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Based on the classification scheme of phase transitions, we study the phase transitions for an ideal Bose gas with a finite number of particles confined in a three-dimensional quartic trap. We show that the phase transition of an ideal Bose gas in the three-dimensional quartic trap is of third order for finite particle numbers, quite different from the fact that the phase transition is of first order in the thermodynamic limit. We discuss the effects of finite particle numbers on the nature of the phase transitions, and determine the dependence of transition temperature on particle number.

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.

We study the performance of minimally nonlinear irreversible refrigerators which works between a hot and cold heat reservoir of constant temperatures [Formula: see text] and [Formula: see text], respectively. Applying optimization criterion [Formula: see text] or [Formula: see text], we analyze the coefficient of performance (COP) at arbitrary value of [Formula: see text] or [Formula: see text] and thus obtain the upper and lower bounds of the COP. Our results show that, for the optimized, tightly-coupled refrigerators, a small loss away from the maximum value of either [Formula: see text] or [Formula: see text] figure of merit means a larger gain in the COP.

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 influence of both the quantum degeneracy and the finite rate heat transfer between the working substance and the cylinder wall on the optimal performance of an Otto engine cycle is investigated. Expressions for several important parameters such as the power output and efficiency are derived. By using numerical solutions, the curves of the power output and efficiency varying with the compression ratio of two isochoric processes are presented. It is found that there are optimal values of the compression ratio at which the power output and efficiency attain their maximum. In particular, the optimal performance of the cycle in strong and weak gas degeneracy and the high temperature limit are discussed in detail. The distinctions and connections between the quantum Otto engine and the classical are revealed. Moreover, the maximum power output and efficiency and the corresponding relevant parameters are calculated, and consequently, the optimization criteria of some important parameters such as the power output, efficiency and compression ratio of the working substance are obtained.

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.

A cycle model of an irreversible heat engine working with harmonic systems is established in this paper. Based on the equation of motion of an operator in the Heisenberg picture and semi-group approach, the first law of thermodynamics for a harmonic system and the time evolution of the system are obtained. The general expressions for several important parameters, such as the work, efficiency, power output, and rate of entropy production are derived. By means of numerical analysis, the optimally operating regions and the optimal values of performance parameters of the cycle are determined under the condition of maximum power output. At last, some special cases, such as high temperature limit and frictionless case, are discussed in brief.

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.

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.

A general irreversible cycle model of a magnetic Ericsson refrigerator is established. The irreversibilities in the cycle model result from the finite-rate heat transfer between the working substance and the external heat reservoirs, the inherent regenerative loss, the additional regenerative loss due to thermal resistances, and the heat leak loss between the external heat reservoirs. The cycle model is used to optimize the performance of the magnetic Ericsson refrigeration cycle. The fundamental optimum relation between the cooling rate and the coefficient of performance of the cycle is derived. The maximum coefficient of performance, maximum cooling rate and other relevant important parameters are calculated. The optimal operating region of the cycle is determined. The results obtained here are very general and will be helpful for the optimal design and operation of the magnetic Ericsson refrigerators.

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

In this paper, we propose a pair of symmetric three-terminal refrigerator models with a hot cavity connected to two colder reservoirs via ideal tunneling quantum dots. The cooling of the refrigerators is achieved by investing thermal power from a hot reservoir and electric power from an applied voltage. Based on the model proposed, we numerically analyze the performance of the refrigerators with different half width of energy levels, and particularly discuss the coefficient of performance for zero applied voltage in the limit of a small half level width. Finally, we optimize with half width of energy levels and get the optimal region of the refrigerators.

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.

Theoretical thermoelectric nanophysics models of low-dimensional electronic heat engine and refrigerator devices, comprising two-dimensional hot and cold reservoirs and an interconnecting filtered electron transport mechanism have been established. The models were used to numerically simulate and evaluate the thermoelectric performance and energy conversion efficiencies of these low-dimensional devices, based on three different types of electron transport momentum-dependent filters, referred to herein as: kx, ky and kr filters. Assuming the Fermi-Dirac distribution of electrons, expressions for key thermoelectric performance parameters were derived for the resonant transport processes, in which the transmission of electrons has been approximated as a Lorentzian resonance function. Optimizations were carried out and the corresponding optimized design parameters have been determined, including but not limited to the universal theoretical upper bound of the efficiency at maximum power for heat engines, and the maximum coefficient of performance for refrigerators. From the results, it was determined that kr filter delivers the best thermoelectric performance, followed by the kx filter, and then the ky filter. For refrigerators with any one of three filters, an optimum range for the full width at half maximum of the transport resonance was found to be, no layout, 20 pages, 8 figures, 2 tables

