Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Progress of Theoretical Physics

A Note on the Cluster Expansion for Quantum Gases

Hiroshi Ichimura

Published Apr 1, 1964• 2 citations
Publisher

Abstract

Combining the techniques which have been introduced by several authors, we derive an expansion formula for the partition function of the quantum gases which will be useful in the case of high degeneracy. Our formula may be regarded as a straightforward extension of the result of Kahn and Uhlenbeck. Diagram techniques are unnecessary in the course of the derivation. Application of our formula to the system of hard sphere fermions reproduces quite easily the results of previous authors.

DOI
10.1143/ptp.31.538

Related papers

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

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

Hiroaki Hara, Hiroshi Ichimura, Tohru Morita

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

A Method in Quantum Statistical Mechanics, III

Hiroshi Ichimura

As an application of the general expansion formula of the grand partition function for the system of interacting particles, we derive the virial expansion formula for quantum gases under the assumption that the degeneracy parameter λ is very small. We get a formula for the second virial coefficient which is effective at extremely low temperatures.

Progress of Theoretical Physics•Jun 1, 1954•0 citations

Pfadintegrale und Cluster

Peter Borrmann

Recursion formulas are derived for Fermi-Dirac and Bose-Einstein statistical partition functions for systems where the energy can be written as a sum of one particle energies. Thus their exact partition functions can be calculated for any N. We consider a system of N particles for which the energy can be written as a sum of one-particle energies such as ideal gases or simple shell models. We aim at a recursion of Z(N)/ in terms of Z for subsystems. The idea may be seen as a vague analogy to cluster expansion methods.

•Jan 1, 1998•0 citations

Simple Derivation of Cluster Expansions for Many-Fermion Systems

J. W. Clark, Dennis A. Chakkalakal, Paul A. Westhaus

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.

Progress of Theoretical Physics•Nov 1, 1965•3 citations
Combining the techniques which have been introduced by several authors, we derive an expansion formula for the partition function of the quantum gases which will be useful in the case of high degeneracy. Our formula may be regarded as a straightforward extension of the result of Kahn and Uhlenbeck. Diagram techniques are unnecessary in the course of the derivation. Application of our formula to the system of hard sphere fermions reproduces quite easily the results of previous authors.
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.
Recursion formulas are derived for Fermi-Dirac and Bose-Einstein statistical partition functions for systems where the energy can be written as a sum of one particle energies. Thus their exact partition functions can be calculated for any N. We consider a system of N particles for which the energy can be written as a sum of one-particle energies such as ideal gases or simple shell models. We aim at a recursion of Z(N)/ in terms of Z for subsystems. The idea may be seen as a vague analogy to cluster expansion methods.
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.