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Physical Review

Quantum Statistics of the Distribution Functions

A. Isihara, D. D. H. Yee

Published Nov 2, 1964• 17 citations
Publisher

Abstract

A statistical-mechanical theory of evaluating the distribution functions of a many-body system is presented. The theory is a natural extension of the quantum statistical theory of Lee and Yang for the grand partition function and gives a new formalism which is different from that developed recently by Fujita, Isihara, and Montroll. The density matrices are first developed in the Uhlenbeck-de Boer $U$ functions. A diagrammatical consideration separates out nonconnected products from connected products of the $U$ functions, yielding an expansion formula which is simpler than that reported by de Boer some time ago. Application of the resulting expression to free bosons and fermions is made. Then the distribution functions are developed in the binary kernel introduced by Yang and Lee. This expansion is used for the evaluation of the pair distribution function of a hard-sphere Bose gas at the lowest temperature. The results improve upon those reported previously by Lee, Huang, and Yang and others. The normalization and divergence difficulties encountered by Fujita and Hirota are removed. Actually, their interpretation of Lee, Huang, and Yang's results in terms of the chain diagrams is not satisfactory. Instead the chain diagram results may be compared with the more recent results by Wu. Use of the approximate pseudopotential $\ensuremath{\varphi}=8\ensuremath{\pi}a\ensuremath{\delta}({r}_{1}\ensuremath{-}{r}_{2})$ is reflected in insufficiency of the chain diagram approximation. Satisfactory and consistent results are obtained when a new set of diagrams is taken into consideration.

DOI
10.1103/physrev.136.a618

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Physical Review•Sep 15, 1959•32 citations
A statistical-mechanical theory of evaluating the distribution functions of a many-body system is presented. The theory is a natural extension of the quantum statistical theory of Lee and Yang for the grand partition function and gives a new formalism which is different from that developed recently by Fujita, Isihara, and Montroll. The density matrices are first developed in the Uhlenbeck-de Boer UUU functions. A diagrammatical consideration separates out nonconnected products from connected products of the UUU functions, yielding an expansion formula which is simpler than that reported by de Boer some time ago. Application of the resulting expression to free bosons and fermions is made. Then the distribution functions are developed in the binary kernel introduced by Yang and Lee. This expansion is used for the evaluation of the pair distribution function of a hard-sphere Bose gas at the lowest temperature. The results improve upon those reported previously by Lee, Huang, and Yang and others. The normalization and divergence difficulties encountered by Fujita and Hirota are removed. Actually, their interpretation of Lee, Huang, and Yang's results in terms of the chain diagrams is not satisfactory. Instead the chain diagram results may be compared with the more recent results by Wu. Use of the approximate pseudopotential \ensuremathφ=8\ensuremathπa\ensuremathδ(r1\ensuremath−r2)\ensuremath{\varphi}=8\ensuremath{\pi}a\ensuremath{\delta}({r}_{1}\ensuremath{-}{r}_{2})\ensuremathφ=8\ensuremathπa\ensuremathδ(r1​\ensuremath−r2​) is reflected in insufficiency of the chain diagram approximation. Satisfactory and consistent results are obtained when a new set of diagrams is taken into consideration.
An alternative approach to the quantum statistics of interacting particles is proposed. It consists of calculating the equilibrium thermodynamical quantities of the many-body system via the pair distribution function with the assumption that the particles interact with each other only through pair central forces. The proposed approach has some advantage over the usual treatment via the partition function in that the pair distribution function is easier to deal with than the partition function in certain circumstances particularly when the collective motion description of the system is desirable. This is because only the pair distribution function can be expressed directly in terms of the collective interaction which is closely connected with collective elementary excitation, such as a plasmon in the electron gas and a phonon in the hard-sphere Boson gas. The equation of state as well as the internal energy are obtained in the form of integrals of the pair distribution function. The close analogy between the pair distribution function and the two-body propagator, which appears in the quantum field theory, makes it possible to analyze the former by the use of Feynman diagrams identical with those usually introduced for the latter. The collective interaction, which is defined by the sum of the direct and the indirect interactions, is introduced as a particular partial sum of the perturbation series of the pair distribution function. This is used in rewriting the pair distribution function in terms of the collective interaction.It is shown that, while the simple chain approximation to the collective interaction in the electron gas is responsible for the transfer of a plasmon, the same approximation to the collective pseudo-interaction in the hard-sphere Boson system has a relation similar to that of transfer of a phonon, both cases occuring at low temperatures. The explicit calculation of the pair distribution functions for these systems at the absolute zero temperature is carried out up to the first order in the collective interaction (simple chain approximation.) These results are used to calculate the ground-state energies. For the electron gas the energy thus obtained confirms Gell-Mann-Bruckner's calculation of the correlation energy. For the hard-sphere Bosons the calculated energy reproduces the result of Lee, Huang, and Yang. The extension of the calculation to the finite temperature case is also indicated. In particular the classical Debye-Huckel equation of state for the electron gas is briefly discussed.
Quantum field theoretical methods are discussed to evaluate the spatial distribution in many-body systems in equilibrium. The pair and triplet distribution functions are explicitly evaluated for noninteracting cases. A general method to evaluate the pair distribution function in the chain and ring diagram approximations for interacting systems is presented in detail. The result is expressed in terms of the eigenvalues of the propagators characteristic of the diagrams. This result generalized the Gell-Mann and Brueckner formula for the equation of state, the derivation of which is also given. The eigenvalues are evaluated for Boltzmann, Bose and Fermi statistics. Application of the result to a hard-sphere Bose gas is made, and the effect of condensation is studied. It is found that the phonon velocity depends on the number of particles in the lowest state. The distribution functions of a degenerate Fermi system are also calculated.
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 formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain UUU functions defined for the same system with Boltzmann statistics. It is then shown that these UUU functions can be evaluated in successive approximations in terms of a binary kernel BBB which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.
A general method of calculation is described for quantum statistical mechanics. It is based on a simplification of the Laplace transform of the density matrix which follows from a theorem due to Hugenholtz. The basic result is that an element of the density matrix can be written as a sum over graphs, with the contribution of each graph factored into contributions from connected or linked graphs. Applied to the grand partition function, the exponential formula of Bloch and DeDominicis is obtained in a simple way. A similar formula is then derived for the canonical ensemble for the case of a nondegenerate gas. In this way the familiar result of Uhlenbeck and Beth is obtained for the second virial coefficient. Techniques are also introduced for evaluating ensemble averages of operators. In this connection, some care must be exercised in the case of diagonal operators. Finally, these methods are used to calculate the pair-correlation function for a system of fermions interacting through short-range forces.