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arXiv:1210.2519 · EPJ Web of Conferences

Large amplitude motion with a stochastic mean-field approach

Denis Lacroix A, Sakir Ayik B, Bulent Yilmaz, Kouhei Washiyama

Published Jan 1, 2012• 0 citations
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Abstract

In the stochastic mean-field approach, an ensemble of initial conditions is considered to incorporate correlations beyond the mean-field. Then each starting point is propagated separately using the Time-Dependent Hartree-Fock equation of motion. This approach provides a rather simple tool to better describe fluctuations compared to the standard TDHF. Several illustrations are presented showing that this theory can be rather effective to treat the dynamics close to a quantum phase transition. Applications to fusion and transfer reactions demonstrate the great improvement in the description of mass dispersion.

DOI
10.1051/epjconf/20123806003

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In the stochastic mean-field approach, an ensemble of initial conditions is considered to incorporate correlations beyond the mean-field. Then each starting point is propagated separately using the Time-Dependent Hartree-Fock equation of motion. This approach provides a rather simple tool to better describe fluctuations compared to the standard TDHF. Several illustrations are presented showing that this theory can be rather effective to treat the dynamics close to a quantum phase transition. Applications to fusion and transfer reactions demonstrate the great improvement in the description of mass dispersion.
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.
The time-dependent Hartree-Fock (TDHF) and density functional theory (DFT) are among the most useful approaches within mean-field theories for studying static and dynamic properties of complex many-body systems in different branches of physics. Despite the fact that they provide a good approximation for the average properties of one-body degrees of freedoms, they are known to fail to include quantal fluctuations of collective observables and they do not provide sufficient dissipation of collective motion. In order to incorporate these missing effects the stochastic mean-field (SMF) approach was proposed (Ayik 2008). In the SMF approach a set of stochastic initial one-body densities are evolved. Each stochastic one-body density matrix consists of a set of stochastic Gaussian random numbers that satisfy the first and second moments of collective one-body observables. Recent works indicate that the SMF approach provides a good description of the dynamics of the nuclear systems (Yilmaz et al. 2018; Ayik et al. 2019). In this work, the one-dimensional Fermi-Hubbard model is simulated with the SMF approach by using different distributions such as Gaussian, uniform, bimodal and two-point distributions. The dissipative dynamics are discussed and the predictive power of the SMF approach with different probability distributions are compared with each other and the exact dynamics. As a result it is shown that by considering different distributions, the predictive power of the SMF approach can be improved.
By extending the stochastic mean-field model by including pairing, an approach is proposed for describing the evolution of complex many-body systems in terms of an ensemble of time-dependent Hartree-Fock Bogoliubov trajectories which is determined by incorporating fluctuations in the initial state. The nonlinear evolution of the initial fluctuations provides an approximate description of quantal correlations and fluctuations of collective observables. Since the initial-state fluctuations break particle-number symmetry, the dynamical description in which pairing correlations play a crucial role is greatly improved as compared to the mean-field evolution. The approach is illustrated for a system of particles governed by a pairing Hamiltonian.
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.
In Chapter 6 we presented the main computational schemes to derive the dynamical laws for the mean field, including the back-reaction from quantum fluctuations. These equations may be derived from the variation of the CTPEA. The result of this approach is a semiclassical theory of a c-number condensate interacting with a quantized fluctuation field. This approach developed at this level of sophistication is limited as it offers no description of the fluctuations themselves. In most applications the magnitude of the fluctuations can be comparable and at times dominates the effects of the mean field in the semiclassical description. One possible way to incorporate fluctuations is to use the 2PI formalism, where the propagators describing the fluctuations are considered as dynamical variables evolving along with the mean fields. In this chapter we shall explore a different strategy, which is to allow for a stochastic component in the mean field. This component arises from both the uncertainty of the initial configuration of the mean field, and from the fluctuations in the back-reaction from the quantized excitations. Both sources of randomness combine so that stochastic averages in the noisy theory reproduce suitable quantum averages in the underlying quantum field theory. Formally, this approach lifts the seemingly overladen CTPEA. So far in this generally complex object, only the real part is enlisted in the derivation of the relevant equations of motion of the mean field.