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arXiv:nucl-th/9703056 · International Journal of Modern Physics E

A Self-Consistent Treatment of Damped Motion for Stable and Unstable Collective Modes

H. Hofmann, Dieter Kiderlen

Published Apr 1, 1998• 42 citations
arXivHTMLPDF

Abstract

We address the dynamics of damped collective modes in terms of first and second moments. The modes are introduced in a self-consistent fashion with the help of a suitable application of linear response theory. Quantum effects in the fluctuations are governed by diffusion coefficients D μν . The latter are obtained through a fluctuation dissipation theorem generalized to allow for a treatment of unstable modes. Numerical evaluations of the D μν are presented. We discuss briefly how this picture may be used to describe global motion within a locally harmonic approximation. Relations to other methods are discussed, like "dissipative tunneling", RPA at finite temperature and generalizations of the "Static Path Approximation".

DOI
10.1142/s0218301398000105

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We address the dynamics of damped collective modes in terms of first and second moments. The modes are introduced in a self-consistent fashion with the help of a suitable application of linear response theory. Quantum effects in the fluctuations are governed by diffusion coefficients D μν . The latter are obtained through a fluctuation dissipation theorem generalized to allow for a treatment of unstable modes. Numerical evaluations of the D μν are presented. We discuss briefly how this picture may be used to describe global motion within a locally harmonic approximation. Relations to other methods are discussed, like "dissipative tunneling", RPA at finite temperature and generalizations of the "Static Path Approximation".
We address the dynamics of damped collective modes in terms of first and second moments. The modes are introduced in a self-consistent fashion with the help of a suitable application of linear response theory. Quantum effects in the fluctuations are governed by diffusion coefficients D_{μν}. The latter are obtained through a fluctuation dissipation theorem generalized to allow for a treatment of unstable modes. Numerical evaluations of the D_{μν} are presented. We discuss briefly how this picture may be used to describe global motion within a locally harmonic approximation.
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Abstract This chapter develops a quantal transport theory for non-linear nuclear collective motion, the coordinates of which are functionals of the nucleonic degrees of freedom, the latter being interpreted as ‘heat bath’. Introduced on the mean field level the collective variables must be quantized. This is accomplished through the Bohm-Pines method, which leads to an extended Hamiltonian for the entire system. On this basis, transport equations are deduced through a non-perturbative Nakajima-Zwanzig procedure. This scheme is carried out within a locally harmonic approximation, in which all collective forces are linearized locally and in which Gaussian forms are used for propagators of Wigner functions. The diffusion coefficients are determined by the quantal fluctuation dissipation theorem, which for unstable modes requires appropriate analytic continuations. This scheme breaks down below a critical value of temperature identical to that found in the Caldeira-Leggett model for real time propagation. Fundamental properties of these propagators are studied.
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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.
Abstract We have previously developed this framework in the context of static response, here we will generalize this perspective to dynamic response, and the evolution between distinct thermal equilibrium states. Of particular importance will be the derivation of the fluctuation-dissipation theorem, which relates the response of a system to its time dependent fluctuations, clarifying an equivalence between how a system responds to a small perturbation and how it regresses back to equilibrium in the absence of a perturbation. With this framework we can develop linear models of relaxation, classically and quantum mechanically, and derive equa- tions of motion suitable for describing the dynamics of thermal ensembles held at fixed temperature rather than fixed energy. These models are Gaussian, and approximate the dynamical correlations of interacting systems. These effective equations of motion are rendered stochastic due to the incomplete description of the microscopic degrees of freedom that evolve the system.