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A pre-embedding mixed A-cation halide strategy is developed to eliminate the residual unstable PbI2 and lattice strain near the buried interface of the perovskite layer for the stability improvement of formamidinium-based perovskite solar cells.

We investigate the finite-time performance of a quantum endoreversible Carnot engine cycle and its inverse operation — Carnot refrigeration cycle, employing a spin-1/2 system as the working substance. The thermal machine is alternatively driven by a hot boson bath of inverse temperature β h and a cold boson bath at inverse temperature β c (>β h). While for the engine model the hot bath is constructed to be squeezed, in the refrigeration cycle the cold bath is established to be squeezed, with squeezing parameter r. We obtain the analytical expressions for both efficiency and power in heat engines and for coefficient of performance and cooling rate in refrigerators. We find that, in the high-temperature limit, the efficiency at maximum power is bounded by the analytical value η + = 1 − sech ( 2 r ) ( 1 − η C ) , and the coefficient of performance at the maximum figure of merit is limited by ε + = sech ( 2 r ) ( 1 + ε C ) sech ( 2 r ) ( 1 + ε C ) − ε C − 1 , where η C = 1 – β h/β c and ε C = β h/(β c – β h) are the respective Carnot values of the engines and refrigerators. These analytical results are identical to those obtained from the Carnot engines based on harmonic systems, indicating that the efficiency at maximum power and coefficient at maximum figure of merit are independent of the working substance.

Hypertrophic scar (HS) is a fibroproliferative disorder that causes cosmetic as well as functional problems; however, to our knowledge, there is no satisfactory treatment for HS to date. Previous studies have indicated that angiogenesis plays a crucial role in HS formation; therefore, anti-angiogenetic therapies are considered effective in improving HS. Although tacrolimus (TAC) has been proven effective in preventing HS formation in vivo and in vitro, its underlying mechanism remains controversial and ambiguous. Because of its anti-angiogenic effects in other diseases, we aimed to determine whether TAC reduces HS by suppressing angiogenesis. Using a rabbit ear HS model that we developed, HS was treated once a week with normal saline, dimethyl sulfoxide, or TAC for 3 weeks. Histological evaluation indicated that TAC significantly reduced collagen deposition and microvessel density in scar tissues. Moreover, immunofluorescence staining for CD31 and vascular endothelial growth factor (VEGF)-A revealed that TAC significantly inhibited HS angiogenesis. In vitro analysis showed that TAC inhibited endothelial cell migration and tubulogenesis as well as the viability and proliferation of human umbilical vascular endothelial cells (HUVECs) and HS fibroblasts (HSFBs). Furthermore, TAC significantly downregulated the expression of the human angiogenetic factors VEGF-A, FGF-2, PDGF-β, and TGF-β1 in HUVECs and HSFBs. Additionally, TAC-mediated inhibition of angiogenesis decreased the gene expression of crucial fibrotic markers, including α- smooth muscle actin and collagens 1 and 3, in HSFBs. This is the first study to demonstrate the inhibitory effects of TAC on HS formation mediated by a mechanism involving the suppression of scar angiogenesis.

OBJECTIVE The purpose of this study was to detect the yield, activity and safety of the stromal vascular fraction (SVF) isolated by a new clinical-grade collagenase with a specific process and to validate the efficacy of the method. MATERIALS AND METHODS For the in vitro study, adipose tissue specimens were obtained from women undergoing elective liposuction (n = 5). The SVF was isolated by new clinical-grade collagenase for the experimental group and research-grade collagenase NB4 for the control group. The surface markers of the SVFs isolated by the two collagenases were detected by flow cytometry. The yield, activity and morphology of the SVFs were compared, together with their proliferation, migration and differentiation. Furthermore, we conducted a clinical trial to evaluate the improvement of skin texture after transplantation of autologous SVF isolated by clinical-grade collagenase. RESULTS There were no statistically significant differences in the yield, viability or morphology of the SVFs isolated by the new collagenase and collagenase NB4. Adipose tissue-derived mesenchymal stem cells (ASCs) surface marker expression was consistent between the two products. The differentiation potential of the cells was not different, which was consistent with previously published results. The SVF isolated by the new collagenase had a higher proliferation and migration. After administration of autologous SVF isolated by the new collagenase, the VISIA score improved significantly with regard to wrinkle scores and texture scores. CONCLUSION We concluded that this new type of clinical-grade collagenase can replace current research-grade products without any negative effect on the yield or function of the SVF, which can be applied to antiaging and rejuvenation of facial skin.

