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Physical Review C

Dynamics of nuclear fluid. III. General considerations on the kinetic theory of quantum fluids

Cheuk-Yin Wong, John A. McDonald

Published Sep 1, 1977• 30 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.16.1196

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Physical Review C•Oct 1, 1979•62 citations

Dynamics of nuclear fluid. II. Normal sound, spin sound, isospin sound, and spin-isospin sound

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Starting with the time-dependent Hartree-Fock equation in density matrix form, we investigate the macroscopic description of the dynamics of the nuclear fluid involving the spin and isospin degrees of freedom, in conjunction with the presence of only central exchange interactions. The time-dependent Hartree-Fock equation can be cast into a set of conservation laws of the classical type coupling spin and isospin densities. With simple zero-range interactions, we obtain the normal modes of density propagations and the corresponding speeds of sound waves. In addition to the normal sound waves in which the total density varies with space and time, there are the spin sound waves in which the difference of the spin-up and the spin-down densities varies with space and time, the isospin sound waves in which the difference of neutron and proton densities varies with space and time, and finally, the spin-isospin sound waves in which the difference of the "parallel" spin and isospin densities and the "antiparallel" spin and isospin densities varies with space and time. It is found that for a zero-range interaction whose density dependence is of the type ${t}_{3}(1+{\mathcal{P}}^{B})\frac{n\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{1}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{2})}{6}$, the speeds of spin sound ${a}_{2}$, isospin sound ${a}_{3}$ and spin-isospin sound ${a}_{4}$ satisfy ${{a}_{2}}^{2}+{{a}_{3}}^{2}=2{{a}_{4}}^{2}$. With the parameters of Golin and Zamick and Vautherin and Brink, we have in addition ${a}_{3}>{a}_{4}>{a}_{2}$, the numerical values of ${a}_{2}$, ${a}_{3}$, and ${a}_{4}$ being in the range of 0.17 to 0.27c.NUCLEAR STRUCTURE Dynamics of nuclear fluid, normal sound wave, spin sound wave, isospin sound wave, and spin-isospin sound wave. Time-dependent Hartree-Fock theory. Estimate speeds of different sound waves.

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A formulation of quantum fluid mechanics and trajectories

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Abstract A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.

