We have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.
The thermodynamical properties of interacting Bose atoms in a harmonic potential are studied within the mean-field approximation. For weak interactions, the quantum statistics is equivalent to an ideal gas in an effective mean-field potential. The eigenvalue of the Gross–Pitaevskii equation is identified as the chemical potential of the ideal gas. The condensation temperature and density profile of atoms are calculated. It is found that the critical temperature Tc decreases as the interactions increase. Below the critical point, the condensation fraction exhibits a universal relation of N0/N = 1-(T/Tc)γ, with the index γ ≈ 2.3 independent of the interaction strength, the chemical potential, as well as the frequency of the confining potential.
We report the effects of interaction on thermodynamic properties of a repulsive Bose-Einstein condensate confined in a harmonic trap by using the correlated potential harmonics expansion method. This many-body technique permits the use of a realistic interactomic interaction, which gives rise to the effective long-range interaction of the condensate in terms of the
s-wave scattering length. We have computed temperature (
T) dependence of the chemical potential, specific heat, condensate fraction, entropy, pressure, and the average energy per particle of a system containing a large number (
A) of
87Rb atoms in the Joint Institute for Laboratory Astrophysics (JILA) trap. The repulsion among the interacting bosons results in a small but measurable drop of condensate fraction and critical temperature (
Tc), compared to those of a noninteracting condensate. These are in agreement with the experiment. Although all thermodynamic quantities have a strong dependence on
A and to a smaller extent on the interatomic interaction, our numerical calculation appears to show that a thermodynamic quantity per particle follows a universal behavior as a function of
T/Tc. This shows the importance of
Tc for all thermodynamic properties of the condensate. As expected, for
T>Tc, these properties follow those of a trapped noncondensed Bose gas.
The critical temperature and condensate fraction of a trapped interacting Bose gas are investigated when both atom-atom interaction and finite-size effects are taken into account. A canonical ensemble is used to obtain the equations on the condensate fraction for the trapped interacting Bose gas near and below the critical temperature. In our approaches corrections due to atom-atom interaction and finite-size effects are obtained simultaneously for the critical temperature and condensate fraction of the system. An analytical high-order correction to the condensate fraction is given.
In this paper the thermodynamical parameters of a condensed Boson gas are calculated from the partial derivative of the grand potential. In particular, the analytical expressions for some important parameters, such as the condensed fraction, specific heat, critical temperature, effective size, and release energy are investigated. The mean effects which can be altered the ideal Bose gas, such as finite size, highly anisotropic of the external potential and interatomic interaction effects are considered simultaneously. Some new characteristics of the trapped interacting Bose gases in a highly anisotropic trap are revealed. The calculated result for the condensed fraction is compared with the ongoing Stuttgart experiment for 52Cr (Griesmaier et al., Phys. Rev. Lett. 94, 160401(2005)) directly. Good agreement between both the theoretical and experimental data are obtained.
We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number
N of atoms, harmonic oscillator length, deformation of the trap,
s-wave scattering length) fix its thermodynamic behavior in the large
N limit through two dimensionless scaling parameters. These are the reduced temperature
t=T/Tc0 and the ratio $\ensuremath{\eta}$ between the
T=0 value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature
Tc0 of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.