Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Bose-Einstein condensation of finite number of confined particles

Wenji Deng, P. M. Hui, New Territories, Hong Kong

Published Jan 1, 1997• 12 citations
OpenAlex

Abstract

The partition function and specific heat of a system consisting of a finite number of bosons confined in an external potential are calculated in different spatial dimensions. Using the grand partition function as the generating function of the partition function, an iterative scheme is established for the calculation of the partition function of system with an arbitrary number of particles. The scheme is applied to finite number of bosons confined in isotropic and anisotropic parabolic traps and in rigid boxes. The specific heat as a function of temperature is studied in detail for different number of particles, different degrees of anisotropy, and different spatial dimensions. The cusp in the specific heat is taken as an indication of Bose-Einstein condensation (BEC). It is found that the results corresponding to a large number of particles are approached quite rapidly as the number of bosons in the system increases. For large number of particles, results obtained within our iterative scheme are consistent with those of the semiclassical theory of BEC in an

Related papers

Multi-step Bose-Einstein Condensation of Trapped Ideal Bose Gases

Wenji Deng, Typeset Using Revtex

Abstract: The Phenomenon of multi-step Bose-Einstein condensation (BEC) of a finite number of non-interacting bosons in anisotropic traps has been demonstrated by studing the populations on eight subsets of states. The cusp in the specific heat is found to be associated with the crossover between subsets of states involving Bose functions gn(z) of different classes, as specified by their behaviour at z = 1.

•Jan 1, 1998•1 citation

Condensation of ideal Bose gas confined in a box within a canonical ensemble

Konstantin Glaum, Hagen M. Kleinert, Axel Pelster

We set up recursion relations for the partition function and the ground-state occupancy for a fixed number of noninteracting bosons confined in a square box potential and determine the temperature dependence of the specific heat and the particle number in the ground state. A proper semiclassical treatment is set up which yields the correct small-$T$ behavior in contrast to an earlier theory in Feynman's textbook on statistical mechanics, in which the special role of the ground state was ignored. The results are compared with an exact quantum-mechanical treatment. Furthermore, we derive the finite-size effect of the system.

arXiv:0707.2715•Physical Review A•Dec 5, 2007•49 citations

Bose-Einstein condensation in anisotropic harmonic traps

Tor Haugset, Hårek Haugerud, Jens O. Andersen

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.

arXiv:cond-mat/9611061•Physical Review A•Apr 1, 1997•61 citations

Bose-Einstein condensation of a finite number of particles confined to harmonic traps

Raj K. Pathria

The onset of Bose-Einstein condensation in a sample of N atoms confined to harmonic traps is critically examined. The monotonic growth of the condensate fraction f and the nonmonotic variation of the specific heat C of the sample are studied in fine detail. While no temperature is truly critical, our analysis does provide a strong theoretical basis for recognizing a temperature ${T}^{*},$ that has been identified and advocated by previous authors, as the transition temperature of the system. The specific-heat maximum ${T}_{m}$ (which happens to be another important ``marker'' in this transition), the value of the specific heat at ${T=T}_{m},$ and the ``jump'' in the specific heat of the system when the number N becomes very large are all explicitly evaluated. Wherever comparison is possible, our analytical results are found to be in perfect agreement with the ones obtained numerically by Kirsten and Toms [Phys. Rev. A 54, 4188 (1996)] and by Haugerud, Haugest, and Ravndal [Phys. Lett. A 225, 18 (1997)].

Physical Review A•Aug 1, 1998•60 citations

Bose-Einstein statistics for a finite number of particles

Pedro Pessoa

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.

arXiv:2110.02890•Physical Review A•Oct 25, 2021•4 citations

Numerical calculation of finite Bose-Einstein condensation in harmonic potentials

Zhan Liu

The chemical potential and its differential coefficients as a function of temperature of an ideal system trapped in three dimensions isotropic harmonic traps with an finite number of particles were calcu- lated from the statistical theory.The results show that,there is no phase transition in a system with an finite number of particles,but the Bose-Einstein condensation occurs at a finite temperature.The critical temperatures were well defined by the minimum temperatures of the second derivative of chemical potential, which were in good agreement with the experimental results.

Journal of Lanzhou University•Jan 1, 2008•0 citations
The partition function and specific heat of a system consisting of a finite number of bosons confined in an external potential are calculated in different spatial dimensions. Using the grand partition function as the generating function of the partition function, an iterative scheme is established for the calculation of the partition function of system with an arbitrary number of particles. The scheme is applied to finite number of bosons confined in isotropic and anisotropic parabolic traps and in rigid boxes. The specific heat as a function of temperature is studied in detail for different number of particles, different degrees of anisotropy, and different spatial dimensions. The cusp in the specific heat is taken as an indication of Bose-Einstein condensation (BEC). It is found that the results corresponding to a large number of particles are approached quite rapidly as the number of bosons in the system increases. For large number of particles, results obtained within our iterative scheme are consistent with those of the semiclassical theory of BEC in an
We set up recursion relations for the partition function and the ground-state occupancy for a fixed number of noninteracting bosons confined in a square box potential and determine the temperature dependence of the specific heat and the particle number in the ground state. A proper semiclassical treatment is set up which yields the correct small-TTT behavior in contrast to an earlier theory in Feynman's textbook on statistical mechanics, in which the special role of the ground state was ignored. The results are compared with an exact quantum-mechanical treatment. Furthermore, we derive the finite-size effect of the system.
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.
The onset of Bose-Einstein condensation in a sample of N atoms confined to harmonic traps is critically examined. The monotonic growth of the condensate fraction f and the nonmonotic variation of the specific heat C of the sample are studied in fine detail. While no temperature is truly critical, our analysis does provide a strong theoretical basis for recognizing a temperature T∗,{T}^{*},T∗, that has been identified and advocated by previous authors, as the transition temperature of the system. The specific-heat maximum Tm{T}_{m}Tm​ (which happens to be another important ``marker'' in this transition), the value of the specific heat at T=Tm,{T=T}_{m},T=Tm​, and the ``jump'' in the specific heat of the system when the number N becomes very large are all explicitly evaluated. Wherever comparison is possible, our analytical results are found to be in perfect agreement with the ones obtained numerically by Kirsten and Toms [Phys. Rev. A 54, 4188 (1996)] and by Haugerud, Haugest, and Ravndal [Phys. Lett. A 225, 18 (1997)].
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.
The chemical potential and its differential coefficients as a function of temperature of an ideal system trapped in three dimensions isotropic harmonic traps with an finite number of particles were calcu- lated from the statistical theory.The results show that,there is no phase transition in a system with an finite number of particles,but the Bose-Einstein condensation occurs at a finite temperature.The critical temperatures were well defined by the minimum temperatures of the second derivative of chemical potential, which were in good agreement with the experimental results.