Ideal Bose-gas with finite particle number N is investigated. It is shown that the well-known Bose–Einstein distribution of particles over their quantum states consists of two terms — Bose–Einstein term and an additional N-dependent one. The statement that the chemical potential of the ideal Bose-gas should be negative requires to be also corrected. It is obtained that the analytically calculated observables rather slowly approach the thermodynamic limit following the law N−1/3lnN.
Ideal Bose-gas with finite particle number N is investigated. It is shown that the well-known Bose–Einstein distribution of particles over their quantum states consists of two terms — Bose–Einstein term and an additional N-dependent one. The statement that the chemical potential of the ideal Bose-gas should be negative requires to be also corrected. It is obtained that the analytically calculated observables rather slowly approach the thermodynamic limit following the law N−1/3lnN.
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.
The calculation of the thermodynamic properties in the Bose-Einstein distribution is presented.The behaviour of the Bose-gas in the Bose-Einstein distribution is analysed based on high and low temperature dependence.At > all particles are in the excited state and the ground state is essentially occupied.But at B T T < particles gradually falls to the ground state and eventually contains all the particles in the system as .0 → TThe analytical expressions for some important parameters such as Bose-Einstein condensation of an ideal Bose gas, the population density of the excited and ground state particles are also presented.Heart capacity of an ideal Bose gas and total internal energy are also calculated and the results presented graphically.The results obtained are consistent with those found in the literatures.
Recursive approaches determining the canonical ideal Bose gas partition function are reviewed that enable the Bose–Einstein condensate occupation probability to be calculated for a finite number of particles ensemble, where the thermodynamic limit approximation fails. In addition to the earlier known method recursive with respect to the number of particles, an iteration procedure with respect to the number of quantum states is proposed. The efficiency of both methods is demonstrated for an ideal Bose gas in a three-dimensional isotropic harmonic trap.
This paper presents a study of the grand canonical Bose-Einstein (BE) statistics for a finite number of particles in an arbitrary quantum system. The thermodynamical quantities that identify BE condensation---namely, the fraction of particles in the ground state and the specific heat---are calculated here exactly in terms of temperature and fugacity. These calculations are complemented by a numerical calculation of fugacity in terms of the number of particles, without taking the thermodynamic limit. The main advantage of this approach is that it does not rely on approximations made in the vicinity of the usually defined critical temperature, rather it makes calculations with arbitrary precision possible, irrespective of temperature. Graphs for the calculated thermodynamical quantities are presented in comparison to the results previously obtained in the thermodynamic limit. In particular, it is observed that for the gas trapped in a three-dimensional box, the derivative of specific heat reaches smaller values than what was expected in the thermodynamic limit---here, this result is also verified with analytical calculations. This is an important result for understanding the role of the thermodynamic limit in phase transitions and makes possible to further study BE statistics without relying neither on the thermodynamic limit nor on approximations near critical temperature.
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.
Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.