摘要： In this paper, first of all, we proved if the ideal Bose gas with a finite volume and number of particles has a non-degenerate single-particle energy level ε_n, the chemical potential μ can take the value μ_n=ε_n and there is a phase transition temperature T_p,n, where n=0,1,2… Taking ε₀≤ε_n<ε_n+1, then T_p,0≥T_p,n>T_p,n+1. When the temperature T>T_p,n or T≤T_p,n+1, μ≠ε_n and the most probable occupation number N_n=0. In the temperature interval T_p,n≥ T>T_p,n+1,μ=ε_n and 0≤N_n=N-Σ_jN_j<～supN_n, where N_j is the most probable occupation number in the degenerate level j. Thus, if the finite ideal Bose gas has some non-degenerate single-particle levels, there exists a characteristic temperature T_p=T_p,0. The chemical potential μ is quantized when T≤T_p, and this leads to the creation of a macroscopic quantum state (pure state) or Bose-Einstein condensation phase. T_p=T_p,0 is a first-order phase transition point, T_p,n≠0 is a zero-order phase transition point. Next, we obtained a new expression of the most probable distribution of the finite ideal Bose gas. In this expression N_j is directly proportional to g_j-1, where g_j and N_j are, respectively, the degeneracy and the most probable occupation number in the degenerate level j. This property agrees with what chemical potential can be quantized if there is a non-degenerate level for the finite ideal Bose gas. Finally, using this expression, we defined a micro-partition function M, obtained the statistical expressions of some thermodynamical quantities.

Abstract: In this paper, first of all, we proved if the ideal Bose gas with a finite volume and number of particles has a non-degenerate single-particle energy level $\varepsilon$_n, the chemical potential $\mu$ can take the value $\mu$_n = $\varepsilon$_n and there is a phase transition temperature T_p,n, where n=0,1,2… Taking $\varepsilon$₀≤$\varepsilon$_n<$\varepsilon$_n+1, then T_p,0≥T_p,n>T_p,n+1. When the temperature T>T_p,n or T≤T_p,n+1, $\mu$≠$\varepsilon$_n and the most probable occupation number N_n=0. In the temperature interval T_p,n≥ T>T_p,n+1, $\mu$ = $\varepsilon$_n and 0≤N_n=N-$\sum$_jN_j<~supN_n, where N_j  is the most probable occupation number in the degenerate level j. Thus, if the finite ideal Bose gas has some non-degenerate single-particle levels, there exists a characteristic temperature T_p=T_p,0. The chemical potential $\mu$ is quantized when T≤T_p, and this leads to the creation of a macroscopic quantum state (pure state) or Bose-Einstein condensation phase. T_p=T_p,0 is a first-order phase transition point, T_p,n≠0 is a zero-order phase transition point. Next, we obtained a new expression of the most probable distribution of the finite ideal Bose gas. In this expression N_j is directly proportional to g_j-1, where g_jand N_j are, respectively, the degeneracy and the most probable occupation number in the degenerate level j. This property agrees with what chemical potential can be quantized if there is a non-degenerate level for the finite ideal Bose gas. Finally, using this expression, we defined a micro-partition function M, obtained the statistical expressions of some thermodynamical quantities.

中图分类号:  (Other Bose-Einstein condensation phenomena)

• 03.75.Nt

05.30.Jp (Boson systems) 05.70.Fh (Phase transitions: general studies) 03.65.Vf (Phases: geometric; dynamic or topological)