AbstractIn 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 Tp,n, where n=0,1,2… Taking $\varepsilon$0≤$\varepsilon$n<$\varepsilon$n+1, then Tp,0≥Tp,n>Tp,n+1. When the temperature T>Tp,n or T≤Tp,n+1, $\mu$≠$\varepsilon$n and the most probable occupation number Nn=0. In the temperature interval Tp,n≥ T>Tp,n+1, $\mu$ = $\varepsilon$n and 0≤Nn=N-$\sum$jNj<~supNn, where Nj 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 Tp=Tp,0. The chemical potential $\mu$ is quantized when T≤Tp, and this leads to the creation of a macroscopic quantum state (pure state) or Bose-Einstein condensation phase. Tp=Tp,0 is a first-order phase transition point, Tp,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 Nj is directly proportional to gj-1, where gjand Nj 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.
Received: 14 September 1998
Revised: 25 February 1999
Accepted manuscript online:
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