protocol of the engine.

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 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 propose a quantum-mechanical Brayton engine model that works between two superposed states, employing a single particle confined in an arbitrary power-law trap as the working substance. Applying the superposition principle, we obtain the explicit expressions of the power and efficiency, and find that the efficiency at maximum power is bounded from above by the function: η+ = θ/(θ + 1), with θ being a potential-dependent exponent.

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 (

We analyze general models of quantum heat engines operating a cycle of two adiabatic and two isothermal processes. We use the quantum master equation for a system to describe heat transfer current during a thermodynamic process in contact with a heat reservoir, with no use of phenomenological thermal conduction. We apply the endoreversibility description to such engine models working in the linear response regime and derive expressions of the efficiency and the power. By analyzing the entropy production rate along a single cycle, we identify the thermodynamic flux and force that a linear relation connects. From maximizing the power output, we find that such heat engines satisfy the tight-coupling condition and the efficiency at maximum power agrees with the Curzon-Ahlborn efficiency known as the upper bound in the linear response regime.

The performance in finite time of a quantum-mechanical Brayton engine cycle is discussed, without introduction of temperature. The engine model consists of two quantum isoenergetic and two quantum isobaric processes, and works with a single particle in a harmonic trap. Directly employing the finite-time thermodynamics, the efficiency at maximum power output is determined. Extending the harmonic trap to a power-law trap, we find that the efficiency at maximum power is independent of any parameter involved in the model, but depends on the confinement of the trapping potential.

Within an exact canonical-ensemble treatment, we investigate the thermodynamics for a finite number of ideal bosons confined in a three-dimensional quartic trap. We calculate several physical quantities including the specific heat C N , chemical potential μ, condensate fraction 〈n 0〉/N, root-mean-square fluctuations δn 0 of the condensate population, and transition temperature T c . We discuss the particle-number dependence of T c through proposing three T c definitions, which are compared with ones derived in the grand canonical ensemble.

An irreversible cycle model of the micro-/nanoscaled Otto engine cycle with internal friction loss is established. The general expressions of the work output and efficiency of the cycle are calculated based on the finite system thermodynamic theory, in which the quantum boundary effect of gas particles as working substance and the mechanical Casimir effect of gas system are considered. It is found that, for a micro-/nanoscaled Otto cycle devices, the work output W and efficiency η of the cycle can be expressed as the functions of the temperature ratio τ of the two heat reservoirs, the volume ratio rV and the surface area ratio rA of the two isochoric processes, the dimensionless thermal wavelength λ and other parameters of cycle, while for a macroscaled Otto cycle devices, the work output W0 and efficiency η0 of the cycle are independent of the surface area ratio rA and the dimensionless thermal wavelength λ. Further, the influence of boundary of cycle on the performance characteristics of the micro-/nanoscaled Otto cycle are analyzed in detail by introducing the output ratio W/W0 and efficiency ratio η/η0. The results present the general performance characteristics of a micro-/nanoscaled Otto cycle and may serve as the basis for the design of a realistic Otto cycle device in micro-/nanoscale.

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 aim of the paper is to present the performance characteristics of a Stirling refrigeration cycle in micro/nano scale, in which the working substance of cycle is an ideal Maxwellian gas. Due to the quantum boundary effect on the gas particles confined in the finite domain, the cycle no longer possesses the condition of perfect regeneration. The inherent regenerative losses, the refrigeration heat and coefficient of performance (COP) of the cycle are derived. It is found that, for the micro/nano scaled Stirling refrigeration cycle devices, the refrigeration heat and COP of cycle all depend on the surface area of the system (boundary of cycle) besides the temperature of the heat reservoirs, the volume of system and other parameters, while for the macro scaled refrigeration cycle devices, the refrigeration heat and COP of cycle are independent of the surface area of the system. Variations of the refrigeration heat ratio r R and the COP ratio r e with the temperature ratio τ and volume ratio r V for the different surface area ratio r A are examined, which reveals the influence of the boundary of cycle on the performance of a micro/nano scaled Stirling refrigeration cycle. The results are useful for designing of a micro/nano scaled Stirling cycle device and may conduce to confirming experimentally the quantum boundary effect in the micro/nano scaled devices.

