Integral Equations for Neutral and Charged Quantum Fluids Including Extension of the Percus-Yevick Equation

Abstract

On the basis of a generalized Hartree equation developed by using Mori 's continued-fraction method, extensions of the direct correlation function and the Ornstein-Zernike relation in a classical fluid to the case of a quantum fluid are obtained. These results give a generalization of Hohenberg-Kohn-Mermin theory for a nonuniform electron gas applicable to a quantum fluid whose inter~tomic potential cannot be Fourier-transformed, and also give an extension of Percus' functional expansion method, which yields integral equations for the radial distribution function g(r) in a. classical fluid, in such a way as to treat a quantum fluid. By applying this method to a neutral quantum fluid, extensions of the Percus-Yevick and the hypernetted chain equations are derived. For a charged quantum fluid (electron gas) new .integral equations are obtained by dividing an interatomic potential into strong short-range and slowly varying long-range parts. These equations give the compressibility sum rule in which the role of the short-range part of the potential is taken into account in such a way as Landau's Fermi liquid theory. The generalized Hartree equation combined with these integral equations for neutral and charged quantum fluids yields extensions of the Landau kinetic equation in the Ferr;ni liquid. theory and of the Landau-Silin equation for the electron gas, respectively, to large wavevectors and frequencies at non-zero temperatures. § 1. Introduction In classical fluids, many integral equations for the radial distribution function g(r), such as the Percus-Yevick (PY) and hypernetted chain (HNC) equations, have been proposed and shown to give ·results in fair agreement with experiment. In quantum fluids, however, it is difficult to derive integral equations for g (r), since, in contrast to the classical case, there' is no fact that the partition function of fluids is represented as a product of the configuration-integral part and the part involving the momentum contribution. There are three method('! for deriving approximate integral equations (for example, the PY equation) for g (r) in classical fluids; (i) the diagrammatic tech-nique/' (ii) the use of collective coordinates 2 ' and (iii) the functional expansion rilethod. 8 ' Thus we may expect to derive integral equations for quantum fluids by extending these methods applicable to quantal systems. In fact, previously Percus 4 ' has tried to extend the PY equation to quantum fluids by the use of collective coordinates, but his attempt is not quite successful.