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arXiv:2301.00202

A transport model description of Time-Dependent Generator Coordinate under Gaussian overlap approximation

Fangyuan Wang, Yingxun Zhang, Z. P. Li

Published Dec 31, 2022• 0 citations
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Abstract

In this work, we derived a transport equation based on a generalized equation of time-dependent generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA). The transport equation is obtained by using quantum-mechanics phase space distributions under a ``quasi-particle" picture and strategy of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hierarchy. The theoretical advantage of this transport equation is that time evolution of $s$-body phase space density distribution is coupled with $s+1$-body phase space density distributions, and thus, non-adiabatic effects and dynamical fluctuations could be involved by more collective degrees and entanglement of phase space trajectories. In future, we will perform the numerical calculations for fission nuclei after obtaining collective inertia and potential energy surface (PES).

DOI
10.48550/arxiv.2301.00202

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In this work, we derived a transport equation based on a generalized equation of time-dependent generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA). The transport equation is obtained by using quantum-mechanics phase space distributions under a ``quasi-particle" picture and strategy of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hierarchy. The theoretical advantage of this transport equation is that time evolution of sss-body phase space density distribution is coupled with s+1s+1s+1-body phase space density distributions, and thus, non-adiabatic effects and dynamical fluctuations could be involved by more collective degrees and entanglement of phase space trajectories. In future, we will perform the numerical calculations for fission nuclei after obtaining collective inertia and potential energy surface (PES).
Time-dependent density functional theory (TDDFT) is widely used for understanding and predicting properties and behaviors of matter. As one of the fundamental theorems in TDDFT, Van Leeuwen theorem [Phys. Rev. Lett. 82, 3863 (1999)PRLTAO0031-900710.1103/PhysRevLett.82.3863] guarantees how to construct a unique potential with the same one-body density evolution. Here we extend Van Leeuwen theorem by exploring truncation criteria in Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Our generalized theorem demonstrates the existence of a unique nonlocal potential to accurately reconstruct the multibody density evolution in binary interacting systems. Under nonstringent conditions, truncation of the BBGKY-hierarchy equations aligns with the behavior of multibody density evolution and maintains consistency in the reduced equations. Based on this theorem, we further construct a static many-body potential and extend the well-known Casida equation for multiple-excitation energy. The extended TDDFT provides an accurate and first-principles framework capable of describing the kinetic processes of correlated systems, including strongly coupled particle transport, multiple-excitation, and ionization processes.
The equations of motion for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. To close the coupled chain at the two-body level, approximations for a three-body density matrix with one-body and two-body density matrices are needed. The time-dependent density-matrix theory (TDDM) assumes that the three-body density matrix is given by the antisymmetrized products of the one-body and two-body density matrices. In this review the truncation schemes of the BBGKY hierarchy beyond TDDM are discussed and a formulation for the study of excited states which is derived from the time-dependent approach is explained. The truncation schemes and the formulation for excited states are applied to the Lipkin model and the Hubbard model to corroborate their validity. Two realistic applications of the TDDM approaches are also presented. One is the dipole and quadrupole excitations of 40^{40}40Ca and 48^{48}48Ca and the other the fusion reactions of 16^{16}16O + 16^{16}16O.
We develop and test efficient approximations to estimate ground state correlations associated with low- and zero-energy modes. The scheme is an extension of the generator coordinate method (GCM) within Gaussian overlap approximation (GOA). We show that the GOA fails in non-Cartesian topologies and presents a topologically correct generalization of the GOA (topGOA). A random-phase-approximation like (RPA-like) correction is derived as the small amplitude limit of topGOA, called topRPA. Using exactly solvable models, the topGOA and topRPA schemes are compared with conventional approaches (GCM-GOA, RPA, Lipkin-Nogami projection) for rotational-vibrational motion and for particle-number projection. The results shows that the new schemes perform very well in all regimes of coupling.
Inspired by the work of McGlynn and Simenel [Phys. Rev. C {\bf 102}, 064614 (2020)], this study investigates the quantum tunneling of two interacting distinguishable particles in two potential wells. We first benchmark the system by reproducing key established results: the exact quantum solution and the spurious self-trapping effect that arises in the real-time mean-field dynamics for strong interactions. To exactly capture the tunneling dynamics, we apply the time-dependent generator coordinate method (TDGCM) to the model. Numerical simulations demonstrate that the TDGCM, by utilizing the real-time mean-field states as generator states, successfully overcomes the self-trapping effect, yielding tunneling dynamics in excellent agreement with the exact solution. Furthermore, we explore the expectation values of the generator coordinates from the correlated TDGCM many-body wave function. While different methods for calculating expectation values show consistent results in some cases, significant discrepancies are observed in others, providing critical insights into the emergence of collective and single-particle behaviors in interacting systems. This work also verifies the TDGCM as a robust framework for describing collective quantum tunneling and opens avenues for its application to more complex and realistic systems.
A numerical procedure is introduced which allows us to extract a collective Hamiltonian, expressed in terms of a collective canonical pair $\stackrel{^}{p}$ and $\stackrel{^}{q}$. The starting point is a microscopic many-body approach, namely the generator coordinate method, and the Weyl formalism permits the description of the collective dynamics in the $p\ensuremath{-}q$ space. As an illustration we compare numerical calculations with exact results, obtained analytically, for the quadratic energy kernel with the Gaussian overlap approximation case and the Goldhaber-Teller dipole model applied to the 4He^{4}\mathrm{He}4He nucleus.NUCLEAR STRUCTURE Numerical derivation of collective Hamiltonians. Generator coordinate method, Weyl procedure. Comparison with analytical results.
The rigorous evolution equation for the one-body density matrix n ( t ) describing a quantum system of interacting particles is not closed according to the investigations of Bogoliubov, Born, Green, Kirkwood, Yvon and others (B-B-G-K-Y hierarchy). The analysis of this equation is made in terms of connected diagrams. It is found that in the limits of large volume and time this equation can be reduced to a generalized Boltzmann equation which is a closed equation with respect to n ( t ) and which can rigorously describe linear and non-linear transport coefficients.