Finite Bose System Is Strictly Ensemble Theoretical Research

Experiments of Bose-Einstein condensation (BEC) in ultracold trapped atomic gases or experimental observations of other mesoscopic systems such as atomic nu-clei, molecules, atomic clusters, and finite polymers, are prototypes of physical inves-tigations of transitions in small systems and have created a renewed interest in critical phenomenon for finite systems. In particular, BEC as a purely quantum-statistical phase transition has opened far-reaching prospects and has become a test laboratory where the atoms can be well manipulated in the modern condensed matter physics. The number of particles in the experiments are roughly fixed and finite, and thus the experimental situation is quite different from the traditional treatments, in which the thermodynamic limit was used.In the finite Bose systems the physical quantities, such as specific heat and conden-sation fraction show a more or less smooth humps extending over some finite temper-ature ranges. This is quite different from the thermodynamic limit which exist a sharp peak or a discontinuity in these physical quantities. Although phase transitions occurs only in the thermodynamic limit, precursors of phase transitions in finite systems far away from the thermodynamic limit do exist, as confirmed by the experiments. Re-search topics in the present dissertation motivated by experiments of finite systems are, within a canonical ensemble treatment, aimed to find finite-size effects on the thermo-dynamics and statistical properties for ideal and weakly interacting Bose gases, espe-cially when these finite Bose systems near the transition regions.In Chapter 2, we discuss briefly the merits as well as flaws of grand canonical and canonical ensembles, respectively. We investigate the thermodynamic behavior of ideal Bose gases with an arbitrary number of particles confined in a harmonic poten-tial. By taking into account the conservation of total number N of particles and using a saddle-point approximation, we derive the simple explicit expression of mean occu-pation number at any state of the finite system, and then numerically obtain the tem-perature dependence of the chemical potential, the specific heat, and the condensate fraction for the trapped gases with a finite number of particles. Comparing the results with corresponding those from the traditional grand canonical treatment, we find that the considerable difference between them show up for at sufficiently low temperatures, specially for the relative small numbers of Bose particles.Within the exact canonical ensemble treatment, in Chapter 3 we investigate the ther-modynamics and finite size scaling for finite ideal and weakly interacting Bose gas in a cubic box or in a harmonic trap. Both in the box trap imposed by either periodic or Dirichlet boundary conditions (BCs) and in the harmonic trap, for an ideal gas we cal-culate several physical quantities such as the chemical potential, the specific heat, the condensate fraction, the root-mean-square fluctuations of the condensate, and the tran-sition temperature etc., and compare these quantities under different traps. We discuss the particle-number dependence of the transition by proposing several transition tem-perature definitions, where the differences among these values are considerable for the finite systems. For a weakly interacting Bose in a box with periodic boundary condi-tions, the thermodynamic properties are investigated theoretically, based on a recursion relation for the canonical ensemble partition function. In a similar manner to the case of an ideal Bose gas, we first establish the recursive scheme of the interacting finite system at finite temperatures in the framework of the Bogoliubov theory. The tempera-ture dependence of the condensate fraction and specific heat with different particles and interactions is obtained numerically. By defining two different transition temperatures of the finite systems, the effect of interactions on the transition temperatures yields the different and non-monotonous behaviors. We discuss the finite-size scaling of conden-sate fraction at the transition temperature for the systems, showing that the calculated finite-size scaling is universal and thus independent on the various system sizes and transition temperatures.Chapter 4 is devoted to study of critical behaviors for an ideal and weakly interact-ing Bose gas. For an ideal Bose gas, we study the trap-size scaling behaviors of the condensate fraction and the specific heat for the three different traps, introducing a trap exponentθin dependence of the trapping potential. In the box trap with periodic and Dirichlet BCs, whereθ→1, we find that the scaling functions governing the various critical behaviors are universal but respective of the BCs. The borders of universality validity are obtained numerically. In the harmonic trap, the critical behavior of the sys-tem is also found to be universal, and the trap exponent is obtained asθ(?) 0.157. For a weakly interacting Bose system, within the exact canonical ensemble treatment we study the finite-size scaling behavior of the specific heat near the critical region, and obtain the specific heat exponent and correlation length exponent which agree well with experimental data and previous theoretical predictions. For fixed interaction parameters nd particle number density, we report for the first time that, the dimensionless specific heats per particle of various system sizes and temperatures collapse onto a single form. From the scaling arguments, we study the interaction-induced shift of the transition temperature (△Tc= Tc-Tc0) and find that△Tc/Tc0= ban(?) with b= 0.42±0.05, where Tc0 is the ideal-gas thermodynamic critical temperature. We also compare our findings with experimental and theoretical results for 4He in cubic lattices, showing the different behaviors for the two systems.In chapter 5, based on the exact canonical ensemble treatment, we generalize the scheme to characterize phase transitions of finite systems in a complex temperature plane, and present the classifications of phase transitions in ideal and weakly interact-ing Bose gases of a finite number of particles, confined in a cubic box with different boundary conditions. In extending the classification parameters to all regions, we pre-dict that for the finite system the phase transition for periodic boundary conditions is of second order, while transition in Dirichlet boundary conditions is of first order. For a weakly interacting Bose gas with periodic boundary conditions, we discuss the ef-fects of finite particle numbers and inter-particle interactions on the nature of the phase transitions. We show that this homogenous weakly interacting Bose gas undergoes a second-order phase transition, which is in accordance with universality arguments for infinite systems. We also discuss the dependence of transition temperature on the inter-action strengths and particle numbers.The research topics about quantum statistical theory, which deserve a deeper study for the finite quantum systems, are listed in the Chapter 6.The results in this dissertation allow to understand well the statistical properties and critical behaviors for the finite quantum systems, and thus provide the theoretical guidelines for the future experiments including finite Fermi systems, atomic clusters, neutron starts, and manipulated condensed matter physics.