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Physical Review C

Self-consistent theory of large amplitude collective motion at finite excitation energy

G. Do Dang, Abraham Klein, P.‐G. Reinhard

Published Apr 1, 1999• 5 citations
Publisher

Abstract

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 $a)$ 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.

DOI
10.1103/physrevc.59.2065

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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.
We illustrate some of the concepts introduced in a recent paper devoted to the formulation of a self-consistent theory of large amplitude collective motion at finite excitation energy. We apply these concepts to a Hamiltonian with the symmetry of $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ (a doubled Lipkin model), in which the latter, exemplifying the slow, collective degrees of freedom, interacts with a set of harmonic oscillators representing the fast, noncollective variables. We pass to a mean-field approximation, which implies a collisionless regime, but show as well how the formalism may be extended to include the limit of collisions that establish instantaneous local equilibrium. In the representation of natural orbitals, the variables split into classical canonical sets and occupation numbers of the orbitals. Careful attention is paid to the problem of eliminating the intrinsic canonical coordinates from the equations of motion in such a way (white noise approximation) as to yield dissipative equations of motion for the collective variables. For the case of local equilibrium, the relaxation problem requires in addition an equation relating the slow change in temperature to the slow change of the collective variables. The two limiting relaxation scenarios are studied numerically.
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 model Hamiltonian describing a two-level system with a crossing plus a pairing force is investigated using technique of large-amplitude collective motion. The collective path, which is determined by the decoupling conditions, is found to be almost identical to the one in the Born-Oppenheimer approximation for the case of a strong pairing force. For the weak pairing case, the obtained path describes a diabatic dynamics of the system. PACS number(s): 21.60.-n, 36.40.Sx Nuclei and metallic clusters are finite fermionic many-body systems which support many kinds of collective motion. While properties of high-frequency vibrations (giant resonances) may be well reproduced by the random-phase approximation (RPA), some low-frequency vibrations exhibit an anharmonic nature that the RPA cannot describe. Shape coexistence and fission phenomena in nuclei and clusters also have such a large amplitude nature. In order to investigate these kinds of large amplitude collective motion, one would like to reduce the number of degrees of freedom to a few judiciously chosen collective coordinates. In order to determine the collective coordinates self-consistently, we use the theory of adiabatic
We study the quantum foundations of a theory of large amplitude collective motion for a Hamiltonian expressed in terms of canonical variables. In previous work the separation into slow and fast (collective and noncollective) variables was carried out without the explicit intervention of the Born-Oppenheimer approach. The addition of the Born-Oppenheimer assumption not only provides support for the results found previously in leading approximation, but also facilitates an extension of the theory to include an approximate description of the fast variables and their interaction with the slow ones. Among other corrections, one encounters the Berry vector and scalar potential. The formalism is illustrated with the aid of some simple examples, where the potentials in question are actually evaluated and where the accuracy of the Born-Oppenheimer approximation is tested. Variational formulations of both Hamiltonian and Lagrangian type are described for the equations of motion for the slow variables.
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.
In the present note, a summary of selected aspects of timedependent mean-field theory is first recalled. This approach is optimized to describe one-body degrees of freedom. A special focus is made on how this microscopic theory can be reduced to a macroscopic dynamic for a selected set of collective variables. Important physical phenomena like adiabaticity/diabaticity, one-body dissipation or memory effect are discussed. Special aspects related to the use of a time-dependent density functional instead of a time-dependent Hartree-Fock theory based on a bare hamiltonian are underlined. The absence of proper description of complex internal correlations however strongly impacts the predictive power of mean-field. A brief overview of theories going beyond the independent particles/quasi-particles theory is given. Then, a special attention is paid for finite fermionic systems at low internal excitation. In that case, quantum fluctuations in collective space that are poorly treated at the meanfield level, are important. Several approaches going beyond mean-field, that are anticipated to improve the description of quantum fluctuations, are discussed: the Balian-V´en´eroni variational principle, the Time-Dependent Random Phase Approximation and the recently proposed Stochastic Mean-Field theory. Relations between these theories are underlined as well as their advantages and shortcomings.