An irreversible cycle model of the quantum Brayton refrigeration cycle is established, in which finite-time processes and irreversibility in the two adiabatic processes are taken into account. On the basis of the thermodynamic properties of an ideal Bose gas, by using the optimal control-theory, the mathematical expressions for several important performance parameters, such as the coefficient of performance, power input and cooling load, are derived and some important performance parameters, e.g., the temperatures of the working substance at several important state-points, are optimized. By means of numerical predictions, the optimal performance characteristic curves of a Bose–Brayton refrigeration cycle are obtained and analyzed. Furthermore, some optimal operating regions including those for the cooling load, coefficient of performance and the temperatures of the cyclic working substance at the two important state-points are determined and evaluated. Finally, several special cases are discussed in detail.

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.

In a recent paper [A. Sisman, I. Muller, Phys. Lett. A 320 (2004) 360] the thermodynamic properties of ideal gases confined in a narrow box were examined theoretically. The so-called “thermosize effects” similar to thermoelectric effects, such as Seebeck-like thermosize effect, Peltier-like thermosize effect and Thomson-like thermosize effect, were analyzed. Like the thermoelectric generator, based on the thermosize effects we have established a model of micro/nano scaled ideal gas heat engine cycle which includes two isothermal and two isobaric processes. The expressions of power output and efficiency of this cycle in the two cases of reversible and irreversible heat exchange are derived and the optimal performance characteristics of the heat engine is discussed by some numerical example. The results obtained here will provide theoretical guidance for the design of micro/nano scaled device.

Based on the thermodynamic properties of a paramagnetic salt, an irreversible model of the magnetic Brayton refrigeration cycle is established, in which the working substance is a special paramagnetic material. The expressions of the important performance parameters, such as the coefficient of performance, refrigeration load and work input, are derived. Moreover, the optimal performance parameters are obtained at the maximum coefficient of performance. The results obtained here may include the ones of the magnetic Brayton refrigeration cycle using the magnetic material obeyed the Curie law as the working substance, the magnetic Brayton refrigeration cycle without regeneration and the eversible magnetic Brayton refrigeration cycle. Therefore, the results obtained here have general significance and will be helpful to deeply understand the performance of a magnetic Brayton refrigeration cycle.

An irreversible cycle model of a quantum heat engine with internal friction is established, which is composed of two adiabatic and two isomagnetic field processes. The working substance of the cycle consists of an ensemble of many non-interacting spin-1/2 systems. Based on a quantum master equation and semi-group approach, the general performance characteristics of the heat engine are investigated. The general expressions for several important parameters, such as the efficiency, power output, and rate of the entropy production, are derived. The performance of the cycle is optimized with respect to the temperatures of the working substance. By numerical solutions, the maximum power output and the corresponding parameters are calculated. The optimal regions of efficiency, temperatures of the working substance, and cycle period are determined. Moreover, the performance of the heat engine in the frictionless case is obtained, which is different from that in the friction case. Finally, the results obtained are generalized to the performance optimization of the heat engine working with spin-J systems.

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)(

We propose a quantum absorption refrigerator driven by photons. The model uses a four-level system as its working substance and couples simultaneously to hot, cold, and solar heat reservoirs. Explicit expressions for the cooling power ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{Q}}_{c}$ and coefficient of performance (COP) ${\ensuremath{\eta}}_{\text{COP}}$ are derived, with the purpose of revealing and optimizing the performance of the device. Our model runs most efficiently under the tight coupling condition, and it is consistent with the third law of thermodynamics in the limit $T\ensuremath{\rightarrow}0$.

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 consider the efficiency at maximum power of a quantum Otto engine, which uses a spin or a harmonic system as its working substance and works between two heat reservoirs at constant temperatures ${T}_{h}$ and ${T}_{c}$ $(l{T}_{h})$. Although the behavior of spin-$1/2$ system differs substantially from that of the harmonic system in that they obey two typical quantum statistics, the efficiencies at maximum power based on these two different kinds of quantum systems are bounded from the upper side by the same expression ${\ensuremath{\eta}}_{\mathrm{mp}}\ensuremath{\le}{\ensuremath{\eta}}_{+}\ensuremath{\equiv}{\ensuremath{\eta}}_{C}^{2}/[{\ensuremath{\eta}}_{C}\ensuremath{-}(1\ensuremath{-}{\ensuremath{\eta}}_{C})ln(1\ensuremath{-}{\ensuremath{\eta}}_{C})]$ with ${\ensuremath{\eta}}_{C}=1\ensuremath{-}{T}_{c}/{T}_{h}$ as the Carnot efficiency. This expression ${\ensuremath{\eta}}_{\mathrm{mp}}$ possesses the same universality of the CA efficiency ${\ensuremath{\eta}}_{\mathrm{CA}}=1\ensuremath{-}\sqrt{1\ensuremath{-}{\ensuremath{\eta}}_{C}}$ at small relative temperature difference. Within the context of irreversible thermodynamics, we calculate the Onsager coefficients and show that the value of ${\ensuremath{\eta}}_{\mathrm{CA}}$ is indeed the upper bound of EMP for an Otto engine working in the linear-response regime.