The Mpemba effect is a counterintuitive relaxation phenomenon whereby a system with a higher initial temperature may cool down to the thermal state faster than an identical system that was initially prepared at a lower temperature. Here, we investigate heat and work in a Markovian state transition system with cyclic switching hot-cold temperatures, which operates as an Otto heat engine working in long but finite time, either with or without the Mpemba effect. Under the condition of the periodic steady state having been reached, the time durations of the heating and cooling relaxation processes are determined by exploring a distance-from-equilibrium equivalent to the Kullback-Leibler divergence. We then numerically evaluate and compare the averages and variances of both the work and the power output of two scenarios with and without the Mpemba effect. The results show that the Markovian Mpemba effect can enhance the machine performance by significantly increasing the power output for a given efficiency without sacrificing the stability.

// Xiaolu Huang 1, * , Yiwen Zhou 1, * , Wenhui Liu 1 , Haizhou Li 1 , Xiao Liang 1 , Rui Jin 1 , Hengyu Du 1 , Jizhou He 1 , Bangda Chai 1 , Ran Duan 1 and Qingfeng Li 1 1 Department of Plastic and Reconstructive Surgery, Shanghai Ninth People’s Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200011, P.R.China * These authors contributed equally to this work and should be considered as co-author Correspondence to: Qingfeng Li, email: dr.liqingfeng@shsmu.edu.cn Keywords: silicone implant, autoimmunity, microarray, WGCNA, hub genes Received: July 04, 2017 Accepted: September 21, 2017 Published: October 06, 2017 ABSTRACT Silicone implants are used widely in the field of plastic surgery and are used in a large population. However, their safety profile, especially the silicone-induced immune response, has been a major concern for plastic surgeons for decades. It has been hypothesized that there is a cause and effect relation between silicone and immunity, but this is controversial. The objective of the present study was to determine the hub genes and key pathways related to silicone implant–induced immune responses in a rat model. In addition to cluster and enrichment analyses, we used weighted gene co-expression network analysis (WGCNA) to examine the gene expression profiles in a systematic context. A total five genes ( Fes , Aif1 , Gata3 , Tlr6 , Tlr2 ) were identified as hub genes that are most likely related to the silicone-induced immune response, four of which ( Aif1 , Gata3 , Tlr6 , Tlr2 ) have been associated with autoimmunity as target genes or disease markers. The Toll-like receptor signaling pathway ( p < 0.01, fold enrichment: 7.01) and systemic lupus erythematosus signaling pathway ( p < 0.05, fold enrichment: 5.01), which are considered strongly associated with autoimmunity, were significantly enriched in the silicone-implanted skin samples. The results indicate that silicone implants might trigger the localized immune response, as various immune reaction genes were detected after silicone implantation. The identified five hub genes will hopefully serve as novel therapeutic targets for silicone-related complications and the associated autoimmune diseases.

BACKGROUND Current rodent models of wound healing and scarring are flawed because of rapid wound contraction and inconspicuous scarring after healing, which is not closely parallel to the physiologic process in humans. This study aimed to establish a novel model of wound healing and scarring in rats. METHODS Excisional wounds were generated in rat tail or dorsal skin and histologic changes and wound contraction were assessed 2, 10, and 16 days after injury. After healing, rat tail scar was investigated for 24 consecutive weeks by histologic and immunohistochemical staining. Finally, a stretched scar model was generated in rat tail with high or low strain after reepithelialization to mimic human hypertrophic scars. The tail hypertrophic scars were analyzed by histology, immunohistochemical staining, and mRNA quantification 0, 2, 6, 12, and 24 weeks after stretching. RESULTS Compared with the dorsal wounds, a larger dermal gap percentage (p < 0.05) and more pronounced granulation were found in rat tail wounds. Tail scars remained conspicuous and underwent maturation over 24 weeks after wound healing. In addition, high mechanical strain induced significantly increased scar area (p < 0.01), scar height (p < 0.05), vessel density (p < 0.01) and hypertrophic scar-related molecule expression, and distorted collagen arrangement in rat tail scars. CONCLUSIONS The rat tail model exhibits minor wound contraction and biological features analogous to both normotrophic and hypertrophic scar in humans when generated with or without stretching, respectively. It is a promising new model for studies of both cutaneous wound healing and scarring.

