The condition which has to be fulfilled in order that the Bose-Einstein condensation be accompanied by a discontinuity of the heat capacity at the condensation temperature is discussed. The nature of the transition depends on the distribution of the energy levels of the individual particles forming the gas. The condensation is studied for several types of individual energy spectra. It is shown that all those Bose-Einstein (B.E.) assemblies which admit a non-ambiguous definition of the statistical distribution function present, beside interesting analogies, a qualitatively different behavior from that observed in liquid helium. It appears that as long as the transition in momentum space of non-ideal B.E. systems is not understood, the possibility of connection between B.E. condensation and the transition observed in liquid helium remains an open problem.
It is shown that for both the Hartree-Fock and Bogoliubov models of interacting bosons associated with the disappearance of the Bose-Einstein condensation at a temperature
Tc, the specific heat
CV has a square root singularity, ${C}_{V}\ensuremath{\sim}A{({T}_{c}\ensuremath{-}T)}^{\ensuremath{-}\frac{1}{2}}$, and the superfluid density ${\ensuremath{\rho}}_{S}$ is discontinuous, with ${\ensuremath{\rho}}_{S}(T)\ensuremath{-}{\ensuremath{\rho}}_{S}({T}_{c}\ensuremath{-}0)\ensuremath{\sim}B{({T}_{c}\ensuremath{-}T)}^{\frac{1}{2}}$ for temperatures $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}0$. Except near ${T}_{c}={T}_{\ensuremath{\lambda}}$ the theoretical and experimental results for ${\ensuremath{\rho}}_{S}$ are in good agreement.
In the formulation of the theory of the interacting many-boson system, the sign of the Fourier Transform of the two-particle interaction in the limit of long-wavelengths plays a dominant role in determining the character of the theory. With this in mind, we have obtained the following result: If the interaction between two neutral atoms is considered in the adiabatic approximation, and if the electronic wave functions correspond to the lowest energy state, then the sign of the scattering length will be negative. For the case of negative scattering lengths, an expansion is given for the ground-state energy of a system of interacting Bose particles. The expansion is performed in a way that eliminates the third-order divergence which results when the entire interaction is used as the perturbation. The structure factor, computed to first order, is in qualitative agreement with experiment, for large momentum values. For non-zero temperatures, expressions are obtained for the depletion of the zero-momentum mode and the specific heat. The results for the specific heat exhibit a discontinuity at the critical temperature. Quantitative estimates for the magnitude of the specific heat at the critical point yield results which are in fair agreement with empirical values for liquid helium. (Author)
We calculate the energy fluctuation of a truly finite number of interacting bosons and study the role of interaction. Although the ideal Bose gas in thermodynamic limit is an exactly solvable problem and analytic expression of various fluctuation measures exists, the experimental Bose-Einstein condensation (BEC) is a nontrivial many-body problem. We employ a two-body correlated basis function and utilize the realistic van der Waals interaction. We calculate the energy fluctuation
(△E2) of the interacting trapped bosons and plot
kB2T2△E2 as a function of
TcT. In the classical limit
△E2 is related to the specific heat per particle
cv through the relation
△E2=kBT2cv. We have obtained a distinct hump in
kB2T2△E2 around the condensation point for three-dimesional harmonically trapped Bose gas when the particle number $N\ensuremath{\simeq}5000$ and above which corresponds to the second-order phase transition. However for finite-size interacting bosons ($N\ensuremath{\simeq}$ a few hundred) the hump is not sharp, and the maximum in
kB2T2△E2 can be interpreted as a smooth increase in the scaled fluctuation below
Tc and then a decrease above
Tc. To illustrate the justification we also calculate
cv, which exhibits the same feature, which leads to the conjecture that for finite-sized interacting bosons phase transition is ruled out.
On a qualitative basis we have examined the identification of the properties of a hard-core Bose–Einstein (BE) gas with those of liquid helium four below the lambda temperature. It is suggested that the use of the generalized condensation of Girardeau together with the inclusion of "three-body ladders" from the perturbation series may improve the agreement between the hard-core BE model system and the physical helium four system to such an extent that reasonable quantitative agreement may be attained.