Statistical Mechanics of a Nonideal Boson Gas: Pair Hamiltonian Model

Abstract

This work is concerned with a microscopic model of a nonideal boson gas. The Hamiltonian is treated in a rigorous manner free of all approximations and, in particular, the Bogoliubov approximation of replacing the zero-momentum single-particle creation and destruction operators by a $c$ number is entirely avoided. Utilizing a method due to Wentzel, the Hamiltonian is replaced by one involving products of two second quantized operators which, in general, can be diagonalized. This replacement is shown to be rigorously valid in the infinite volume limit for all temperatures despite the avoidance of the Bogoliubov approximation. It is found that the assumption of a positive-definite quasiparticle excitation spectrum $\ensuremath{\epsilon}(\mathrm{k})$ becomes untenable for temperatures $T$ less than a critical value ${T}_{c}$, if the interparticle potential is described by a certain kernel which possesses positive eigenvalues only, corresponding to a sufficiently repulsive interparticle force. Assuming that $\ensuremath{\epsilon}(\mathrm{k})=0$ only for k=0, it is found that for $T<{T}_{c}$ the system undergoes an Einstein condensation into the k=0 single-particle state, and the integral equations characterizing the system are precisely the same as would be obtained if the Bogoliubov approximation were made, proving within the framework of this model the strict validity of this approximation procedure. The well-known criterion for an Einstein condensation $v(0)>0$, where $v(\mathrm{k})$ is the Fourier transform of the interparticle potential, is shown to be a weaker statement of our eigenvalue criterion. Further, in agreement with Girardeau and Arnowitt, it is found that an energy gap separates $\ensuremath{\epsilon}(0)$ and $\ensuremath{\epsilon}(\mathrm{k})$ for $k>0$. An approximation method is developed for solving the integral equations which describe those systems which undergo an Einstein condensation, and it is shown that this method can be justified if $|T\ensuremath{-}{T}_{c}|\ensuremath{\lesssim}\frac{{T}_{c}}{10}$, and in cases of short-range repulsive interparticle forces and systems for which $\ensuremath{\rho}v(0)$ possesses a certain prescribed upper bound, $\ensuremath{\rho}$ being the number of particles per unit volume. The transition temperature is found to be lower than the corresponding quantity for the ideal boson gas of the same density. A detailed discussion of the thermal properties of the system in the vicinity of ${T}_{c}$ is also given. In an Appendix a general type of smeared Einstein condensation is assumed, and it is found that for systems in which this property occurs $\ensuremath{\epsilon}(\mathrm{k})$ is linear for small $k$. This ansatz is shown to be tenable only if $v(0)<0$. Finally, by assuming an appropriate pseudopotential representation of a hard repulsive core and by including a weak, longranged attractive force between particles, $\ensuremath{\epsilon}(\mathrm{k})$ shows a nonmonotonic behavior of the same qualitative type as observed in liquid ${\mathrm{He}}^{4}$.