In this work we show that the introduction of a U(1) symmetry breaking field in the energy operator of the boson-free gas, is equivalent, in the thermodynamic limit, to the inclusion, in the Hamiltonian of the ideal gas, of a non-linear function of the number operator associated with the zero mode. In other words, the limit pressures coincide. Moreover, both models undergo non conventional Bose-Einstein condensation (BEC) for strictly negative values of the chemical potential
μ. Finally, the proof of equivalence of limit pressures is extended to a class of full-diagonal models.
We have recently demonstrated that, when investigating the internal energy and pressure of an ideal boson gas at temperatures below the Bose–Einstein temperature, it is necessary to include the contribution from the kinetic energy of the particles in the condensate. Here we express the same requirement in terms of the constraints of the Heisenberg uncertainty principle applied to the particles confined in the condensate. The problem may be used by undergraduate students as an additional example of the connection between the ground state energy of a system and the Heisenberg uncertainty principle.
We present theoretical aspects concerning the thermodynamics of an ideal bosonic gas trapped by a harmonic potential. Working in the Grand Canonical ensemble we are able to properly identify the extensive thermodynamic variable equivalent to the volume and the intensive thermodynamic variable equivalent to the pressure. These are called the "harmonic volume" and the "harmonic pressure" and their physical meaning is discussed. With these variables, the problem of Bose-Einstein condensation is studied in terms of the behavior of the corresponding equation of state and in terms of measurable susceptibilities such as the heat capacities, the isothermal compressibility and the coefficient of thermal expansion. From the analysis, an interesting analogy with Black-Body radiation emerges, showing that at and below the critical temperature, the non-condensate fraction of atoms behaves thermodynamically like a gas of massless particles.
The existence of a Bose-Einstein condensation in an interacting many-boson system at T = 0°K is proved under certain conditions on the particle density and the interparticle potential. Starting with the tentative assumption that the condensation exists, we study the fluctuation in the occupation number of the condensate with due regard to its interactions (1) with particles outside the condensate as well as (2) with the fluctuation itself. If the condensate fluctuation has a normalizable ground state, then the assumed existence of the condensation is tenable. For the case of the pair-Hamiltonian model satisfying the conditions for condensation, the interactions of the second category are of no importance. In the limit of infinite volume, this Hamiltonian can be diagonalized in an irreducible representation of a Bose-field operatorφ(x), whereφ(x) has nonvanishing ground state expectation value, in accordance with the usual c-number replacement of creation and destruction operators for the condensate particles. The full Hamiltonian for a system of pairwise interacting bosons is studied only in a low-density limit. Bose-Einstein condensation exists when the over-all space integral of the interparticle potential is positive. In this case the interactions of the second category play an important role in ensuring a normalizable ground state for the condensate fluctuation. There is an indication that in the limit of infinite volume the Hamiltonian cannot be diagonalized in any irreducible representation of the field operatorφ. Yet the c-number replacement of the condensate operators is legitimate as far as states of particles outside the condensate are concerned. Some speculations are made as to what may happen for systems of moderate density.
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
Tc, 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<Tc 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(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
Tc 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
He4.
We present an exactly soluble boson model which manifests two kinds of condensation. They occur in two stages: for intermediate densities one has a non-conventional Bose condensation in the lowest mode k = 0, which is due to a diagonal perturbation of the imperfect Bose gas Hamiltonian, whereas for large densities , this condensation coexists with conventional (generalized, non-extensive) Bose-Einstein condensation in non-zero modes condensation, corresponding to the standard mechanism of saturation.