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Progress of Theoretical Physics

A Note on the Modified Cluster Expansion for Quantum System

Hiroaki Hara

Published Aug 1, 1976• 0 citations
Publisher

Abstract

A systematic treatment of the rearrangement formula of WN in the modified cluster expansion method due to Ichimura is given. This method is easily generalized to the case of a multi-component system. By the use of the present procedure and the corresponding diagrammatic representations, it is shown that the h-particle reduced density matrix ρ(h) and the grand canonical distribution Ξ (T, V, µ) for the multi-component system are easily obtained in a similar manner to the single-component case, where A(=h or µ) denotes a set of {Aα} and Aα is a quantity concerned with the species α(= a, b, …, f).

DOI
10.1143/ptp.56.450

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Progress of Theoretical Physics•Dec 1, 1972•0 citations
A systematic treatment of the rearrangement formula of WN in the modified cluster expansion method due to Ichimura is given. This method is easily generalized to the case of a multi-component system. By the use of the present procedure and the corresponding diagrammatic representations, it is shown that the h-particle reduced density matrix ρ(h) and the grand canonical distribution Ξ (T, V, µ) for the multi-component system are easily obtained in a similar manner to the single-component case, where A(=h or µ) denotes a set of {Aα} and Aα is a quantity concerned with the species α(= a, b, …, f).
A simple alternative derivation of conventional cluster expansion formulas of Iwamoto and Yamada and others for the expectation value ≪H > of the energy of a many-Fermion system is offered. The assumed trial function is of the general form FΦp/(FΦp, FΦp)1/2, where Φp is a Slater determinant of appropriate single-particle functions and the factor F is attached to account for the possibility of strong interparticle correlations. The proposed scheme is applicable to finite as well as infinite systems and is easily generalized to the calculation of non-diagonal as well as diagonal matrix elements, with respect to the members of the set {FΦp/(FΦp, FΦp)1/2}. Explicit results for ≪H > are displayed to second cluster order, contributions being classified according to their assigned order of magnitude in a correlation parameter ξ=|Nω/Ω| of the sort introduced by Iwamoto and Yamada and Clark and Westhaus.
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 first of two papers in which a systematic scheme for renormalizing the reduced density matrices of an initially inhomogeneous quantum fluid is introduced. In this paper, an expression for the perturbation Liouville expansion of the n-particle reduced density matrix is formulated in the momentum representation using the grand ensemble. A generating function for the reduced density matrices is introduced and the theory is formulated in such a way that the binary Liouville expansion may easily be introduced in place of the perturbation expansion.
We develop diagrammatic methods which give the explicit expression for the n th order term of the linked Iwamoto-Yamada cluster expansion in terms of the cluster integrals. The method applies independently of the number A of particles of the system. We carry through partial summations which for finite A result in a representation of the expansion by skeleton diagrams, whereas in the limit A → ∞ the expansion is reduced to two analytical equations. In addition, for finite A the renormalization establishes a connection between the expectation value of an arbitrary operator and the probability to find n hole states, 1 ≤ n ≤ A, occupied. Simple considerations raise some doubts concerning the convergence of the unrenormalized expansion.