The memory kernel of the kinetic equation for the time correlation function of a quantum fluid is determined both in third order of the interaction strength and in the low-density approximation. The results are obtained with the help of a diagram representation for the kernel. The connexion with the kinetic theory for classical fluids is established. The properties of a nonequilibrium fluid may be studied with the help of the kinetic theory of time correlation functions. These functions are determined by specifying the memory kernel in the kinetic equation that governs their time evolution. For classical fluids a suitable formalism that permits the approximate evaluation of this kernel has been developed several years ago (for a review see ref. 1). The generalization of these methods to the case of quantum fluids was considered in a few recent papers2f3). In particular, the memory kernel was established for a normal quantum fluid in the weakcoupling approximation. In the long-time and large-distance limit this kernel reduces to that of the linearized Uehling-Uhlenbeck equation4) in which the scattering amplitude is replaced by its first Born approximation. The purpose of the present paper is to extend the available results by employing higher-order perturbation theory for the evaluation of the memory kernel. An essential ingredient of the treatment will consist in the introduction of four-point diagrams that represent the various contributions to the kernel. With the use of this diagram representation we shall derive the memory kernel in third order of the interaction strength. In this order the kernel still has the same general form as the weak-coupling kernel, but with shifted energies and a generalized transition probability. Its relation to the kernel of the linearized Bogoliubov-Gurov’) equation will be established. An alternative approach to the approximate evaluation of the memory kernel makes use of an expansion in the density. The low-density limit of the
The memory kernel of the kinetic equation for the time correlation function of a quantum fluid is determined both in third order of the interaction strength and in the low-density approximation. The results are obtained with the help of a diagram representation for the kernel. The connexion with the kinetic theory for classical fluids is established. The properties of a nonequilibrium fluid may be studied with the help of the kinetic theory of time correlation functions. These functions are determined by specifying the memory kernel in the kinetic equation that governs their time evolution. For classical fluids a suitable formalism that permits the approximate evaluation of this kernel has been developed several years ago (for a review see ref. 1). The generalization of these methods to the case of quantum fluids was considered in a few recent papers2f3). In particular, the memory kernel was established for a normal quantum fluid in the weakcoupling approximation. In the long-time and large-distance limit this kernel reduces to that of the linearized Uehling-Uhlenbeck equation4) in which the scattering amplitude is replaced by its first Born approximation. The purpose of the present paper is to extend the available results by employing higher-order perturbation theory for the evaluation of the memory kernel. An essential ingredient of the treatment will consist in the introduction of four-point diagrams that represent the various contributions to the kernel. With the use of this diagram representation we shall derive the memory kernel in third order of the interaction strength. In this order the kernel still has the same general form as the weak-coupling kernel, but with shifted energies and a generalized transition probability. Its relation to the kernel of the linearized Bogoliubov-Gurov’) equation will be established. An alternative approach to the approximate evaluation of the memory kernel makes use of an expansion in the density. The low-density limit of the
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 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.
Renormalized expressions for the transport coefficients of a normal quantum fluid are derived from a nonlocal kinetic equation. As was shown in the classical case by Forster and Martin and by R\'esibois, the expressions for the transport coefficients separate naturally into "kinetic" and "direct" parts. The kinetic terms are proportional to matrix elements of the inverse of the kinetic kernel, and have the same general structure as the "bare" transport coefficients obtained from a local Boltzmann equation. The direct terms are proportional to matrix elements of the kernel itself, and have no counterpart in calculations based on a local kinetic equation, as in the kinetic theory of gases. We evaluate the transport coefficients using the weak-coupling approximation to the kernel derived in a previous paper. Results are given, first, for both Bose and Fermi fluids at arbitrary temperature, and then for the Fermi fluid near T=0, where complete solutions are obtained. It is found that the direct parts of the shear viscosity and thermal conductivity are of higher order in T than the kinetic parts, and are therefore negligible at very low temperature. The kinetic parts have the same leading temperature dependence as the predictions of the Landau theory. For the bulk viscosity, however, the direct and kinetic contributions begin at the same order (T2) in the temperature.
From the quantum kinetic equation of Bogoliubov, Gurov, Born, and Green, the equations of motion for macroscopic variables are examined in many different ways. First, in the configuration space a hierarchy of generalized fluid-dynamical equations can be obtained by taking the appropriate limits of the quantum kinetic equation. The resultant equations of continuity, of momentum, and of energy are similar in form to those one encounters in classical fluid dynamics with the exception of additional terms proportional to ${\ensuremath{\hbar}}^{2}$ Secondly, the quantum kinetic equation is examined in phase space. The same set of equations of continuity, of momentum, and of energy can be derived by taking the first three moments of the quantum kinetic equation. The exact results we obtained are utilized to form the starting point for many simplifying approximations for the investigation of the dynamics of quantum many-body systems such as the elastic response limit, the hydrodynamical limit, Landau's Fermi-liquid theory, and finally the time-dependent Hartree-Fock and the multideterminant time-dependent Hartree-Fock approximations. The fact that all these different dynamical descriptions can be traced to a common origin provides a unifying viewpoint to the present approach with the quantum kinetic equation.NUCLEAR STRUCTURE Dynamics of nuclear fluid, quantum kinetic equation, and generalized fluid-dynamical equation. Elastic response, hydrodynamical limit, Landau's Fermi-liquid theory, time-dependent Hartree-Fock approximation, and multideterminant time-dependent Hartree-Fock approximation.
In the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved. \textcopyright{} 1996 The American Physical Society.
We consider an analytically solvable model of two interacting electrons that allows for the calculation of the exact exchange-correlation kernel of time-dependent density functional theory. This kernel, as well as the corresponding density response function, is studied in the limit of large repulsive interactions between the electrons and we give analytical results for these quantities as an asymptotic expansion in powers of the square root of the interaction strength. We find that in the strong interaction limit the three leading terms in the expansion of the kernel act instantaneously while memory terms only appear in the next orders. We further derive an alternative expansion for the kernel in the strong interaction limit on the basis of the theory developed previously [Phys. Chem. Chem. Phys. 18, 21092 (2016)] using the formalism of strictly correlated electrons in the adiabatic approximation. We find that the first two leading terms in this series, corresponding to the strictly correlated limit and its zero-point vibration correction, coincide with the two leading terms of the exact expansion. We finally analyze the spatial nonlocality of these terms and show when the adiabatic approximation breaks down. The ability to reproduce the exact kernel in the strong interaction limit indicates that the adiabatic strictly correlated electron formalism is useful for studying the density response and excitation properties of other systems with strong electronic interactions.