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

Abstract

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.