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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 \ensuremathℏ2{\ensuremath{\hbar}}^{2}\ensuremathℏ2 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 time-dependent Hartree-Fock approximation, the fermions are assumed to interact only through the mean field and the collisions between particles are neglected. We formulate an extended time-dependent Hartree-Fock approximation which incorporates particle collisions due to the residual interaction, with the usual time-dependent Hartree-Fock approximation as the collisionless limit. It is obtained by a truncation of the Martin-Schwinger hierarchy at the second level and by using a simple representation of the one-body Green's function in terms of time-dependent occupation numbers. The final set of coupled equations consists of a modified time-dependent Hartree-Fock equation and a master equation for the occupation numbers. These results are physically transparent and describe properly the physics of the collision process. They may also be simple enough to be of practical use to study heavy-ion collisions or the dynamics of other fermion systems. Furthermore, as the configuration-space analog of the quantum Boltzmann equation, many important results concerning statistical dynamics are obtained. Concepts such as entropy, temperature, and local and thermal equilibrium can be quantitatively introduced. The well-known HHH theorem that entropy never decreases can be readily recovered. With the collision term explicitly exhibited, the macroscopic equations (equations of continuity, momentum flux, and energy) and their associated conservation theorems can also be derived. Analytic solutions for the master equation for simple cases lead to new "level crossing" formulas having characteristics distinctly different from the Landau-Zener level-crossing formula and illuminate the salient features as to how a nonequilibrium fermion system approaches thermal equilibrium.NUCLEAR REACTIONS Extension of time-dependent Hartree-Fock approximation. Collisions between particles. Master equation for occupation probabilities. Entropy, temperature, and thermal equilibrium. HHH theorem. Analytic solution of the master equation. New types of level crossing formula.
Starting with the time-dependent Hartree-Fock equation in density matrix form, we investigate the macroscopic description of the dynamics of the nuclear fluid involving the spin and isospin degrees of freedom, in conjunction with the presence of only central exchange interactions. The time-dependent Hartree-Fock equation can be cast into a set of conservation laws of the classical type coupling spin and isospin densities. With simple zero-range interactions, we obtain the normal modes of density propagations and the corresponding speeds of sound waves. In addition to the normal sound waves in which the total density varies with space and time, there are the spin sound waves in which the difference of the spin-up and the spin-down densities varies with space and time, the isospin sound waves in which the difference of neutron and proton densities varies with space and time, and finally, the spin-isospin sound waves in which the difference of the "parallel" spin and isospin densities and the "antiparallel" spin and isospin densities varies with space and time. It is found that for a zero-range interaction whose density dependence is of the type ${t}_{3}(1+{\mathcal{P}}^{B})\frac{n\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{1}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{2})}{6}$, the speeds of spin sound a2{a}_{2}a2​, isospin sound a3{a}_{3}a3​ and spin-isospin sound a4{a}_{4}a4​ satisfy a22+a32=2a42{{a}_{2}}^{2}+{{a}_{3}}^{2}=2{{a}_{4}}^{2}a2​2+a3​2=2a4​2. With the parameters of Golin and Zamick and Vautherin and Brink, we have in addition a3>a4>a2{a}_{3}>{a}_{4}>{a}_{2}a3​>a4​>a2​, the numerical values of a2{a}_{2}a2​, a3{a}_{3}a3​, and a4{a}_{4}a4​ being in the range of 0.17 to 0.27c.NUCLEAR STRUCTURE Dynamics of nuclear fluid, normal sound wave, spin sound wave, isospin sound wave, and spin-isospin sound wave. Time-dependent Hartree-Fock theory. Estimate speeds of different sound waves.
By means of a quantum mechanical phase space distribution function introduced by von Roos, the Schroedinger equation for a non-relativistic system of N identical particles with scalar interactions is transformed into a quantum mechanical generalization of the Liouville equation, thereby formulating the problem in terms of a generalized density in phase space, a quantity of primary interest in most treatments of the corresponding classical system (or plasma). This transformation permits a parallel development of the theories of classical and quantum plasmas and thus allows the quantum many-body problem to be discussed virtually completely in classical terms. In particular, a kinetic theory of quantum plasmas is obtained by deriving the quantum analogue of the BBGKY hierarchy, and applying thereto approximation techniques similar to those of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from similar previous studies based on the Wigner distribution function is that the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations, in which case the quantum BBGKY system reduces to a simple quantum generalization of the Vlasov equation. This equation is used to study the response of spatially homogeneous systems to weak external forces, and the associated problems of plasmon and spin-wave excitations. It is also used to derive the quantum and exchange corrected equations of inviscid hydrodynamical transport which are then applied to the problem of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the many-electron atom in the Hartree and Hartree-Fock approximations. The relevant quantum Vlasov equations lead naturally to a statistical theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and exchange corrections to these models are simultaneously generated. The quantum hydrodynamical theory developed earlier is used to determine the influence of these corrections on the boundary conditions of the model, and a theory of the compressed atom is consequently obtained. Considered in somewhat less detail are the effects of non-zero temperature, net orbital angular momentum, relativity and correlations, as well as time dependent processes. The final part deals with the problem of the degenerate electron gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with parallel and anti-parallel spin are obtained by neglecting three particle correlations. From these functions, a quantum-mechanical collision integral is derived which differs from that obtained by Silin and Guernsey and conjectured by Wyld and Pines in that dynamical exchange effects are included. Also obtained from the pair function is an expression for the correlation which reduces in the high density limit to the result of Gell-Mann and Brueckner. At intermediate densities an additional term appears in the energy due to the screening of the exchange interaction by the dielectric properties of the medium. It is evaluated in the high density limit and found to be -0.151 r_s ln r_s Ryd/electron in marked disagreement with the corresponding value obtained by DuBois.
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.
Abstract A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.