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Journal of Mathematical Physics

Reduced Density Matrices of Quantum Gases. II. Cluster Property

J. Ginibre

Published Feb 1, 1965• 58 citations
Publisher

Abstract

The reduced density matrices of quantum gases are studied by means of a Wiener integral representation described in a previous paper. They are shown to satisfy a cluster property in the form of an absolute integrability condition of the natural quantum analogues of the Ursell functions, considered as functions of the differences of their arguments. Use is made of the natural transposition to the quantum case of the algebraic formalism introduced by Ruelle in the classical case. By-products are two results on the signs of the coefficients of the Mayer expansion, in the case of Maxwell-Boltzmann and Fermi-Dirac statistics, respectively.

DOI
10.1063/1.1704276

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Finite-size effects on the cluster expansions for quantum gases in restricted geometries

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Abstract We have analytically obtained 1-particle density matrices for ideal Bose and Fermi gases in both the 3-D box geometries and the harmonically trapped geometries for the entire range of temperature. We have obtained quantum cluster expansions of the grand free energies in closed forms for the same systems in the restricted geometries. We have thoeremed (with a proof) about the generic form of the quantum cluster integral. We also have considered short ranged interactions in our analyses for the quasi 1-D cases of Bose and Fermi gases in the box geometries. Our theoretical results are exact, and are directly useful for understanding finite-size effects on quantum cluster expansion of Bose and Fermi gases in the restricted geometries. Our results would be relevant in the context of experimental study of spatial correlations in ultra-cold systems of dilute Bose and Fermi gases of alkali atoms (i) in 3-D magneto-optical box traps with quasi-uniform potential around the center [1], and (ii) in 3-D harmonic traps [2, 3].

arXiv:1908.02106•Physica Scripta•Apr 28, 2020•4 citations
The reduced density matrices of quantum gases are studied by means of a Wiener integral representation described in a previous paper. They are shown to satisfy a cluster property in the form of an absolute integrability condition of the natural quantum analogues of the Ursell functions, considered as functions of the differences of their arguments. Use is made of the natural transposition to the quantum case of the algebraic formalism introduced by Ruelle in the classical case. By-products are two results on the signs of the coefficients of the Mayer expansion, in the case of Maxwell-Boltzmann and Fermi-Dirac statistics, respectively.
In two previous papers, the reduced density matrices of quantum gases, in the grand canonical formalism and for suitably restricted interaction potentials, have been shown to be analytic vector-valued functions of the activity in a neighborhood of the origin, to tend in some sense to well-defined limits as the volume of the system becomes infinite, and to satisfy a cluster decomposition property. The same results are extended here by the same methods to a wider class of potentials, including hard-core, and allowing attractive interactions, which were excluded previously.
We apply the formalism of Ursell operators, introduced in a previous article, to the calculation of reduced density operators (one and two-body density operators) in a dilute quantum gas; the calculation is not a fugacity expansion and is therefore not limited to low degrees of degeneracy. We obtain quantum corrections for the one-particle density operator as a function of the second Ursell operator. For the two-body density operator, we examine how statistics and interactions combine their effects on the correlations between particles; in particular we discuss in detail how hard cores potentials affect the short range correlations, a non-perturbative effect.
A systematic generalization of the Mayer cluster integral theory has been developed to deal with the quantum statistics of interacting particles. The grand partition function appears in a natural way and the cluster integrals are integrals over propagators which are derived from the Green's function solution of the Bloch equation (which follows from the Schroedinger equation by replacing it/ℏ by β = 1/kT). Every cluster integral can be represented by a hybrid of a Mayer graph and a Feynman diagram in (β, r) space. The generalization of classical ring cluster integrals has been analyzed. It is shown that in the case of the electron gas the classical limit of the contribution of these integrals to the grand partition function yields the Debye-Huckel theory while the low temperature limit leads to the Gell-Mann—Brueckner equation for the correlation energy of the ground state. A prescription is given for the construction of the cluster integral associated with any given diagram.
A new cluster development for the logarithm of the grand partition function of a system of interacting particles is derived. The leading term in this expansion is the pressure exerted by an ideal Bose or Fermi gas at the same temperature and absolute activity ZZZ as the actual system. Succeeding terms involve quantum cluster integrals which themselves depend upon ZZZ, unlike their classical analogs. The definition of these cluster integrals follows in a natural fashion using techniques illustrated by construction (in closed form) of the successive ZZZ derivatives of the Bose and Fermi ideal gas grand partition functions. It is not possible (except in the classical limit) to eliminate ZZZ explicitly between the pressure and density series, so that the equation of state must remain in parametric form.
Abstract We have analytically obtained 1-particle density matrices for ideal Bose and Fermi gases in both the 3-D box geometries and the harmonically trapped geometries for the entire range of temperature. We have obtained quantum cluster expansions of the grand free energies in closed forms for the same systems in the restricted geometries. We have thoeremed (with a proof) about the generic form of the quantum cluster integral. We also have considered short ranged interactions in our analyses for the quasi 1-D cases of Bose and Fermi gases in the box geometries. Our theoretical results are exact, and are directly useful for understanding finite-size effects on quantum cluster expansion of Bose and Fermi gases in the restricted geometries. Our results would be relevant in the context of experimental study of spatial correlations in ultra-cold systems of dilute Bose and Fermi gases of alkali atoms (i) in 3-D magneto-optical box traps with quasi-uniform potential around the center [1], and (ii) in 3-D harmonic traps [2, 3].