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In this work, the oxygen defect strategy was conducted to promote photo-splitting H2S into hydrogen (H2) on the typical anatase TiO2 with active {0 0 1} facet for the first time. Density functional theory (DFT) calculation results illustrate that surface oxygen vacancy can efficiently facilitate the electron transition across the forbidden band and enhance the surface adsorption ability with more negative adsorption energies of −0.82 to −2.85 eV for H2S and its fragments than that of 0.24 to −0.90 eV on the perfect surface. Importantly, the energy barrier is reduced by 1.58 eV in maximum along the reaction paths on the defected surface, and the changes of rate-determining step lead to H2 as the final product. In addition, the time-resolved fluorescence tests, photocurrent measurements and electrochemical impedance spectroscopy demonstrate that the oxygen vacancies can effectively separate photo-generated electron-hole pairs. As a result, the photocatalytic activities of H2 evolution from H2S on TiO2 {0 0 1} doped with oxygen vacancies have been significantly enhanced from 21.44 μmol g−1 h−1 to 95.25 μmol g−1 h−1. This enhanced photoactivity is due to the low recombination of photo-generated carriers, the favorable surface adsorption and reaction activity induced by oxygen vacancies.

Based on a two-qubit Heisenberg XY model, we construct a four-level entangled quantum heat engine (QHE). It is an interesting quantum Otto cycle where the exchange constant is fixed and only the magnetic field is varied during the adiabatic steps. The expressions for several thermodynamic quantities such as the heat transferred, the work and the efficiency are derived. Moreover, the influence of the entanglement on the thermodynamic quantities is investigated numerically. Several interesting features of the variation of the heat transferred, the work and the efficiency with the concurrences of the thermal entanglement of different thermal equilibrium states are obtained. Finally, we discussed the maximum efficiency of the QHE.

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.

Based on the classification scheme of phase transitions, we study the phase transitions for an ideal Bose gas with a finite number N of particles trapped in a d-dimensional quartic potential. We find that the presence and nature of phase transition depend on the dimensionality of the quartic potential. Proposing three different definitions of transition temperature, we discuss either N or d dependence of transition temperature for the ideal Bose condensate in the d-dimensional quartic potential.

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.

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 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.

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.

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.

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.

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.

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.

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 investigate the thermodynamic performance of a quantum-dot refrigerator consisting of a single orbital interacting quantum dot embedded between two electron reservoirs at different temperatures and chemical potentials. Based on the quantum master equation the expressions for the cooling power and the coefficient of performance are derived. The characteristic curves between the cooling power and the coefficient of performance are plotted, and the optimal regions of the performance parameters are determined. Moreover, the optimal performance parameters are calculated numerically. Finally, the influence of the Coulomb interaction and the temperature ratio on optimal performance parameters are discussed in detail. The results obtained here can provide some theoretical guidelines for the design and operation of the practical quantum-dot refrigerator. Our work is not restricted to the linear-response regime.

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.