With the help of thermodynamic properties of paramagnetic materials based on statistical mechanics, the performance of an irreversible regenerative magnetic Brayton refrigeration cycle is investigated. Expressions of some important parameters such as the coefficient of performance, refrigeration load and work input are derived analytically and are used to reveal the general performance characteristics of the cycle. Moreover, the minimum ratio and the lower bound of the optimal ratio of two magnetic fields are determined and the optimal criteria of the cyclic parameters obtained. Several special cases are discussed in detail. The results obtained will be helpful to deeply understand the performance of an irreversible regenerative magnetic Brayton refrigeration cycle.

The performance of a ferroelectric Ericsson refrigeration cycle is investigated on the basis of the statistic relation between the electrical polarization and the electric field strength of the ferroelectric materials. The inherent regenerative losses in the cycle are calculated. The coefficients of performance of the cycle are derived. Moreover, the performance of the Ericsson refrigeration cycle using other dielectric materials as the working substance is discussed. The results obtained here may reveal the general characteristics of the electrocaloric Ericsson refrigeration cycle.

The thermodynamic properties of the dielectric materials are analyzed in detail, based on the Curie law, Curie–Weiss law and other relations between the electrical polarization and the electric field strength. It is proven that the specific heat at constant electrical polarization is only a function of temperature, while the specific heat at constant electric field strength is dependent on the electric field strength and temperature. Moreover, the regenerative characteristics of the electrocaloric Stirling and Ericsson refrigeration cycles are discussed. Some important conclusions are obtained.

The performance of the Ericsson refrigeration cycle using an ideal Fermi gas as the working substance, which is simply referred to as the Fermi Ericsson refrigeration cycle, is examined, based on the equation of state of an ideal Fermi gas. The inherent regenerative losses and the coefficient of performance (COP) of the cycle are calculated. Three special cases are discussed in detail. It is found that under the conditions of low and high temperatures, the COP of the cycle is only a function of temperature, while under other conditions, the COP of the cycle is not only a function of temperature, but also is dependent on the pressures and other parameters. The results obtained here may reveal the general performance characteristics of the Fermi Ericsson refrigeration cycle.

The influence of quantum degeneracy on the performance of a Stirling refrigeration cycle is investigated, based on the equation of state of an ideal Fermi gas. The inherent regenerative losses and the coefficient of performance (COP) of the cycle are calculated. It is found that, under the condition of strong gas degeneracy, the COP of the cycle in the first approximation is a function only of the temperatures of the heat reservoirs, while under other conditions, the COPs of the cycle depend on the temperatures of the heat reservoirs and other parameters of the cycle. The results obtained here reveal the general performance characteristics of a Stirling refrigeration cycle having a Fermi gas as its working substance.

In this paper, the energy selective electron (ESE) refrigerator with an ideal energy filter based on resonant tunneling is established. It consists of two infinitely large electron reservoirs with different temperatures and chemical potentials, and electrons can be exchanged between the two reservoirs through the ideal energy filter. According to Landauer formula and the assumption of being coupled tightly with the electron current, the expressions for the heat flux, the cooling rate and the coefficient of performance (COP) are derived analytically. The performance characteristic curves such as the cooling rate versus coefficient of performance, the cooling rate and coefficient of performance versus the position of energy level are plotted by numerical calculation. The optimal regions of the cooling rate and the COP are determined. Moreover, we optimize the cooling rate and the COP respectively with respect to the position of energy level. The influence of the width of energy level on performance of the ESE refrigerator is discussed. Finally, based on the optimization criterion for refrigerator, i.e. the product of the COP times the cooling rate, the optimal performance of the ESE refrigerator is discussed in detail.

