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Physical Review C

Dynamics of nuclear fluid. V. Extended time-dependent Hartree-Fock approximation illuminates the approach to thermal equilibrium

Cheuk-Yin Wong, Henry H. K. Tang

Published Oct 1, 1979• 62 citations
Publisher

Abstract

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 $H$ 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. $H$ theorem. Analytic solution of the master equation. New types of level crossing formula.

DOI
10.1103/physrevc.20.1419

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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.
The region of validity of the TDHF approximation is extended by including the collisions between particles. The resultant equations of motion consist of a modified TDHF equation and a master equation for the occupation numbers. Concepts such as entropy, temperature, and thermal equilibrium can be quantitatively introduced. The well-known H-theorem that entropy never decreases is proved for nuclear systems. New types of level crossing formulas are obtained to shed light on how a nonequilibrium finite nuclear system approaches thermal equilibrium. 2 figures.
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.
A summation of selected graphs in the perturbation expansion for the statistical operator and the operator of state leads to a simplified equation of motion for the nuclear many-body problem. It contains as a special limit the time-dependent Hartree-Fock and the quantum-statistical master equation. In a pictorial way one could label the approach as "Hartree-Fock plus collision terms." The formalism allows one to understand the progression of the nucleus from the boiling of the particle-hole excitations to the collective state of motion. The work is restricted presenting the formalism. Various applications are possible and indicated, but apart from an illustrative example no concrete comparison with the experiment has been performed.NUCLEAR REACTIONS Formalism developed for unifying collective and single-particle aspects; mean field theory, Hartree-Fock plus collision terms.
Introduction In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving . A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n (x, t ) and a current density J (x, t ) which satisfy the continuity equation, or a total momentum P ( t ) and a Lorentz force F ( t ) which is the time-derivative of P ( t ), or a total energy E ( t ) and a total power which is the time-derivative of E ( t ), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.
The presence of pairing correlations within the time-dependent density-functional-theory (TDDFT) extension to superfluid systems is tantamount to the presence of a quantum collision integral in the evolution equations, which leads to an obviously non-Markovian behavior of the single-particle occupation probabilities, unexpected in a traditional quantum extension of kinetic equations. The quantum generalization of the Boltzmann equation, based on a collision integral in terms of phase-space occupation probabilities, is the most used approach to describe nuclear dynamics and which by construction has a Markovian character. By contrast, the extension of TDDFT to superfluid systems has similarities with the Baym and Kadanoff kinetic formalism, which, however, is formulated with much more complicated evolution equations with long-time memory terms and nonlocal interactions. The irreversibility of quantum dynamics is properly characterized using the canonical wave functions/natural orbitals and the associated canonical occupation probabilities, which provide the smallest possible representation of any fermionic many-body wave function. In this basis, one can evaluate the orbital entanglement entropy, which is an excellent measure of the nonequilibrium dynamics of an isolated system. To explore the phenomena of memory effects and irreversibility, we investigate the use of canonical wave functions/natural orbitals in nuclear many-body calculations, assessing their utility for static calculations, dynamics, and symmetry restoration. As the number of single-particle states is generally quite large, it is highly desirable to work in the canonical basis whenever possible, preferably with a cutoff. We show that truncating the number of canonical wave functions can be a valid approach in the case of static calculations, but that such a truncation is not valid for time-dependent calculations, as it leads to the violation of continuity equation, energy conservation, and other observables, as well as an inaccurate representation of the dynamics. Indeed, in order to describe the dynamics of a fissioning system within a TDDFT framework all the canonical states must be included in the calculation. Finally, we demonstrate that the canonical representation provides a very efficient basis for performing symmetry projections.