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Physical Review C

Lifetime of interference effects in collisional time-dependent Hartree-Fock dynamics

Claudio Oscar Dorso, E. Susana Hernández

Published Aug 1, 1982• 16 citations
Publisher

Abstract

A kinetic equation in operator form for the one body density matrix is derived by means of a convenient truncation of the quantal Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Using projection techniques in a procedure similar to the derivation of the master equation, the evolution of the diagonal and the nondiagonal terms of the density matrix can be disentangled. A quantal, non-Hermitian master equation for the diagonal part is extracted and it is seen that corrections to the collision frequency or to the relaxation time arise as a consequence of the creation and destruction of incoherent events.NUCLEAR REACTIONS Reduced one body density matrix, BBGKY hierarchy, quantal kinetic equation, coupling between probabilistic and interference matrix element, collision frequency and relaxation time, dynamical quantal corrections.

DOI
10.1103/physrevc.26.528

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Physical Review C•Feb 1, 1983•13 citations
A kinetic equation in operator form for the one body density matrix is derived by means of a convenient truncation of the quantal Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Using projection techniques in a procedure similar to the derivation of the master equation, the evolution of the diagonal and the nondiagonal terms of the density matrix can be disentangled. A quantal, non-Hermitian master equation for the diagonal part is extracted and it is seen that corrections to the collision frequency or to the relaxation time arise as a consequence of the creation and destruction of incoherent events.NUCLEAR REACTIONS Reduced one body density matrix, BBGKY hierarchy, quantal kinetic equation, coupling between probabilistic and interference matrix element, collision frequency and relaxation time, dynamical quantal corrections.
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 show that the dynamics of interacting fermions can be exactly replaced by a quantum jump theory in the many-body density matrix space. In this theory, jumps occur between densities formed of pairs of Slater determinants, ${D}_{\mathit{ab}}=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|$, where each state evolves according to the stochastic Schr\"odinger equation given by O. Juillet and Ph. Chomaz [Phys. Rev. Lett. 88, 142503 (2002)]. A stochastic Liouville-von Neumann equation is derived as well as the associated. Bogolyubov-Born-Green-Kirwood-Yvon hierarchy. Due to the specific form of the many-body density along the path, the presented theory is equivalent to a stochastic theory in one-body density matrix space, in which each density matrix evolves according to its own mean-field augmented by a one-body noise. Guided by the exact reformulation, a stochastic mean-field dynamics valid in the weak coupling approximation is proposed. This theory leads to an approximate treatment of two-body effects similar to the extended time-dependent Hartree-Fock scheme. In this stochastic mean-field dynamics, statistical mixing can be directly considered and jumps occur on a coarse-grained time scale. Accordingly, numerical effort is expected to be significantly reduced for applications.
In the time-dependent Hartree-Fock approximation, the fermions are assumed to interact only through the mean field and the collisions between particles are neglected. We formulate an extended time-dependent Hartree-Fock approximation which incorporates particle collisions due to the residual interaction, with the usual time-dependent Hartree-Fock approximation as the collisionless limit. It is obtained by a truncation of the Martin-Schwinger hierarchy at the second level and by using a simple representation of the one-body Green's function in terms of time-dependent occupation numbers. The final set of coupled equations consists of a modified time-dependent Hartree-Fock equation and a master equation for the occupation numbers. These results are physically transparent and describe properly the physics of the collision process. They may also be simple enough to be of practical use to study heavy-ion collisions or the dynamics of other fermion systems. Furthermore, as the configuration-space analog of the quantum Boltzmann equation, many important results concerning statistical dynamics are obtained. Concepts such as entropy, temperature, and local and thermal equilibrium can be quantitatively introduced. The well-known HHH theorem that entropy never decreases can be readily recovered. With the collision term explicitly exhibited, the macroscopic equations (equations of continuity, momentum flux, and energy) and their associated conservation theorems can also be derived. Analytic solutions for the master equation for simple cases lead to new "level crossing" formulas having characteristics distinctly different from the Landau-Zener level-crossing formula and illuminate the salient features as to how a nonequilibrium fermion system approaches thermal equilibrium.NUCLEAR REACTIONS Extension of time-dependent Hartree-Fock approximation. Collisions between particles. Master equation for occupation probabilities. Entropy, temperature, and thermal equilibrium. HHH theorem. Analytic solution of the master equation. New types of level crossing formula.
In this work, we analyze the Born, Bogoliubov, Green, Kirkwood, and Yvon (BBGKY) hierarchy of equations for describing the full time evolution of a many-body fermionic system in terms of its reduced density matrices (at all orders). We provide an exhaustive study of the challenges and open problems linked to the truncation of such a hierarchy of equations to make them practically applicable. We restrict our analysis to the coupled evolution of the one- and two-body reduced density matrices, where higher-order correlation effects are embodied into the approximation used to close the equations. We prove that within this approach, the number of electrons and total energy are conserved, regardless of the employed approximation. Further, we demonstrate that although most of the truncation schemes available in the literature give acceptable ground-state energy, when applied to describe driven electron dynamics, they exhibit undesirable and unphysical behavior, e.g., violation and even divergence in local electronic density, both in weakly and strongly correlated regimes. We illustrate and analyze these problems within the few-site Hubbard model. The model can be solved exactly and provides a unique reference for our detailed study of electron dynamics for different values of interaction, different initial conditions, and the large set of approximations considered here. Moreover, we study the role of compatibility between two hierarchical equations and positive semidefiniteness of reduced density matrices in the instability of electron dynamics. We show that even if the used approximation holds the compatibility, electron dynamics can still diverge when positive definitiveness is violated. We propose some partial solutions of such problems and point out the main paths for future work in order to make this approach applicable for the description of the correlated electron dynamics in complex systems.
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.
We present a formal derivation of the exact dynamics of the one-body density matrix. Its essential ingredients are shown to be the following: (a) a mean field unitary time evolution; (b) irreducible nonunitary corrections to the unitary evolution (collision effects); and (c) effects due to the time evolution of initial state correlation [which contribute to both (a) and (b)]. In the weak coupling limit the collision effects give rise to a term of Boltzmann form. A qualitative discussion of the importance of collision effects for the expectation values of one-body operators and a quantitative illustration in the framework of an exactly soluble model are given. In this case one finds large nonunitary contributions.NUCLEAR STRUCTURE Exact microscopic description of the time evolution of the one body density matrix.