Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Extended Time-Dependent Hartree-Fock Approximation with Particle Collisions

Cheuk-Yin Wong, Henry H. K. Tang

Published Apr 17, 1978• 149 citations
Publisher

Abstract

We formulate an extended time-dependent Hartree-Fock approximation which includes particle collisions. As the configuration-space analog of the quantum Boltzmann equation, it can be utilized to study the dynamics of nuclear or other fermion systems when irreversible dissipation is present.
DOI
10.1103/physrevlett.40.1070

Related papers

Dynamics of nuclear fluid. V. Extended time-dependent Hartree-Fock approximation illuminates the approach to thermal equilibrium

Cheuk-Yin Wong, Henry H. K. Tang

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 $H$ 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. $H$ theorem. Analytic solution of the master equation. New types of level crossing formula.

Physical Review C•Oct 1, 1979•62 citations

The TDHF Approximation for Hamiltonians with mmm-particle Interaction Potentials

Claude Bardos, Bernard Ducomet A, François Golse +2 more

According to a theory of H. Spohn, the time-dependent Hartree (TDH) equation governs the 1-particle state in N -particle systems whose dynamics are prescribed by a non-relativistic Schrödinger equation with 2-particle interactions, in the limit N tends to infinity while the strength of the 2-particle interaction potential is scaled by 1/N .In previous work we have considered the same mean field scaling for systems of fermions, and established that the error of the time-dependent Hartree-Fock (TDHF) approximation tends to 0 as N tends to infinity.In this article we extend our results to systems of fermions with m-particle interactions (m > 2).

Communications in Mathematical Sciences•Jan 1, 2007•14 citations

Extended Time-Dependent Hartree-Fock Approximation and Dissipative Diabatic Dynamic

Shun-Jin Wang

The extended time-dependent Hartree-Fock approximation (ETDHF) is derived in a new approach in which the equations of motion for many-body correlation matrices are given up to the second order. The eigen representation of the ETDHF is established and compared with other representations. A possible parametrization of the ETDHF is presented in the limit of low collective velocity and the diabatic shell model single particle bases are discussed from the viewpoint of the parametrized ETDHF. The dissipative diabatic dynamics(DDD) is derived from the ETDHF and it 1s proved to be a realization of the ETDHF in the time-dependent shell model representation. The formal solutions of the occupation number are given in reasonable approximations and they are proved to bear the basic features of the DDD. Finally the decaying diabatic potential proposed phenomenologically in DDD is derived microscopically.

Communications in Theoretical Physics•Nov 1, 1985•0 citations

Time-Dependent Hartree–Fock Approximation and Mean-Field Decoupling

Giancarlo Calvanese Strinati

Mean-field decouplings can also be utilized in time-dependent (nonequilibrium) situations. This chapter considers the time-dependent Hartree–Fock approximation for fermions in the normal phase, as obtained in terms of a time-dependent mean-field decoupling (postponing to Part II the Gor’kov generalization to the superfluid phase). A connection is also established with a more standard formulation of the time-dependent Hartree–Fock approximation in terms of a set of time-dependent single-particle wave functions.

Cambridge University Press eBooks•Feb 6, 2026•0 citations

Gaussian phase-space representation of fermion dynamics: Beyond the time-dependent Hartree-Fock approximation

Saar Rahav, Shaul Mukamel

A Gaussian operator representation for the many-body density matrix of fermionic systems, developed by Corney and Drummond [Phys. Rev. Lett. 93, 260401 (2004)], is used to derive approximate decoupling schemes for their dynamics. In this approach the reduced single electron density-matrix elements serve as stochastic variables which satisfy an exact Fokker-Planck equation. The number of variables scales as $\ensuremath{\sim}{N}^{2}$ rather than $\ensuremath{\sim}\text{exp}(N)$ with the basis set size, and the time-dependent Hartree-Fock approximation (TDHF) is recovered in the ``classical'' limit. An approximate closed set of equations of motion for the one- and two-particle reduced density matrices provides a direct generalization of the TDHF.

arXiv:0902.3227•Physical Review B•Apr 10, 2009•10 citations

Extended Time-Dependent Hartree-Fock Approximation with Particle Collisions.

Cheuk-Yin Wong, Henry H. K. Tang

Physical Review Letters•Jun 5, 1978•1 citation
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.
According to a theory of H. Spohn, the time-dependent Hartree (TDH) equation governs the 1-particle state in N -particle systems whose dynamics are prescribed by a non-relativistic Schrödinger equation with 2-particle interactions, in the limit N tends to infinity while the strength of the 2-particle interaction potential is scaled by 1/N .In previous work we have considered the same mean field scaling for systems of fermions, and established that the error of the time-dependent Hartree-Fock (TDHF) approximation tends to 0 as N tends to infinity.In this article we extend our results to systems of fermions with m-particle interactions (m > 2).
The extended time-dependent Hartree-Fock approximation (ETDHF) is derived in a new approach in which the equations of motion for many-body correlation matrices are given up to the second order. The eigen representation of the ETDHF is established and compared with other representations. A possible parametrization of the ETDHF is presented in the limit of low collective velocity and the diabatic shell model single particle bases are discussed from the viewpoint of the parametrized ETDHF. The dissipative diabatic dynamics(DDD) is derived from the ETDHF and it 1s proved to be a realization of the ETDHF in the time-dependent shell model representation. The formal solutions of the occupation number are given in reasonable approximations and they are proved to bear the basic features of the DDD. Finally the decaying diabatic potential proposed phenomenologically in DDD is derived microscopically.
Mean-field decouplings can also be utilized in time-dependent (nonequilibrium) situations. This chapter considers the time-dependent Hartree–Fock approximation for fermions in the normal phase, as obtained in terms of a time-dependent mean-field decoupling (postponing to Part II the Gor’kov generalization to the superfluid phase). A connection is also established with a more standard formulation of the time-dependent Hartree–Fock approximation in terms of a set of time-dependent single-particle wave functions.
A Gaussian operator representation for the many-body density matrix of fermionic systems, developed by Corney and Drummond [Phys. Rev. Lett. 93, 260401 (2004)], is used to derive approximate decoupling schemes for their dynamics. In this approach the reduced single electron density-matrix elements serve as stochastic variables which satisfy an exact Fokker-Planck equation. The number of variables scales as $\ensuremath{\sim}{N}^{2}$ rather than $\ensuremath{\sim}\text{exp}(N)$ with the basis set size, and the time-dependent Hartree-Fock approximation (TDHF) is recovered in the ``classical'' limit. An approximate closed set of equations of motion for the one- and two-particle reduced density matrices provides a direct generalization of the TDHF.