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Physical Review Letters

Is There a Consistent Theory of Large-Amplitude Collective Motion?

G. Do Dang, Abraham Klein

Published Nov 18, 1985• 11 citations
Publisher

Abstract

Villars's equations, first derived from the adiabatic limit of time-dependent Hartree-Fock theory, are presented here as a set of exact conditions for the decoupling (in the adiabatic limit) of a sub-system from a classical Hamiltonian system. It is emphasized that these conditions do not incorporate all the required decoupling conditions. By combining the additional requirements with Villars's equations, we obtain a mathematically complete system, which will yield exact solutions where such exist, but which can also be applied to cases of approximate decoupling.

DOI
10.1103/physrevlett.55.2265

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Equations of Collective Submanifold for Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom. III

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From the viewpoint defferent from that of Part I, the time-dependent Harteree-Fock theory is formulated for describing large amplitude collective motion and its coupling with intrinsic degrees of freedom. The basic idea consists of the introduction of an additional degree of freedom for the collective motion and the use of Dirac’s canonical theory with constraints. The formalism is essentially equivalent to the ordinary time-dependent Hartree-Fock theory presented in Part I. The result given in Part I is completely reproduced.

Progress of Theoretical Physics•Jan 1, 1987•5 citations
Villars's equations, first derived from the adiabatic limit of time-dependent Hartree-Fock theory, are presented here as a set of exact conditions for the decoupling (in the adiabatic limit) of a sub-system from a classical Hamiltonian system. It is emphasized that these conditions do not incorporate all the required decoupling conditions. By combining the additional requirements with Villars's equations, we obtain a mathematically complete system, which will yield exact solutions where such exist, but which can also be applied to cases of approximate decoupling.
The starting point for most studies of large amplitude collective motion in nuclear physics, the time-dependent Hartree-Fock equations, can be mapped to a problem in classical Hamiltonian mechanics, which is the form of the problem studied in this work. For a system with N degrees of freedom, collective motion is identified with motion completely confined to a surface in K<N dimensions. Conditions for the existence of such decoupled surfaces are worked out and a procedure for constructing them is formulated. For most practical problems, where such surfaces do not strictly exist, a concept of approximate decoupling is developed and a test of its accuracy is described. Our results generalize the method of maximum global decoupling found in the earlier literature.
A classical theory of large amplitude, adiabatic collective motion, developed and applied previously to test examples, is transcribed into the language of time-dependent Hartree-Fock theory in order to initiate the application of the new method to problems in nuclear physics. The formulation which emerges can be characterized as a generalized cranking theory, in the sense that the cranking operator cannot be chosen arbitrarily, as in the conventional formulation of this method, but is instead fully constrained by the formalism. A procedure for obtaining approximate solutions to the new equations is described and illustrated with several simplified many-body models. It is inferred that traditional cranking calculations can serve as starting points for more realistic models. This paper also includes additional discussion concerning the calculation of collective masses and the problem of local stability of collective motion, not covered in our previous work.
The theory of large amplitude adiabatic motion is applied to a recently proposed model of tunneling in many body systems. We construct the corresponding collective Hamiltonian and study the applicability of the decoupling procedure, both by comparing the results of the large amplitude theory with exact results and by studying different measures of the decoupling procedure, including the stability of the collective motion against noncollective vibrations. We also analyze briefly another approximation scheme especially suited for the problem at hand. The theory of large amplitude collective motion provides an extremely accurate description of the collective motion and of the tunneling process.
We formulate a theory for the transfer of energy from collective degrees of freedom to internal degrees of freedom when the former belong to the adiabatic large-amplitude regime. The framework is that of an extended mean-field theory including dissipation, with the following salient features: (i) The Born-Oppenheimer approximation is extended to finite excitation energy by the introduction of the concept of thermal state, that is pure with respect to collective coordinates and a (square root of a)a)a) density matrix with respect to internal coordinates. (ii) By assuming that these states form an approximate complete set for application of the Kerman-Klein method, the self-consistent theory of large amplitude collective motion is extended to finite excitation energy. (iii) The mean-field limit is taken, and further study is carried out in the basis of natural orbitals that diagonalizes the one-particle density matrix. In the approximation maintained uniformly in this work, the equations of motion for the orbitals are shown to be of Hamiltonian form; to these are conjoined the master equations for the time rate of change of the occupation numbers. The latter are studied in two extreme limits. In the collisionless limit, dissipation still arises from the response of the mean field to the collective motion (one-body friction). At the opposite extreme, collisions are assumed to be so effective as to force the system always to be in a state of local (constrained) equilibrium. (iv) We review a procedure by which the noncollective variables in the Hamiltonian may be eliminated, leading to equations of motion for the collective variables with dissipative terms that in general depend on the history of the system. The limit of instantaneous friction can be justified in an adiabatic approximation. (v) A decoupling procedure is developed for deriving the form of these equations of motion from the mean-field theory, with the ultimate aim of obtaining the macroscopic parameters that appear in the classical equations of motion for the collective variables. This procedure generalizes that developed in extensive previous work on large amplitude collective motion at zero temperature. (vi) Associating the dissipative equations of motion for the collective variables with one of the assumptions concerning the rates of change of the occupation numbers provides a description of the relaxation of a system initially perturbed from a state of dynamic equilibrium.
In recent years we have developed a new mathematical treatment of large amplitude collective motion in the adiabatic limit and formulated a successful approximation method, called the generalized valley approximation. In this paper we discuss its application to adiabatic time-dependent Hartree theory, for which the method is ideally suited. We apply the method first to an exactly solvable limiting case (the Suzuki model), for which we have shown in a previous paper that the usual form of adiabatic time-dependent Hartree theory is not general enough to yield the exact solution. We introduce an extended theory that remedies this deficiency. The modified theory has also been applied to monopole models close to the Suzuki model that are not exactly solvable. The algorithm developed for this case is sufficiently general to serve as a prototype for those necessary to study more complex realistic models of collective motion.
From the viewpoint defferent from that of Part I, the time-dependent Harteree-Fock theory is formulated for describing large amplitude collective motion and its coupling with intrinsic degrees of freedom. The basic idea consists of the introduction of an additional degree of freedom for the collective motion and the use of Dirac’s canonical theory with constraints. The formalism is essentially equivalent to the ordinary time-dependent Hartree-Fock theory presented in Part I. The result given in Part I is completely reproduced.