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arXiv:1211.4119

Bose gas in power-like spherically symmetric potential in arbitrary spatial dimensionality

A. A. Kozhevnikov

Published Nov 17, 2012• 2 citations
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Abstract

Temperature of the Bose -- Einstein condensation and the temperature behavior of the chemical potential and other thermodynamical functions of the ideal Bose gas are found for the arbitrary power-like spherical-symmetric potential at an arbitrary space dimension. It is shown that the recently observed Bose -- Einstein condensation of photons in the cavity is the phase transition of the third kind.

DOI
10.48550/arxiv.1211.4119

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Temperature of the Bose -- Einstein condensation and the temperature behavior of the chemical potential and other thermodynamical functions of the ideal Bose gas are found for the arbitrary power-like spherical-symmetric potential at an arbitrary space dimension. It is shown that the recently observed Bose -- Einstein condensation of photons in the cavity is the phase transition of the third kind.
Thermodynamic properties of ideal Bose gas trapped in an external generic power law potential are investigated systematically from the grand thermodynamic potential in ddd dimensional space. The most general conditions for Bose-Einstein condensate and the discontinuous conditions of heat capacity at the critical temperature in presence of generic power law potential are presented in this manuscript. The dependence of the physical quantities on external potential, particle characteristics and space dimensionality are discussed. The more general results obtained in this paper presents an unified illustration of Bose-Einstein condensation of ideal Bose systems as they reduces to the expressions and conclusions available in the literature with appropiate choice of power law exponent.
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The properties of a trapped ideal Bose gas in n -dimensional space are studied. General analytic expressions of the critical temperature of Bose-Einstein condensation, the jump of heat capacity at and the fraction of condensation at temperatures below have been derived. How these physical quantities depend on the external potential, particle characteristics and space dimensionality are discussed. We find that when a proper external potential is taken, Bose-Einstein condensation may occur in any dimensional space.
We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.
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 n, the chemical potential μ can take the value μn = n and there is a phase transition temperature Tp,n, where n = 0,1,2 Taking 0 ≤ n Tp,n+1. When the temperature T > Tp,n or T ≤ Tp,n+1, μ # n and the most probable occupation number Nn = 0. In the temperature interval Tp,n ≥ T > Tp,n+1, μ = n and 0 ≤ Nn = N - NjsupNn, 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 μ 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 gj and 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.