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Progress of Theoretical Physics Supplement

Spatial Correlation of Particles in Many-Body Systems

A. Isihara

Published Jan 1, 1969• 19 citations
Publisher

Abstract

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.

DOI
10.1143/ptps.44.1

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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.

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•Jan 1, 1963•0 citations
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.
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.
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 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 $\ensuremath{\varphi}=8\ensuremath{\pi}a\ensuremath{\delta}({r}_{1}\ensuremath{-}{r}_{2})$ 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.
By means of a quantum mechanical phase space distribution function introduced by von Roos, the Schroedinger equation for a non-relativistic system of N identical particles with scalar interactions is transformed into a quantum mechanical generalization of the Liouville equation, thereby formulating the problem in terms of a generalized density in phase space, a quantity of primary interest in most treatments of the corresponding classical system (or plasma). This transformation permits a parallel development of the theories of classical and quantum plasmas and thus allows the quantum many-body problem to be discussed virtually completely in classical terms. In particular, a kinetic theory of quantum plasmas is obtained by deriving the quantum analogue of the BBGKY hierarchy, and applying thereto approximation techniques similar to those of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from similar previous studies based on the Wigner distribution function is that the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations, in which case the quantum BBGKY system reduces to a simple quantum generalization of the Vlasov equation. This equation is used to study the response of spatially homogeneous systems to weak external forces, and the associated problems of plasmon and spin-wave excitations. It is also used to derive the quantum and exchange corrected equations of inviscid hydrodynamical transport which are then applied to the problem of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the many-electron atom in the Hartree and Hartree-Fock approximations. The relevant quantum Vlasov equations lead naturally to a statistical theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and exchange corrections to these models are simultaneously generated. The quantum hydrodynamical theory developed earlier is used to determine the influence of these corrections on the boundary conditions of the model, and a theory of the compressed atom is consequently obtained. Considered in somewhat less detail are the effects of non-zero temperature, net orbital angular momentum, relativity and correlations, as well as time dependent processes. The final part deals with the problem of the degenerate electron gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with parallel and anti-parallel spin are obtained by neglecting three particle correlations. From these functions, a quantum-mechanical collision integral is derived which differs from that obtained by Silin and Guernsey and conjectured by Wyld and Pines in that dynamical exchange effects are included. Also obtained from the pair function is an expression for the correlation which reduces in the high density limit to the result of Gell-Mann and Brueckner. At intermediate densities an additional term appears in the energy due to the screening of the exchange interaction by the dielectric properties of the medium. It is evaluated in the high density limit and found to be -0.151 r_s ln r_s Ryd/electron in marked disagreement with the corresponding value obtained by DuBois.