We consider a quantum endoreversible Carnot engine cycle and its inverse operation–Carnot refrigeration cycle, working between a hot bath of inverse temperature [Formula: see text] and a cold bath at inverse temperature [Formula: see text]. For the engine model, the hot bath is constructed to be squeezed, whereas for the refrigeration cycle, the cold bath is set to be squeezed. In the high-temperature limit, we analyze efficiency at maximum power and coefficient of performance at maximum figure of merit, revealing the effects of the times allocated to two thermal-contact and two adiabatic processes on the machine performance. We show that, when the total time spent along the two adiabatic processes is negligible, the efficiency at maximum power reaches its upper bound, which can be analytically expressed in terms of squeezing parameter [Formula: see text]: [Formula: see text], with the Carnot efficiency [Formula: see text] and the coefficient of performance at maximum figure of merit is bounded from the upper side by the analytical function: [Formula: see text], where [Formula: see text].

Based on the thermosize effects of a confined ideal Maxwellian gas, we established a model of micro/nano-scaled heat pump including two isothermal and two isobaric processes, where the Joule-like heating effect was analyzed in detail by using the nonequilibrium thermodynamic theory. Further, in order to improve the performance of a single thermosize device, a model of two-stage micro/nano-scaled heat pump with multiple micro/nano components is for the first time put forward. The general expressions of heating load and coefficient of performance for the single and the two-stage heat pump are derived. The influences of the internal and external irreversibilities of the system, i.e., Fourier heat flow and the Joule-like heating and the finite rate heat transfer with the reservoirs, on the performance characteristic of thermosize devices are studied in detail. Moreover, we also discuss the effect of the allocation the thermosize components on the optimal performance of the two-stage micro/nano-scaled heat pump cycle. It is found that in the presence of the internal and external irreversibilities, the two kinds of the micro/nano-scaled heat pump can achieve specific heating performance. In particular, when the total number M of the thermosize components is fixed, both the maximum of the optimal coefficient of performance Φ opt , max and the maximum of the heating load Π opt , max corresponding to the optimal coefficient of performance appear near n = M / 2 with n being the number of the thermosize elements of the top stage. The results obtained here show several main irreversibility source of the micro/nano-scaled heat pump and have potential applications in the efficient utilization and conversion of energy in the microchannel system using gas.

We derive the general probability distribution function of stochastic work for quantum Otto engines in which both the isochoric and driving processes are irreversible due to finite time duration. The time-dependent work fluctuations, average work, and thermodynamic efficiency are explicitly obtained for a complete cycle operating with an analytically solvable two-level system. The effects of the irreversibility originating from finite-time cycle operation on the thermodynamic efficiency, work fluctuations, and relative power fluctuations are discussed.

A three-terminal refrigerator based on resonant-tunneling quantum wells is proposed. With the help of the Landauer formula, the expressions for the cooling rate and the coefficient of performance (COP) are derived. The working regions of the refrigerator are determined and the three-dimensional projection graphs of the cooling rate and the COP varying with the positions of the two energy levels are plotted. Moreover, the influence of the bias voltage, the asymmetric factor, and the temperature difference on the optimal performance parameters is analyzed in detail. Finally, the performance characteristics of the refrigerator in the case of negative temperature difference are discussed.

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

Finite-time cycle period for a quantum Otto machine implies that either an adiabatic stroke or an isochoric process proceeds in finite time duration. The quantum Otto refrigerators under consideration consist of two adiabatic strokes, where the system (isolated from the heat reservoir) undergoes finite-time unitary transformation, and two isochoric steps, where the system may not reach thermal equilibrium even at the respective ends of the two stages due to finite-time interaction intervals. Using two-time projective measurement method, we find the probability distribution functions of both coefficient of performance and cooling load, which are dependent on the time duration along each process. With these distributions we find the analytical expressions for the performance parameters as well as their fluctuations. We then numerically determine the performance and fluctuations for the refrigerator operating with a two-level system employed in a recent experimental implementation. Our results clarify the role of finite-time durations of four processes on the performance and fluctuations of the quantum Otto refrigerators.

Resection of a prominent mandibular angle is commonly used in Eastern society to improve the lower one third facial proportion. Historically, this procedure had a high complication rate, such as severe bleeding, asymmetry of the angle reduction, and "second mandibular angle." A safer and more effective way of performing such procedures is needed. The aim of this study is to introduce 3 instruments, a tunable guide handpiece, milling cutter, and flywheel, which were invented by the author, as well as a related ostectomy technique for correcting prominent mandibular angles using a modified full-thickness marginal ostectomy of the mandibular corpus angle, named the "stamp perforation" technique. This technique has 4 highlights: First, it ensures a smooth symmetric contour. Second, it prevents the risks of rupture of the inferior alveolar vessel and facial artery, ensuring the safety of this approach. Third, the "stamp perforation" technique eases the removal of bone fragments, shortening the operation time. Fourth, the recovery time of patients treated with this approach is much shorter than with the traditional approaches. From January 2006 to January 2016, 1106 patients underwent the surgery to contour the prominent mandible angles, and satisfactory results were achieved. Thus, we recommend the instruments as well as the "stamp perforation" technique for correcting prominent mandibular angles, and we hope that our 10 years of experience could provide a reference for other plastic surgeons.

