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Progress of Theoretical Physics

Cluster Expansion for Quantum Gases. II

Hiroshi Ichimura

Published Mar 1, 1967• 0 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.37.484

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Journal of Mathematical Physics•Feb 1, 1965•58 citations
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.
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.
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).
The method developed in the previous paper for the treatment of the quantum-mechanical NNN-body hard-sphere problem is applied to a calculation of the grand partition function of an imperfect Bose gas with hard-sphere interactions. The grand partition function is calculated to second order in an expansion in powers of $\frac{a}{\ensuremath{\lambda}}$, where aaa is the hard-sphere diameter and $\ensuremath{\lambda}$ the thermal wavelength. The approximate equation of state for the gas phase is thereby obtained by calculating all the virial coefficients to order ${(\frac{a}{\ensuremath{\lambda}})}^{2}$.The first-order energy levels obtained in the previous paper embody some interesting physical properties. A fictitious system with exactly such energy levels is considered. The partition function for such a system can be calculated exactly and the exact equation of state obtained. It is shown that there is a phase transition, which more closely resembles an ordinary gas-liquid transition than the Bose-Einstein condensation.
The reduced density matrices of quantum gases are studied by means of a Wiener integral representation described in a previous paper. They are shown to satisfy a cluster property in the form of an absolute integrability condition of the natural quantum analogues of the Ursell functions, considered as functions of the differences of their arguments. Use is made of the natural transposition to the quantum case of the algebraic formalism introduced by Ruelle in the classical case. By-products are two results on the signs of the coefficients of the Mayer expansion, in the case of Maxwell-Boltzmann and Fermi-Dirac statistics, respectively.