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 study the thermodynamic performance of a nanosized photoelectric refrigerator consisting of three coupled single-level quantum dots embedded between two reservoirs at different temperatures. Based on the quantum master equation, we derive expressions for the cooling power and coefficient of performance (COP) of the refrigerator and plot the characteristic curves between the cooling power and the COP. We analyze the optimal performance parameters under conditions of maximum cooling power and maximum COP, and we discuss the influence of the energy level difference and the temperature ratio on the optimal performance parameters in detail.

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.

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 $\ensuremath{\sigma}$.

Based on counting statistics and Bogoliubov theory, we present a recurrence relation for the microcanonical partition function for a weakly interacting Bose gas with a finite number of particles in a cubic box. According to this microcanonical partition function, we calculate numerically the distribution function, condensate fraction, and condensate fluctuations for a finite and isolated Bose-Einstein condensate. For ideal and weakly interacting Bose gases, we compare the condensate fluctuations with those in the canonical ensemble. The present approach yields an accurate account of the condensate fluctuations for temperatures close to the critical region. We emphasize that the interactions between excited atoms turn out to be important for moderate temperatures.

A new cycle model of an irreversible quantum harmonic heat engine is established in this paper, which is composed of two isothermal processes, an adiabatic process and a constant frequency process. Based on the quantum master equation and semi-group approach, the general optimal performance characteristics of the cycle are investigated. Expressions for several important performance parameters such as the efficiency, the power output and the ecological function are derived analytically in high temperature limit. By numerical calculation, a set of performance characteristic curves is plotted, such as the optimal curves between the power output and efficiency, the ecological function and efficiency. A comparison of the corresponding efficiencies at the maximum ecological function and power output is also presented.

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.

An irreversible Braysson heat engine cycle model, in which the heat transfer obeys a linear heat transfer law between the working fluid and the external heat reservoirs, with heat resistance, heat leak and internal irreversibility is established in this paper. The analytical expressions of power output, efficiency and ecological objective function are derived. These performance parameters are maximized with respect to the cycle temperatures along with the isobaric temperature ratio. The performance characteristic curves are obtained by numerical calculation. The optimal operating regions of the important parameters are determined.

We study a Feynman-like thermal Brownian motor, in which the asymmetric corrugated cylinder as the ratchet is coupled with a corrugated plate by the noncontact Casimir interaction between them. The source of driving of the system is the thermal fluctuations and its dynamic evolution is described by a set of Langevin equations. Further, the mean velocity and thermal efficiency of the motor in the overdamped limit are studied in detail as a function of the temperature of the baths, external load applied, magnitude of the Casimir interaction, and other relevant parameters by the numerical stimulation. The transport properties attained here and the essential roles of the Casimir interaction can be explicitly demonstrated by designing a Casimir Brownian motor with present nanotechnology.

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.

An irreversible model of the air-standard Diesel heat engine cycle is established, in which the primary irreversibilities, such as the heat leak loss through the cylinder wall, friction loss, internal irreversibility and temperature-variable heat capacities, are considered. The analytic expressions of the power output and efficiency are derived. The relations between the power output and the ratio of the highest to the lowest pressure of the working fluid, between the efficiency and the ratio of the highest to the lowest pressure of the working fluid, as well as the optimal relation between the power output and the efficiency of the Diesel cycle are shown by numerical example. Moreover, the influence of these primary irreversible factors on the performance parameters of the Diesel heat engine is discussed. The results obtained here may provide some significant guidelines for the optimal design and operation of the practical Diesel heat engines.

The simple system of two two-level identical atoms couple to single-mode optical cavity in the resonance case is studied for investigating the thermal entanglement. It is interesting to see that the critical temperature is only dependent on the coefficient of atom-atom dipole-dipole interaction. Based on the mode, we construct and investigate a entangled quantum Otto engine (QOE). Expressions for several important performance parameters such as the heat transferred, the work done in a cycle, and the efficiency of the entangled QOE in zero G are derived in terms of thermal concurrence. Some intriguing features and their qualitative explanations are given. Furthermore, the validity of the second law of thermodynamics is confirmed in the entangled QOE. The results obtained here have general significance and will be helpful to understand deeply the performance of an entangled QOE.

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 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.