Abstract
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 $^{40}$Ca and $^{48}$Ca and the other the fusion reactions of $^{16}$O + $^{16}$O.
Highlights
The time-dependent Hartree-Fock theory (TDHF) is the basis of the mean-field theories such as the Hartree-Fock theory (HF) and the random-phase approximation (RPA): The HF ground state is given as a stationary solution of the TDHF equation and RPA can be formulated as the small amplitude limit of the TDHF equation
The equations of motions for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. In this time-dependent density-matrix approach (TDDMA) the truncation of the BBGKY hierarchy is applied at the twobody level by approximating the correlated part of the threebody density matrix (C3)
TDDMA has great advantages that a correlated ground state is obtained as a stationary solution of the TDDMA equations and that the small amplitude limit of the TDDMA equations gives the most general form of beyond the random-phase approximation (RPA)
Summary
The time-dependent Hartree-Fock theory (TDHF) is the basis of the mean-field theories such as the Hartree-Fock theory (HF) and the random-phase approximation (RPA): The HF ground state is given as a stationary solution of the TDHF equation and RPA can be formulated as the small amplitude limit of the TDHF equation. The TDDM truncation scheme has been used for simulations of heavy-ion collisions [7,8,9] where the TDDM equations are formulated by using the time-dependent s.p. states which obey a TDHF-like equation. The application of such a TDDM approach to the fusion reactions of 16O + 16O is presented. The paper is organized as follows: The equations of motion for the one-body and two-body density matrices formulated by using a time-independent s.p. basis are given in section 2 and the truncation schemes of the BBGKY hierarchy and the formulation of ERPA are discussed.
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