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Acta Physica Sinica (Overseas Edition)

Chemical potential quantization and Bose-Einstein condensation

Zheng Jiu-ren

Published Oct 1, 1999• 1 citation

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 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.

DOI: 10.1088/1004-423x/8/10/001

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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.

The properties of the ideal Bose gas are calculated from the integral equations for the energy and the number of particles as a function of the temperature and chemical potential. It is shown that the integral equations break down below the Einstein temperature that corresponds to the transition to the low-temperature state. The lowest single-particle energy level must be treated explicitly to get the proper equations. With the inclusion of the lowest single-particle energy level, the low-temperature behavior is calculated. The occupation of the lowest level becomes comparable to the total number of particles in the system below the Einstein temperature, and equal to the total number of particles at zero temperature. A numerical solution to the properties of the Bose gas is discussed, and the detailed calculations are assigned to the problems at the end of the chapter.

The paper is devoted to the theory of an ideal Bose-gas with a finite number N of particles. The exact expressions for the partition functions and occupation numbers of the model in the grand canonical, canonical, and microcanonical ensembles are found. From the calculations, it is followed that, oppositely to the accepted opinion that the chemical potential μ of an ideal Bose-gas is only negative, it can take values in the range −∞ < μ < ∞. The asymptotic expressions (in the case N ≫ 1) for the partition functions and occupation numbers for all above-mentioned thermodynamic ensembles are also evaluated.

Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.

When the chemical potential μ limits to zero, the steady increase of particles in the ground state with e=0 under Bose-Einstein gas in equilbirium distribution. This is called Bose-Einstein condensation. Though Monte Carlo simulation in quantum ensenble, when the number of particles in the system in small we find: First, the fluctuation of the number of particles in the ground state is very quick and is analogous to energy level distribution. Second, when TT c , the system appears Bose-Einstein condensation, but whether T c is congruous with the theoretical value should be further verified.

The ideal Bose-gas with finite number

N

of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp\-totic expressions (in the case

N\gg1

) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential

μ

of the ideal Bose-gas can lie in the range $-\infty

Abstract The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be εi≡εs1,....,s10=constanth2ms1α−1a12+...+swαaw2(1≤α≤2) the quantum numbers being s1 ,w= 1, 2, ..., and a1, ..., aw being certain characteristic lengths. (For α = 2, the potential field is that of the w-dimensional rectangular box; for α = 1, we obtain the w-dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w/α determines the appearance of an Einstein transition temperature T0·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ- per particle and the specific heat dϵ-/dT are continuous at T = T0· For q > 2, the specific heat is discontinuous at T = T0, giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T0 is finite only if the quantity v = N/(a1 .... aw)2/ α¯is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, then T0 likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T 0' there is a finite fraction N0 /N of the particles, given by N 0/N = 1-(T/T0)q, in the lowest state. London’s formula (1938 b) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.