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International Journal of Modern Physics E

GENERALIZED HUGENHOLTZ–VAN HOVE THEOREM FOR MULTI-COMPONENT FERMI SYSTEMS WITH MULTI-BODY FORCES

Ramesh Chandra Nayak, S. PATTNAIK

Published Jul 21, 2011• 0 citations
Publisher

Abstract

We show here that the Hugenholtz–Van Hove theorem for an infinite Fermi system remains valid even with all possible multi-body interaction terms taken into account either separately or in any combination under Hartree–Fock approximation. We further extend the theorem to obtain its most generalized form that would hold good for any multi-component Fermi system.

DOI
10.1142/s0218301311020253

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We show here that the Hugenholtz–Van Hove theorem for an infinite Fermi system remains valid even with all possible multi-body interaction terms taken into account either separately or in any combination under Hartree–Fock approximation. We further extend the theorem to obtain its most generalized form that would hold good for any multi-component Fermi system.
The second term being the pressure of the system would vanish for a saturating system at ground state i.e. at equilibrium resulting in the Eq. E/A = ǫf . This is a rare theorem in many-body Physics, which has been rigorously shown[1] to be true by its original authors Hugenholtz and Van Hove by taking all orders of perturbation in the frame-work of timeindependent perturbation theory[1–5]. However it should be mentioned here that prior to this rigorous proof Bethe[6] had also visualized the theorem under HF approximation. It is valid for any interacting infinite Fermi system and thereby applicable to liquid He and in particular to nuclear matter. With its
A method for choosing and deriving a suitable single-particle potential for many-body calculations at finite temperature is suggested. With this potential one can readily determine the self-consistent correlated chemical potential and consequently the resulting equations of state rigorously satisfy a number of basic thermodynamic relations. Application of our method to a Brueckner-Hartree-Fock nuclear matter calculation yields results in remarkably good agreement with the Hugenholtz--Van Hove theorem.
This paper investigates the criteria for maintenance of the macroscopic conservation laws of number, momentum, and energy by approximate two-particle correlation functions in many-body systems. The methods of generating such approximations are the same as in a previous paper. However, the derivations of the conservation laws given here clarify both why the approximation method works and the connection between the macroscopic conservation laws and those at the vertices.Conserving nonequilibrium approximations are based on self-consistent approximations to the one-particle Green's function. The same condition that ensures that the nonequilibrium theory be conserving also ensures that the equilibrium approximation has the following properties. The several common methods for determining the partition function from the one-particle Green's function all lead to the same result. When applied to a zero-temperature normal fermion system, the approximation procedure maintains the Hugenholtz-Van Hove theorem. Consequently, the self-consistent version of Brueckner's nuclear matter theory obeys this theorem.
New forms of the linked cluster expansion for the ground-state energy of a normal spherical system are given in terms of the exact single-particle propagator and of the associated self-energy operator. These forms are shown to be stationary with respect to variations of the latter quantities. A new concise proof of the Hugenholtz-Van Hove theorem on single-particle energies is then possible. The variational principle also explains the insensitivity of recent binding energy calculations to the choice of single-particle energies. It is shown how all results may be extended to systems with anisotropic features.