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Journal of Physics Conference Series

Testing a variational method for fluctuations

P. D. Stevenson, J. M. A. Broomfield

Published Jan 1, 2010• 0 citations
Publisher

Abstract

A variational approach for many-body systems due to Balian and Veneroni, which goes beyond the Harrtree-Fock mean field, has been implemented in the case of nuclear Skyrme effective interactions used in examining large amplitude collective motion. An evaluation of the numerical issues involved with the method is presented, in which it is found that the effect of model parameters is generally under control, but long-time running leads to unphysical results.

DOI
10.1088/1742-6596/205/1/012027

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A variational approach for many-body systems due to Balian and Veneroni, which goes beyond the Harrtree-Fock mean field, has been implemented in the case of nuclear Skyrme effective interactions used in examining large amplitude collective motion. An evaluation of the numerical issues involved with the method is presented, in which it is found that the effect of model parameters is generally under control, but long-time running leads to unphysical results.
The Balian-Vénéroni (BV) variational principle, which optimizes the evolution of the state according to the relevant observable, is used at the mean-field level to determine the particle-number fluctuations in fragments of many-body systems. For fermions, the numerical evaluation of such fluctuations requires the use of a time-dependent Hartree-Fock (TDHF) code. Proton, neutron, and total nucleon number fluctuations in fragments produced in collisions of two (40)Ca are computed well above the fusion barrier. For deep-inelastic collisions, the fluctuations calculated from the BV variational principle are much larger than standard TDHF results, and closer to experimental fluctuations. Correlations between proton and neutron numbers are determined. These correlations are shown to be larger with exotic systems where charge equilibration occurs.
Standard methods used for computing the dynamics of a quantum many-body system are the mean-field (MF) approximations such as the time-dependent Hartree-Fock (TDHF) approach. Even though MF approaches are quite successful, they suffer some well-known shortcomings, one of which is insufficient dissipation of collective motion. The stochastic mean-field approach (SMF), where a set of MF trajectories with random initial conditions are considered, is a good candidate to include dissipative effects beyond mean field. In this approach, the one-body density matrix elements are treated initially as a set of stochastic Gaussian c numbers that are adjusted to reproduce first and second moments of collective one-body observables. It is shown that the predictive power of the SMF approach can be further improved by relaxing the Gaussian assumption for the initial probabilities. More precisely, using Gaussian or uniform distributions for the matrix elements generally leads to overdamping for long times, whereas distributions with smaller kurtosis lead to much better reproduction of the long time evolution.
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.
The background field formalism based on effective actions is a compelling framework for developing an effective field theory for nuclear density functional theory. Among the challenges in carrying out this development is handling both the particle-hole and pairing channels beyond the mean-field level, which includes how to incorporate collective degrees of freedom. Here we use the exactly solvable one-dimensional Gaudin-Yang model as a theoretical laboratory to explore candidate approaches. We compare Variational Perturbation Theory (VPT) to ordinary many-body perturbation theory and the inversion method, all to second order in their respective expansions, and verify issues with Hubbard-Stratonovich auxiliary fields. VPT outperforms the other approaches at this level over a wide range of densities. The next steps to extend this approach toward nuclei are outlined.
At present there are two vastly different ab initio approaches to the description of the many-body dynamics: the density functional theory (DFT) and the functional integral (path integral) approaches. On one hand, if implemented exactly, the DFT approach can allow in principle the exact evaluation of arbitrary one-body densities. However, when applied to large amplitude collective motion (LACM), this approach needs to be extended in order to accommodate the phenomenon of surface-hopping, when adiabaticity is strongly violated and the description of a system using a single (generalized) Slater determinant is not valid anymore. The functional integral approach on the other hand does not appear to have such restrictions, but its implementation does not appear to be a straightforward endeavor. However, within a functional integral approach one seems to be able to evaluate in principle any kind of observable, such as the fragment mass and energy distributions in nuclear fission. These two radically different approaches can likely be brought together by formulating a stochastic time-dependent DFT approach to many-body dynamics.
Large amplitude collective motions in a many-Fermion system are discussed from a standpoint of the variational method. In our method, the variational space itself is also determined by a variational procedure which is an extension of the Peierls-Yoccoz method. The Schrödinger equation for large amplitude collective motions is obtained under the additional approximations: ( i ) The variational space forms a Lie group and ( ii ) it is separated adiabatically from the remaining space. Our method may be regarded as the Born-Oppenheimer approximation in the unclear physics. Finally it is applied to the Lipkin model.