Thermodynamics of an Ideal Bose Gas with a Finite Number of Particles Confined in a Three-Dimensional Quartic Trap

Abstract

Within an exact canonical-ensemble treatment, we investigate the thermodynamics for a finite number of ideal bosons confined in a three-dimensional quartic trap. We calculate several physical quantities including the specific heat C _N , chemical potential μ, condensate fraction 〈n ₀〉/N, root-mean-square fluctuations δn ₀ of the condensate population, and transition temperature T _c . We discuss the particle-number dependence of T _c through proposing three T _c definitions, which are compared with ones derived in the grand canonical ensemble.

Appendix: Thermodynamics of an IBG in the Thermodynamic Limit

For an IBG in a quartic potential, one has the density of states [25]

(13)

The total number N of the particles as a function of density of states is given by

(14)

which, after integration, becomes

(15)

where N ₀ is the particle number of the ground state, z=e ^μ/kT a fugacity, and $g_{n}(z) =\sum_{l=1}^{\infty}\frac{z^{l}}{l^{n}}$ denote the Bose-Einstein functions. The mean energy U of the N-particle system is

(16)

where the energy of the ground state ϵ ₀=0 has been used. The specific heat capacity can be determined by differentiation with respect to temperature, namely, $C_{N}=\frac{\partial U}{\partial T}$, which leads to the specific heat above the critical temperature $T_{c}^{0}$

(17)

(18)

and the specific heat below $T_{c}^{0}$

(19)

When $T>T_{c}^{0}$, we obtain by using the condition $\frac{\partial N}{\partial T}=0$

(20)

It is not verify to find that

(21)

Clearly, in the thermodynamic limit the specific heat becomes discontinuous at the critical temperature $T_{c}^{0}$ and its maximum is equal to 5.0805. The magnitude of the jump $\frac{\Delta C_{N}}{N k}=\frac{9}{4} \frac{\varGamma(13/4)}{\varGamma(9/4)} \frac{\zeta(\frac{9}{4})}{ \zeta(\frac{5}{4})}\simeq1.6087$ is quite significant. It indicates that the specific heat of quartically trapped, ideal bosons at the onset of condensation displays some resemblance to the specific heat of liquid ⁴He in the vicinity of the λ-point. As in the case of harmonic trap, the reason for the emergence of the λ-like phase transition for the quartic trap is attributed to the trapping potential instead of the interactions between atoms in the case of ⁴He.