Abstract We discuss thermodynamic properties of harmonically trapped imperfect quantum gases. The spatial inhomogeneity of these systems imposes a redefinition of the mean-field interparticle potential energy as compared to the homogeneous case. In our approach, it takes the form , where N is the number of particles, —the harmonic trap frequency, d —system’s dimensionality, and a is a parameter characterizing the interparticle interaction. We provide arguments that this model corresponds to the limiting case of a long-ranged interparticle potential of vanishingly small amplitude. This conclusion is drawn from a computation similar to the well-known Kac scaling procedure, which is presented here in a form adapted to the case of an isotropic harmonic trap. We show that within the model, the imperfect gas of trapped repulsive bosons undergoes the Bose–Einstein condensation provided d > 1. The main result of our analysis is that in d = 1 the gas of attractive imperfect fermions with is thermodynamically equivalent to the gas of repulsive bosons with provided the parameters and fulfill the relation . This result supplements similar recent conclusion about thermodynamic equivalence of two-dimensional (2D) uniform imperfect repulsive Bose and attractive Fermi gases.
We consider two-dimensional imperfect attractive Fermi and repulsive Bose gases consisting of spinless point particles whose total interparticle interaction energy is represented by
aN2/2V with $a=\ensuremath{-}{a}_{F}\ensuremath{\le}0$ for fermions and $a={a}_{B}\ensuremath{\ge}0$ for bosons. We show that, in spite of the attraction, the thermodynamics of a
d=2 imperfect Fermi gas remains well defined for $0\ensuremath{\le}{a}_{F}\ensuremath{\le}{a}_{0}={h}^{2}/2\ensuremath{\pi}m$, and is exactly the same as the one of the repulsive imperfect Bose gas with ${a}_{B}={a}_{0}\ensuremath{-}{a}_{F}$. In particular, for
aF=a0 one observes the thermodynamic equivalence of the attractive imperfect Fermi gas and the ideal Bose gas.
The effects of a finite number of particles on the thermodynamic properties of ideal Bose and Fermi gases trapped in any-dimensional harmonic potential are investigated. The orders of relative corrections to the thermodynamic quantities due to the finite number of particles are estimated in different situations. The results obtained for the two trapped quantum gases are compared, and consequently, it is shown that the finite-particle-number effects for the condensed Bose gas (a Bose gas with Bose–Einstein Condensation (BEC) occurring in the system) are much more significant than those for the Fermi gas and normal Bose gas (a Bose gas without BEC).
Motivated by quantum statistical mechanics, we propose an accurate analytical solution to the problem of Bose–Einstein condensation (BEC) of ideal bosons in a two-dimensional anisotropic harmonic trap. The study reveals that the number of noncondensed bosons is characterized by an analytical function, which relates to a series expansion of q-digamma functions in mathematics. The q-digamma function is a function of temperature, boson number, and anisotropic parameter. The analytical solution describes fully the experimental results of the BEC of ideal bosons in a two-dimensional anisotropic harmonic trap. We derive the analytical expressions of the critical temperature and the condensate fraction in the thermodynamic limit. The first main conclusion is that for a fixed temperature and boson number, there is a critical anisotropic parameter, which is the precise onset of BEC in this harmonically trapped two-dimensional system. The second main conclusion is that the critical temperature in a two-dimensional anisotropic harmonic trap is larger than that in a two-dimensional isotropic harmonic trap.
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.
We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose-Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit.
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.