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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Many-body partition function and thermal Hartree-Fock approximations

Behnam Farid, N. H. March, A. K. Theophilou

Published Jul 1, 2000• 3 citations
Publisher

Abstract

The hierarchy equations for the quantum thermal density matrices of an assembly of interacting particles in an external potential are derived in a compact form. This approach can complement and finally replace the phenomenological treatment of the thermodynamic properties of quantum gasses, liquids, and solids. We also derive a temperature Hartree-Fock approximation for fermions based on the symmetry properties of the exact hierarchy and finally we discuss the present approximation in relation to that of Kaplan and Argyres [T. A. Kaplan and P. N. Argyres, Ann. Phys. (N.Y.) 92, 1 (1975)].

DOI
10.1103/physreve.62.134

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The hierarchy equations for the quantum thermal density matrices of an assembly of interacting particles in an external potential are derived in a compact form. This approach can complement and finally replace the phenomenological treatment of the thermodynamic properties of quantum gasses, liquids, and solids. We also derive a temperature Hartree-Fock approximation for fermions based on the symmetry properties of the exact hierarchy and finally we discuss the present approximation in relation to that of Kaplan and Argyres [T. A. Kaplan and P. N. Argyres, Ann. Phys. (N.Y.) 92, 1 (1975)].
Several closely related ab initio thermal mean-field theories for fermions, both well-established and new ones, are compared with one another at the formalism level and numerically. The theories considered are Fermi-Dirac theory; thermal Hartree-Fock (HF) theory; two modifications of the thermal single-determinant approximation of Kaplan and Argyres, Ann. Phys. 92, 1-24 (1975); and the first-order finite-temperature many-body perturbation theory based on a zero-temperature or thermal HF reference. Thermal full-configuration-interaction theory is used as the benchmark.
Starting with Schr\"odinger's time-independent equation for NNN indistinguishable particles, an exact equation is derived for the reduced density matrix of order sss, ${\ensuremath{\rho}}_{s}$. The equation contains terms dependent on ${\ensuremath{\rho}}_{s+1}$ and ${\ensuremath{\rho}}_{s+2}$. The Hartree-Fock equation is derived, without recourse to the variational method, by assuming that ${\ensuremath{\rho}}_{2}$ and ${\ensuremath{\rho}}_{3}$ are particular functionals of ${\ensuremath{\rho}}_{1}$.
The new approach in quantum statistical mechanics proposed by the author in previous publications is extended to systems of nonzero-spin particles. It is shown that properties of particles relevant to their spin can well be incorporated into the approach proposed. In so doing the principal ideas of the approach undergo no changes and lead to a hierarchy of equations for reduced density matrices at thermodynamic equilibrium that generalizes the hierarchy obtained earlier for spinless particles. Thermodynamics based on the hierarchy is worked out as well. The hierarchy and relevant thermodynamics enable one to calculate both thermodynamical properties of a system of particles and structural ones. With the help of perturbation expansions it is shown that the hierarchy has a solution and the solution is unique at least in the case of a weakly interacting system. Using the example of a hard-sphere system wherein triplet correlations are disregarded it is demonstrated that the hierarchy may serve as basis for deriving various results. Comparison of these results with results of other theoretical treatments evidences complete agreement within the limits allowed by the approximation used.
Abstract We analyze the correspondence between two formalisms describing an interacting Bose gas, namely the standard Feynman diagrammatic expansion on the one hand, and the hierarchy equations for the imaginary-time Green functions on the other hand. We show that the Hartree–Fock approximation, as well as its first corrections at low density derived by Baym et al (2001 Eur. Phys. J. B 24 107–24), can be equivalently formulated in both formalisms. Within the hierarchy approach, the two-body correlations are expressed at lowest order in the interactions in terms of the full one-body Green function. This sheds light on the physical content of the corrections to the Hartree–Fock theory. Moreover, this representation of can be extended to higher orders, opening the way to systematic calculations of further corrections to the ideal critical temperature when the density increases.