A two-level atomic system as a working substance is used to set up a refrigerator consisting of two quantum adiabatic and two isochoric processes (two constant-frequency processes $\omega_a$ and $\omega_b$ with $\omega_a

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 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)(

本文研究了一种由周期性非均匀线性温场所驱动的布朗制冷机性能特征. 利用布朗粒子的动力学方程——Langevin equation, 获得了稳态时布朗制冷机的粒子流、制冷率和制冷系数的解析表达式. 通过数值计算分析了势垒高度、非对称性、外力和温比等参数对制冷机性能的影响. 在最大品质因子条件下, 对制冷机性能参数进行优化, 画出了优化的制冷系数和制冷率随温比的演化曲线. 研究表明: 在准静态极限下, 制冷系数趋于用等效温度给出的卡诺制冷系数, 制冷率趋于零; 制冷系数随制冷率增加而减少; 优化的制冷系数随温比减小而下降, 而对应的制冷率却相反; 相比于在分段均匀温度场中, 非均匀线性温度场将导致制冷系数的提高.

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 establish a quantum Otto engine (QOE) of a two-level atom, which is confined in a one-dimensional (1D) harmonic trap and is coupled to single-mode radiation fields. Besides two adiabatic processes, the QOE cycle consists of two isochoric processes, along one of which the two-level atom as the working substance interacts with a single-mode radiation field. Based on the semigroup approach, we derive the time for completing any adiabatic process and then present a performance analysis of the heat engine model. Furthermore, we generalize the results to the performance optimization for a QOE of a single two-level atom trapped in a 1D power-law potential. Our result shows that the efficiency at maximum power output is dependent on the trap exponent $\ensuremath{\theta}$ but is independent of the energy spectrum index $\ensuremath{\sigma}$.

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.

The Ericsson refrigeration cycle working with an ideal Bose gas is called the Bose Ericsson refrigeration cycle. The effect of quantum degeneracy on the performance of the cycle is investigated, based on the thermodynamicproperties of an ideal Bose gas. The inherent regenerative losses of the cycle are analyzed, and the coefficient of performance and the refrigeration load of the cycle are calculated. The results obtained here are compared with those derived from the classical Ericsson refrigeration cycle using an ideal gas as the working substance General performance characteristics of the Bose Ericsson refrigeration cycle are revealed.

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.

In this paper, we establish a nanothermoelectric engine consisting of two discrete energy levels embedded between two reservoirs at different temperatures and chemical potentials. Based on master equation, the expressions for the power output and efficiency of the nanothermoelectric engine are derived. The characteristic curves between the power output and the efficiency are plotted. Moreover, the optimal performance parameters are obtained by the numerical calculation. The influence of the strength of variations in electron–electron interactions on the optimal performance parameters is analyzed in detail.

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.

An energy selective electron (ESE) refrigerator with heat leaks is established in a one-dimensional system. Based on the theory of electronic transport, the expressions of the heat flux into hot and cold electron reservoirs are derived. When the heat leaks between two electron reservoirs via phonons are taken into account, the cooling rate, coefficient of performance (COP) and input power are obtained. The performance characteristic curves such as the cooling rate versus the COP, the cooling rate and the COP versus the center position of the resonance energy level are plotted by numerical calculation. The influence of the center position and width of the resonance energy level on the performance of the ESE refrigerator is analyzed in detail. Lastly, the influence of heat leaks and average temperature on the performance of the ESE refrigerator is discussed. The results obtained here have theoretical significance for understanding the thermodynamic performance of the practical ESE refrigerator.

An irreversible cycle model of a Bose–Otto engine is established, in which finite time thermodynamic processes and the irreversibility result from the nonisentropic compression and expansion processes are taken into account. Based on the model, expressions for the power output and efficiency of the Bose–Otto engine are derived. On the basis of the thermodynamic properties of ideal Bose gas, the effects of the irreversibility and the compression ratio of the two isochoric processes on the performance of the Bose–Otto engine are revealed and some important performance parameters are optimized. Furthermore, some optimal operating regions including those for the power output, efficiency, and the temperatures of the cyclic working substance at two important state points are determined and evaluated. Finally, several special cases are discussed in detail.

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,...

Based on the so-called thermosize effects similar to thermoelectric effects, the cycle model of a micro-/nanothermosize refrigerator composed of two isothermal and two isobaric processes is established. The working fluid in this cycle consists of atoms in an ideal gas confined in micro-/nanoscale. The detailed expressions for the rate of refrigeration and coefficient of performance are derived in the two cases of reversible and irreversible heat exchanges. Some important performance characteristic curves are plotted. The optimal performance parameters of the micro-/nanothermosize refrigerator are obtained by numerical calculation. The results obtained here will provide theoretical guidance for designing of a micro-/nanoscaled refrigerator.