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

We investigate the finite-power performance of quantum heat engines working in the linear response regime where the temperature gradient is small. The engine cycles with working substances of ideal harmonic systems consist of two heat transfer and two adiabatic processes, such as the Carnot cycle, Otto cycle, and Brayton cycle. By analyzing the optimal protocol under maximum power we derive the explicitly analytic expression for the irreversible entropy production, which becomes the low dissipation form in the long duration limit. Assuming the engine to be endoreversible, we derive the universal expression for the efficiency at maximum power, which agrees well with that obtained from the phenomenological heat transfer laws holding in the classical thermodynamics. Through appropriate identification of the thermodynamic fluxes and forces that a linear relation connects, we find that the quantum engines under consideration are tightly coupled, and the universality of efficiency at maximum power is confirmed at the linear order in the temperature gradient.

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

We 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.

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.

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

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

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.

We derive an exact recursive scheme to determine exactly the microcanonical partition function of a finite Bose system. Such a recursive approach is identical to that previously obtained within the context of counting statistics. Within the exact microcanonical ensemble, we study microcanonical finite-size scaling behaviors of condensate fraction and specific heat around the critical energy ϵ c for the finite ideal Bose system. We show that the microcanonical scaling functions governing the various critical behaviors are universal in the ideal Bose-Einstein condensates.

A numerical model of a nano-scaled thermoelectric heat engine with InP/InAs/InP trilayer quantum well (QW) is investigated. The expressions of those performance parameters, such as current, power output, and efficiency are expressed. By numerical calculation, the resonant tunneling behavior of electrons in the QW is described, which seems like a very good energy selective electron mechanism for the heat engine. After considering the radiation heat leakage, for fixed layer thicknesses of the QW, the optimum working regions of the heat engine with respect to the chemical potentials and the bias voltage are obtained numerically under the economic criterion. From these results, the power output can be increased by narrowing down the layer thicknesses. In addition, owing to the radiant heat leakage, the efficiency initially increases in the working regions and then decreases when the layer thicknesses increase gradually, from which one can obtain a maximum efficiency by optimizing layer thicknesses of QW. These results calculated here may provide a guide for the optimum designs of tunneling thermoelectric devices.

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.

A model of thermionic heat engine with the energy selective electron mechanism is studied. Analytical expressions of the power output and efficiency of this device are derived at low temperature, where the chemical potentials of the reservoirs are assumed to be constant. After discussing the impact of the energy spectrum width of the energy selective electron mechanism, we find two bounds ( η ± ) of efficiency at maximum power exist naturally. When the energy spectrum width increases gradually from zero and then to the semi-infinite case with the infinite upper limit, the efficiency at maximum power decreases monotonously from the upper bound η + to the lower bound η − at a given temperature ratio of the cold and hot reservoirs. The two bound are given by numerical simulation and by an analytical expression respectively. These results may provide some guidance for the application of the practical energy selective electron heat engines.

The unsteady Reynolds-averaged Navier-Stokes equations coupled with the k-ω SST turbulence model are solved to obtain the steady and unsteady aerodynamic forces for airfoils and wings. The effects of vibration types and amplitudes on aerodynamic forces of airfoils and wings are studied. The deformation characteristics of a swept wing induced by steady aerodynamic load are presented. It is found that for a vibrating elastic wing at small and medium incidences, its mean aerodynamic loads are almost the same as those obtained from the static one. On the contrary, at high incidences especially around the stall incidence, the vibration may change the mean values. In addition, the larger amplitude is, the larger discrepancy will be. For a swept wing, the steady aerodynamic loads usually lead to the “pitching down” effect on the wing tip which delays the stall compared with a rigid one; But this phenomenon dose not occur on a aeroelastic wing which can induce the separation ahead and trigger the stall. The above conclusions are in good agreement with the scatter characteristics of wind-tunnel data. The reason why the data obtained from wind tunnel and CFD are different is also analyzed. Meanwhile, it can be an explanation for scatter phenomenon of wind-tunnel data, especially for high incidence cases, which remains a puzzle so far.

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.

Based on a two-qubit isotropic Heisenberg XXX model with a constant external magnetic field, we construct a four-level entangled quantum heat engine (QHE). The expressions for several thermodynamic quantities such as the heat transferred, the work and efficiency are derived. Moreover, the influence of the entanglement on the thermodynamic quantities is investigated analytically and 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.

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.