In the linear-response regime, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. We derive a general expression for this kernel and evaluate it explicitly to second order in the interparticle potential. The result is a wave-vector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born-approximation cross section. Our theory is developed in terms of a second-quantized form of the Wigner representation. Convenient expressions are obtained for the commutators and anticommutators of the phase-space density operators, and the equilibrium averages of these operators are analyzed in terms of momentum-dependent generalizations of the classical pair distribution function
h(k) and direct correlation function
c(k). The central quantity in this study is a two-particle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ gives the symmetrized scattering function $S(k\ensuremath{\omega})$ by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ to the same order and that the result satisfies the appropriate positivity and symmetry conditions.
Close to equilibrium, a normal Bose or Fermi fluid can be
described by an exact kinetic equation whose kernel is nonlocal in space and time. The general expression derived for the kernel is evaluated to second order in the interparticle potential. The result is a wavevector- and frequency-dependent generalization of the linear
Uehling-Uhlenbeck kernel with the Born approximation cross section. The theory is formulated in terms of second-quantized phase space operators whose equilibrium averages are the n-particle Wigner distribution functions. Convenient expressions for the commutators and anticommutators of the phase space operators are obtained. The two-particle equilibrium distribution function is analyzed in terms of momentum-dependent quantum generalizations of the
classical pair distribution function h(k) and direct correlation function c(k). The kinetic equation is presented as the equation of motion of a two -particle correlation function, the phase space density-density anticommutator, and is derived by a formal closure of the quantum BBGKY hierarchy. An alternative derivation using a projection operator is also given. It is shown that the method used for approximating the kernel by a second order expansion preserves all the sum rules to the same order, and that the second-order kernel satisfies the appropriate positivity and symmetry conditions.
In describing transport phenomena, it is vital to build the conservation laws of number, energy, momentum, and angular momentum into the structure of the approximation used to determine the thermodynamic many-particle Green's functions. A method for generating conserving approximations has been developed. This method is based on a consideration, at finite temperature, of the equations of motion obeyed by the one-particle propagator
G, defined in the presence of a nonlocal external scalar field
U. Approximations for
G(U) are obtained by replacing the
G2(U) which appears in these equations by various functionals of
G(U). If the approximation for
G2(U) satisfies certain simple symmetry conditions, then the
G(U) thus defined obeys all the conservation laws. Furthermore, the two-particle correlation function, generated as ${(\frac{\ensuremath{\delta}G}{\ensuremath{\delta}U})}_{U=0}\ensuremath{\equiv}\ifmmode\pm\else\textpm\fi{}L$, in terms of which all linear transport can be described, will obey all the conservation laws as well as several essential sum rules, such as the longitudinal
f-sum rule.Several examples of conserving approximations are described. The Hartree approximation,
G2(U)=G(U)G(U), generates the random-phase approximation for
L. The Hartree-Fock approximation for
G(U) leads to a natural generalization of the random-phase approximation in which hole-particle ladder diagrams are summed. Another conserving approximation for
G(U) is obtained by expanding the self-energy to first order in the many-particle scattering matrix
T(U). This
T is obtained by summing ladder diagrams in which the sides of the ladder are composed of
G(U)′s. The resulting
L equation, which involves coefficients proportional to
∣T∣2, is analogous to the linearized version of the usual Boltzmann equation. Finally, in order to obtain a description of collisions in a plasma, the self-energy is expanded to first order in a dynamically shielded potential,
Vs(U). This potential is obtained by summing bubbles composed of two
G(U)′s. The resulting
L equation is similar in structure to a Boltzmann equation in which the collision cross section is proportional to
∣Vs∣2.
We derive an exact expression for the second term in the activity expansion of S (Q,ω), the Fourier transform of Van Hove’s correlation function, for a quantum system of bosons or fermions (spin neglected). The results for both the self and distinct parts of this function are given in terms of standard two-particle T matrices. The detailed-balance condition and zeroth and first moment relations are confirmed. An expression for the second term in the activity expansion of S (Q), the structure factor, is obtained. The derivation follows directly from the definition of S (Q,ω) using formal operator techniques; in addition, we show that the conserving T approximation for two-particle Green’s functions yields the same result.
The method of functional differentiation, used in classical statistical mechanics to obtain approximate integral equations for the pair distribution function, is extended to quantum systems obeying Maxwell-Boltzmann statistics. The grand partition function is written as a path integral, and space-temperature distributions are generated from it by successive functional differentiation. Distributions can be expanded in powers of an external potential via a functional Taylor series. When the series is truncated so that no distributions higher than second order enter and the external potential is specialized to one arising from a fixed particle in the system, coupled integral equations result for the two kinds of pair distributions. The first-order Ursell function calculated from these equations yields the Montroll-Ward ring summation for the grand potential. This approximate Ursell function determines also the Fourier transform of the momentum density (useful in calculating the p = 0 occupation number). Although divergences appear in the low-temperature limit, they can be removed by a simple modification of the independent functional. An extension to Fermi-Dirac and Böse-Einstein statistics is indicated at appropriate places in the text.
We study the temperature dependence of various spatial correlation functions of normal fluid and liquid
4He along an isochoric line of particle number density $\ensuremath{\varrho}=0.02185\text{ }{\text{\AA{}}}^{\ensuremath{-}3}$. The formal and numerical investigation employs correlated density-matrix theory as currently developed. The formalism permits a clear analysis of particle exchange (statistical) effects and so-called direct-direct quantum effects that appear only in a many-body system of interacting particles. At supercritical temperatures $T\ensuremath{\ge}12\text{ }\text{K}$ the helium fluid can be characterized as a perfect quantum Boltzmann system. In such a fluid, quantum statistical correlations are absent and the particle exchange follows classical Boltzmann statistics despite direct-direct quantum effects being present. At temperatures below 12 K, statistical quantum correlations begin to appear and are largest at the Bose-Einstein transition temperature
TBE=2.17 K. However, these correlations are considerably smaller than the statistical exchange correlations existing in a free Bose gas of bosons with helium mass. We present analytical and numerical results on the radial distribution function, the exchange correlation function, the phase-phase correlation function, the one-body reduced density matrix, and gross thermal quantities such as specific heat and total exchange energy. The role of particle exchange in determining their quantum properties is analyzed and discussed in detail.