https://doi.org/10.1140/epjb/e20020052
Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles
Laboratoire Kastler Brossel (The “Laboratoire Kastler Brossel (LKB)” is
“Unité Mixte de Recherche du CNRS (UMR 8552) et de l'Université
Pierre et Marie Curie (Paris)”.) , Département de Physique de l'ENS, 24 rue Lhomond, 75005 Paris, France
Corresponding author: a franck.laloe@lkb.ens.fr
Received:
18
October
2001
Published online: 15 February 2002
The purpose of this article is to discuss cluster expansions in dense quantum systems, as well as their
interconnection with exchange cycles. We show in general how the Ursell operators of order contribute to
an exponential which corresponds to a mean-field energy involving the second operator U2, instead of the
potential itself as usual – in other words, the mean-field correction is expressed in terms of a modification of a
local Boltzmann equilibrium. In a first part, we consider classical statistical mechanics and recall the relation
between the reducible part of the classical cluster integrals and the mean-field; we introduce an alternative
method to obtain the linear density contribution to the mean-field, which is based on the notion of tree-diagrams
and provides a preview of the subsequent quantum calculations. We then proceed to study quantum particles with
Boltzmann statistics (distinguishable particles) and show that each Ursell operator Un with
contains
a “tree-reducible part”, which groups naturally with U2 through a linear chain of binary interactions; this
part contributes to the associated mean-field experienced by particles in the fluid. The irreducible part, on the
other hand, corresponds to the effects associated with three (or more) particles interacting all together at the
same time. We then show that the same algebra holds in the case of Fermi or Bose particles, and discuss
physically the role of the exchange cycles, combined with interactions. Bose condensed systems are not considered
at this stage. The similarities and differences between Boltzmann and quantum statistics are illustrated by this
approach, in contrast with field theoretical or Green's functions methods, which do not allow a separate study of
the role of quantum statistics and dynamics.
PACS: 05.30.-d – Quantum statistical mechanics / 05.20.Jj – Statistical mechanics of classical fluids
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2002