Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

摘要

The purpose of this article is to discuss cluster expansions in dense quantumsystems as well as their interconnection with exchange cycles. We show ingeneral how the Ursell operators of order 3 or more contribute to anexponential which corresponds to a mean-field energy involving the secondoperator U2, instead of the potential itself as usual. In a first part, weconsider classical statistical mechanics and recall the relation between thereducible part of the classical cluster integrals and the mean-field; weintroduce an alternative method to obtain the linear density contribution tothe mean-field, which is based on the notion of tree-diagrams and provides apreview of the subsequent quantum calculations. We then proceed to studyquantum particles with Boltzmann statistics (distinguishable particles) andshow that each Ursell operator Un with n greater or equal to 3 contains a``tree-reducible part'', which groups naturally with U2 through a linear chainof binary interactions; this part contributes to the associated mean-fieldexperienced by particles in the fluid. The irreducible part, on the other hand,corresponds to the effects associated with three (or more) particlesinteracting all together at the same time. We then show that the same algebraholds in the case of Fermi or Bose particles, and discuss physically the roleof the exchange cycles, combined with interactions. Bose condensed systems arenot considered at this stage. The similarities and differences betweenBoltzmann and quantum statistics are illustrated by this approach, in contrastwith field theoretical or Green's functions methods, which do not allow aseparate study of the role of quantum statistics and dynamics.