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Progress of Theoretical Physics

Cluster Expansion for Degenerate Bose Fluids

T. Morita

Published May 1, 1968• 1 citation
Publisher

Abstract

The diagram expansions which are applicable to the degenerate Bose fluids at finite temperatures are derived within the framework of the canonical ensemble. They are natural extensions of the diagram expansions due to Bloch and Dominicis, Lee and Yang, and Montroll and Ward for the non-degenerate case. The formulation is performed as carefully as Kromminga and Bolsteri treated the Bose fluids at zero temperature.

DOI
10.1143/ptp.39.1100

Related papers

Theory of Degenerate and Non-Degenerate Bose Fluids and Calculation of the Pair Distribution Function

Tohru Morita

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.

Progress of Theoretical Physics Supplement•Jan 1, 1969•2 citations

Cluster Expansion for Quantum Gases. IV

Hiroaki Hara, Hiroshi Ichimura

A form of the cluster expansion which will be suitable for degenerate quantum gases is introduced based on the grand canonical distribution and its relation to the usual Green's function formalism is discussed. The formula for the temperature dependence of the “condensed part” of the hard-sphere Bose gas is derived and the location of the Bose-Einstein condensation temperature of the system is discussed.

Progress of Theoretical Physics•Dec 1, 1972•0 citations

Cluster Expansion for Quantum Gases. III

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The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.

Progress of Theoretical Physics•Apr 1, 1968•1 citation

Long-Distance Behaviors of the Pair Distribution Function of Degenerate and Non-Degenerate Bose Fluids

Tohru Morita, Hiroaki Hara

The formalism proposed for degenerate Bose fluids by one of the present authors [T. Morita, Prog. Theor. Phys. 39 (1968), 1100J is extended for the calculation of the one- and two-particle distribution functions. With the aid of the diagram expansion, it is shown that the long-distance l/r and 1/r2 behaviors of the, pair distribution function of the degenerate ideal Bose are exactly canceled in degenerate interacting Bose fluids in general. In the proof, it is assumed that vk/k->= as k-70 for the weakly interacting case and.sa(k, 0) /k-0= as k-00 for the general case, where Vk and .saCk, 0) are the Fourier transforms of interacting potential and of a generalized binary collision kernel, respectively. It is stressed that the Hartree-Fock approximation is insufficient in showing the cancellation of the l/r2 term. In fact, the cancellation is attained by taking account of the averages of the type: <afo-I-k ak ap) and <ap* ak* ap-I,k). For non-degenerate case, the 1/7-2 behavior, which appears Just above the A-point of the ideal Bose gas is als9 shown to be canceled in interacting Bose fluids.

Progress of Theoretical Physics•Jan 1, 1969•5 citations

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

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

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.

Physical Review A•Sep 1, 1970•4 citations

Cluster Expansion for Quantum Gases. II

Hiroshi Ichimura

A modified cluster expansion formula of the reduced density matrix for quantum gases is derived following the procedure analogous to the method which has been introduced in a previous paper (Prog. Theor. Phys. 31 (1964), 538) in connection with the cluster expansion of the partition function. This new cluster expansion formula is useful in highly degenerate cases. As an example, the effect of the hard-sphere interaction on the Bose-Einstein condensation is discussed.

Progress of Theoretical Physics•Mar 1, 1967•0 citations
The diagram expansions which are applicable to the degenerate Bose fluids at finite temperatures are derived within the framework of the canonical ensemble. They are natural extensions of the diagram expansions due to Bloch and Dominicis, Lee and Yang, and Montroll and Ward for the non-degenerate case. The formulation is performed as carefully as Kromminga and Bolsteri treated the Bose fluids at zero temperature.
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.
The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.
The formalism proposed for degenerate Bose fluids by one of the present authors [T. Morita, Prog. Theor. Phys. 39 (1968), 1100J is extended for the calculation of the one- and two-particle distribution functions. With the aid of the diagram expansion, it is shown that the long-distance l/r and 1/r2 behaviors of the, pair distribution function of the degenerate ideal Bose are exactly canceled in degenerate interacting Bose fluids in general. In the proof, it is assumed that vk/k->= as k-70 for the weakly interacting case and.sa(k, 0) /k-0= as k-00 for the general case, where Vk and .saCk, 0) are the Fourier transforms of interacting potential and of a generalized binary collision kernel, respectively. It is stressed that the Hartree-Fock approximation is insufficient in showing the cancellation of the l/r2 term. In fact, the cancellation is attained by taking account of the averages of the type: <afo-I-k ak ap) and <ap* ak* ap-I,k). For non-degenerate case, the 1/7-2 behavior, which appears Just above the A-point of the ideal Bose gas is als9 shown to be canceled in interacting Bose fluids.
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.