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Journal of Mathematical Physics

Perturbation Theory of a Nonideal Bose Gas. I

Indira Ramarao, Franz Mohling

Published Aug 1, 1971• 3 citations
Publisher

Abstract

With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the average energy and the momentum distribution through second order in the two-body interaction potential. A phenomenological theory of helium II is proposed. The chief underlying feature of this phenomenological theory, which is shown to have a firm basis in the microscopic theory, is the presence of a double quasiparticle spectrum. Phenomenological expressions for the average energy and other quantities of interest are also proposed. Simple prescriptions are obtained for calculating the quasiparticle energies and the chemical potential from the average energy. Via these prescriptions, the quasiparticle energies and the chemical potential are calculated through second order in the interaction potential at T ≠ 0 °K, and the results are compared with earlier work in the literature.

DOI
10.1063/1.1665786

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With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the average energy and the momentum distribution through second order in the two-body interaction potential. A phenomenological theory of helium II is proposed. The chief underlying feature of this phenomenological theory, which is shown to have a firm basis in the microscopic theory, is the presence of a double quasiparticle spectrum. Phenomenological expressions for the average energy and other quantities of interest are also proposed. Simple prescriptions are obtained for calculating the quasiparticle energies and the chemical potential from the average energy. Via these prescriptions, the quasiparticle energies and the chemical potential are calculated through second order in the interaction potential at T ≠ 0 °K, and the results are compared with earlier work in the literature.
With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the radial distribution function and the structure factor through second order in the two-body interaction potential for a many-boson system below the Bose-Einstein transition temperature. For the special case of a dilute hard-sphere Bose gas at T=0° K, the structure factor is calculated in the low-momentum limit keeping terms through second order in the hard-sphere gas parameter (n a3)1/2, where n is the gas density and a is the hard-sphere diameter. The resulting expression for the structure factor is shown to satisfy the Feynman-Bijl relation explicitly.
The wave functions of the ground (Ψ 0 ) and the first excited (Ψ k ) states of He II in the second-order approximation, i.e., up to the first two corrections to the corresponding solutions for a weakly nonideal Bose gas, are determined by the collective variable method, which was proposed by Bogolyubov and Zubarev and developed in the studies by Yukhnovskii and Vakarchuk. The functions Ψ 0 and Ψ k = ψ k Ψ 0 are determined as the eigenfunctions of the N -particle Schrödinger equation from a system of coupled equations for Ψ 0 , Ψ k , and the quasiparticle spectrum E ( k ) of helium II. The results consist in the following: (1) these equations are solved numerically for a higher order approximation compared with those investigated earlier (the first-order approximation), and (2) Ψ 0 and ψ k are derived from a model potential of interaction between He 4 atoms (rather than from the structure factor as earlier) in which the potential barrier is joined with the attractive potential found from experiment. The height V 0 of the potential barrier is a free parameter. Except for V 0 , the model does not have any free parameters or functions. The calculated values of the structure factor, the ground-state energy E 0 , and the quasiparticle spectrum E ( k ) of He II are in agreement with the experimental values for V 0 ≈ 100 K. The second-order correction to the logarithm of Ψ 0 significantly affects the value of E 0 and provides the asymptotics E ( k → 0) = ck , while the second-order correction to ψ k slightly affects the E ( k ). The second-order corrections to Ψ 0 and ψ k have a smaller effect on the results compared with the first-order corrections, whereby the theory is in agreement with experiment; therefore, one may assume that the truncated Ψ 0 and ψ k well describe the microstructure of He II. Thus, the series for Ψ 0 and Ψ k can be truncated in spite of the fact that the expansion parameter is not very small (∼1/2).
Bose system of zero spin particles is considered in the presence of the Bose–Einstein condensate in the vicinity of the phase transition point. The system is investigated in the framework of the Bogolyubov model with the separated condensate. In this model an effective Hamiltonian of the system is introduced by replacing condensate creation and annihilation operators in system Hamiltonian by n01/2 where n0 is occupation number of the condensate state. According to Bogolyubov, the grand canonical thermodynamic potential related to the effective Hamiltonian is considered as nonequilibrium thermodynamic potential. In the present paper this potential is investigated as a function of the small variable n0. With the help of the thermodynamic perturbation theory it is shown that it is expanded in a series over integer powers of n0. This corresponds to the basic idea of the Landau theory of the phase transitions of the second kind. Coefficients at terms of the first and second orders in n0 in the expansion are calculated for Bose gas in the main approximation in small interaction. Calculation of the coefficients at terms of the third and fourth orders needs accounting contributions of the thermodynamic perturbation theory at least of the 4th order and will be done elsewhere. It is established that the results obtained for Bose gas do not fit into the Landau theory of phase transitions of the second kind. Some comments that discuss the situation are given.
The correct qualitative features of the helium II elementary excitation spectrum are derived microscopically for a realistic interparticle potential using a form of pair Hamiltonian. The strong repulsive core is included by using reaction matrix elements in the Hamiltonian, and the attractive well is successfully included by assuming a generalized Bose-Einstein condensation. The pair Hamiltonian is diagonalized by the thermodynamically equivalent Hamiltonian method of Wentzel. Numerical solution of the equations yields spectra with phonon and roton regions. Since the energies in the spectrum are too high when the Yntema-Schneider potential is used, another potential is constructed which still fits the second-virial-coefficient data and gives better results. Spectra calculated for a series of attractive well strengths show the effect of weaker wells.
The analysis of the preceding two papers in this series is applied to the specific case of a degenerate Bose fluid, in which there can be macroscopic occupation of the zero-momentum state (in a system at rest). The quasiparticle ensemble of Mohling and Tuttle is incorporated into the theory in order to guarantee that Nernst's theorem for the entropy is satisfied. It is shown that a double-quasiparticle model emerges naturally from the theory, as developed earlier, and this feature is then incorporated into the quasiparticle ensemble. The general theory is prepared, with the aid of the $\ensuremath{\Lambda}$ transformation, for calculational applications. A phenomenological theory of the degenerate Bose fluid, based on the double-quasiparticle model, is presented, but no significant physical implications are deduced from it.
The simultaneous effects of the condensate and of the long-range nature of the interaction have made the high-density charged Bose gas an interesting theoretical problem. In this paper we present a straight-forward procedure which takes these effects into account. It is based on the perturbation theory in the Bogoliubov representation and the principle of the compensation of dangerous diagrams. Both the ground-state energy and the excitation spectrum are studied. Numerical results are given for the first two orders in rs34{{r}_{s}}^{\frac{3}{4}}rs​43​. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.