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Journal of Mathematical Physics

Reduced Density Matrices of Quantum Gases. I. Limit of Infinite Volume

J. Ginibre

Published Feb 1, 1965• 108 citations
Publisher

Abstract

The reduced density matrices for quantum gases are studied by Banach space techniques. For suitably restricted interactions, they are shown to be analytic functions of the activity. As the volume of the system becomes infinite, they tend in some sense to well-defined limits for which the same analyticity properties hold. As a consequence, the virial expansion is shown to be convergent in a neighborhood of the origin.

DOI
10.1063/1.1704275

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preprint: SISSA 39/2007/mp Emergence and applications of the ubiquitous tool of reduced density matrices in the rigorous analysis of Bose Einstein condensation is reviewed, and new related results are added. The need and the nature of scaling limits of infinitely many particles is discussed, which imposes that a physically meaningful and mathematically well-posed definition of asymptotic condensation is placed at the level of marginals. The topic of correlations in the condensed state is addressed in order to show their influence at this level of marginals, both in the true condensed state and in the suitable trial functions one introduces to approximate the many-body structure and energy. Complete condensation is shown to be equivalently defined at any fixed kbody level, both for pure and mixed states. Further, it is proven to be equivalent to some other characterizations in terms of asymptotic factorization of the many-body state, which are currently present in the literature.

•Aug 7, 2007•6 citations
The reduced density matrices for quantum gases are studied by Banach space techniques. For suitably restricted interactions, they are shown to be analytic functions of the activity. As the volume of the system becomes infinite, they tend in some sense to well-defined limits for which the same analyticity properties hold. As a consequence, the virial expansion is shown to be convergent in a neighborhood of the origin.
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.
Abstract We consider infinite volume limit Gibbs states of a nonrelativistic quantum Bose gas consisting of one species of spinless particles with positive interaction potentials. The finite volume reduced density matrices are dominated by the corresponding matrices for the noninteracting gas, and as a consequence all infinite volume limit states are regular, locally normal, and analytic on the appropriate CCR algebra. For sufficiently short range repulsive two-body interactions, the cyclic vector associated with the limit state is separating for the σ-weak closure of the algebra in the associated representation.
The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality ddd. In particular, the convergence radius ρc(d)ρ_c(d)ρc​(d) of the expansion is particulary large at {\em exactly\/} d=3d=3d=3 dimensions, ρc(3)=7.1068…×lim⁡d→3ρc(d)ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)ρc​(3)=7.1068…×limd→3​ρc​(d). The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.
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.
preprint: SISSA 39/2007/mp Emergence and applications of the ubiquitous tool of reduced density matrices in the rigorous analysis of Bose Einstein condensation is reviewed, and new related results are added. The need and the nature of scaling limits of infinitely many particles is discussed, which imposes that a physically meaningful and mathematically well-posed definition of asymptotic condensation is placed at the level of marginals. The topic of correlations in the condensed state is addressed in order to show their influence at this level of marginals, both in the true condensed state and in the suitable trial functions one introduces to approximate the many-body structure and energy. Complete condensation is shown to be equivalently defined at any fixed kbody level, both for pure and mixed states. Further, it is proven to be equivalent to some other characterizations in terms of asymptotic factorization of the many-body state, which are currently present in the literature.