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Physical Review A

Microscopic Theory of Quantum Fluids. III. Quasiparticle Ensemble for Degenerate Bose Fluid

Franz Mohling, Indira Ramarao, Dion W. J. Shea

Published Sep 1, 1970• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.2.1002

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Physical Review A•Jun 1, 1970•1 citation
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 \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ 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.
Recently the present author developed a diagram formalism applicable to the degenerate Bose fluids.The present author and Hara applied the formalism to show that the long distance 1/r and 1/r 2 behaviors of the pair distribution function are exactly cancelled for interacting Bose fluids.The basic ideas of the formalism have been given in separate articles.It is the purpose of the present article to present the formalism in a unified form.The formalism is developed in the canonical ensemble.However the generating function is used.Hence a criterion of the equivalence of the canonical and grand canonical ensembles is given and used in the discussions.Replacement of operators ao and at by a c•number na' 2 is performed with the same degree of care as Kromminga and Bolsteri took in treating the zero-temperature problem.Hugenholtz and Pines's relation is generalized to finite temperatures.The cancellation of the long-distance 1/r and 1/r 2 behaviors at and below the 'A temperature is briefly discussed.For some of details, readers are requested to read the preceding articles.§ 1. § 2. § 3. § 4. § 5. § 6. § 7. § 8. § 9. §10.§11.§12.§13.§14.§15.§16.Al.Contents Introduction, 70 Operators for the one-and two-particle distribution functions, 73 Second quantization, 73 Canonical and grand canonical ensemble averages, 76 Equivalence of the canonical and grand canonical ensembles, 80 Ideal Bose gas, 83 Diagram expansion for non-degenerate interacting Bose fluids, 87 Lee and Yang's expansion and binary collision expansion, 90 Equivalent Hamiltonian and the averages equivalent to the canonical averages of degenerate Bose fluids, 91 Diagram expansion for degenerate interacting Bose fluids, 95 Distribution functions of the ideal Bose gas, 99 Chain approximation for the degenerate case, 100 Bogolyubov Hamiltonian and Bogolyubov transformation, 102 Cancellation of 1/r and 1/r 2 terms in pC 2 )(r) for the degenerate case 104 Cancellation of 1/r 2 terms in pC 2 )(r) just above TJ.., 107 Discussions and comparison with related works, 108 Calculation for the second quantization, 111 T. Morita A2.Alternative formulation of the diagram expansiOn for nondegenerate Bose fluids, 112 A3.Alternative formulation of the diagram expansion for degenerate Bose fluids, 115 References, 116 )Comparing this fact with the formula (1•2), Hiroike 4 ) noticed that the divergence of the isothermal compressibility is due to this long-range behavior of the pair distribution function.*)We now compare the above situation with the phase transition of liquid helium.The phase diagram is shown in Fig. 2. The p-v isotherms for *' In a recent paper, Luban and Rezven 5 > expressed a similar point of view.
In the first paper of this series, the master-graph formulation of the Lee-Yang quantum statistical theory for a degenerate Bose system was derived. In the present paper, this entire theory is transformed from a free-particle description to a quasiparticle description by means of a $\ensuremath{\Lambda}$ transformation. This transformation leaves the master-graph formulation of the theory essentially unchanged, while at the same time introducing the quasiparticle energy-momentum relation and the quasiparticle interaction function into the theory. The transformation is motivated by a study of the very low-temperature behavior of the fundamental integral equation of the theory. An interesting feature of the transformed theory is that it contains two quasiparticle energy-momentum relations, ${\ensuremath{\epsilon}}_{+}(k)$ and ${\ensuremath{\epsilon}}_{\ensuremath{-}}(k)$. The understanding of this result is not achieved in this paper, although the possibility of a double quasiparticle spectrum has previously been suggested by Lieb.
A simplified derivation of the master-graph formulation of the quantum-statistical theory of an interacting quantum fluid is presented. A complete analysis of the self-energy structures arising from both particle statistics and particle interactions simplifies the master-graph expressions for the thermodynamic properties of the fluid. The xxx-ensemble formulation of Lee and Yang is used to treat the degenerate Bose fluid below its Bose-Einstein condensation temperature. The results for a Bose fluid above its Bose-Einstein condensation temperature and for a Fermi fluid are obtained as special cases.
Abstract In this chapter we outline the microscopic theory of an interacting Bose fluid. The aim is to provide a basis for a microscopic understanding of the excitations in superfluid and normal 4He discussed in the following chapters. The microscopic picture can be omitted on a first reading and referred to when desired.We assume that the reader is familiar with the theory of Bose-Einstein condensation in a Bose gas introduced in §7.2 and presented in many texts on statistical mechanics. In §7.2 we also discussed the generalization of Bose condensation to an interacting fluid by Penrose and Onsager (1956).Condensation was identified with macroscopic occupation of one of the eigenstates of the fluid and the appearance of an infinitely long tail in the one-body density matrix, denoted as off-diagonal long range order (ODLRO). For a homogeneous fluid it is natural to take momentum states as the eigenstates and to assume that the macroscopically occupied state is the state with momentum p = 0. In this case the criterion reduces to that for a gas but with a firm foundation.
In a previous paper, a Hartree-Fock theory of the quantum Ursell expansion for Fermi fluids was developed. The same techniques are used in this paper to derive a theory of degenerate Bose fluids from the xxx-ensemble theory of Lee and Yang. It is shown that this theory is equivalent to the $\ensuremath{\Lambda}$-transformed theory recently derived by Mohling, RamaRao, and Shea. The present approach considerably simplifies the formalism and eliminates the need for the $\ensuremath{\Lambda}$ transformation. By working entirely in a quasiparticle framework, we also avoid the problems with unphysical line factors which have appeared in earlier theories. Finally, a greatly simplified $\ensuremath{\Lambda}$ transformation is obtained for use in problems in which the potential contains an infinitely repulsive core.