Working within a time-dependent Hartree-Fock framework, one develops a semiclassical approximation appropriate for large systems. It is demonstrated that the standard semiclassical approach, the Thomas-Fermi approximation, is inconsistent with Hartree-Fock theory when the basic two-body interaction is short-ranged (as in nuclear systems, for example). However, by introducing a simple extension of the Thomas-Fermi approximation, one overcomes this problem. One also discusses the infinite nuclear matter problem and point out that time-dependent Hartree-Fock theory yields collective modes of the zero sound variety instead of ordinary hydrodynamic (first) sound. One thus emphasizes that one should be extremely circumspect when attempting to cast the equations of motion of time-dependent Hartree-Fock theory into a hydrodynamic-like form.
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.
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.
Some limitations of the conventional time-dependent Hartree-Fock method for describing complex reactions are noted, and one particular ubiquitous defect is discussed in detail: the post-breakup spurious cross channel correlations which arise whenever several asymptotic reaction channels must be simultaneously described by a single determinant. A reformulated time-dependent---
S-matrix Hartree-Fock theory is proposed, which obviates this difficulty. Axiomatic requirements minimal to assure that the time-dependent---
S-matrix Hartree-Fock theory represents an unambiguous and physically interpretable asymptotic reaction theory are utilized to prescribe conditions upon the definition of acceptable asymptotic channels. That definition, in turn, defines the physical range of the time-dependent---
S-matrix Hartree-Fock theory to encompass the collisions of mathematically well-defined "time-dependent Hartree-Fock droplets." The physical properties of these objects then circumscribe the content of the Hartree-Fock single determinantal description. If their periodic vibrations occur for continuous ranges of energy then the resulting "classical" time-dependent Hartree-Fock droplets are seen to be intrinsically dissipative, and the single determinantal description of their collisions reduces to a "trajectory" theory which can describe the masses and relative motions of the fragments but can provide no information about specific asymptotic excited states beyond their constants of motion, or the average properties of the limit, if it exists, of their equilibrization process. If, on the other hand, the periodic vibrations of the time-dependent Hartree-Fock droplets are discrete in energy, then the time-dependent---
S-matrix Hartree-Fock theory can describe asymptotically the time-average properties of the whole spectrum of such periodic vibrations of these "quantized" time-dependent Hartree-Fock droplets which are asymptotically stationary on a time-averaged basis. Such quantized time-dependent Hartree-Fock droplet spectra promise the closest analog to the rich array of asymptotic channel eigenstates in the exact Schr\"odinger theory which single time-dependent Hartree-Fock self-consistent determinants might describe. Whether the time-dependent Hartree-Fock droplets are classical or quantized is determined by the mathematical properties of the periodic solutions of the time-dependent Hartree-Fock equation. If the droplets are in fact classical, then the question remains open whether an explicit requantization, by assumption, could consistently restore the close structural analogy with the exact theory. We argue that if the statistical interpretation of the single determinantal wave function, which is central to the present restructuring of the conventional time-dependent Hartree-Fock description, were to be found inadmissible, then no basis would remain for considering the determinant as a "wave function" in the Schr\"odinger sense. Finally, we note that the conceptual basis of this time-averaged
S-matrix theory need not be restricted to the time-dependent Hartree-Fock theory, but might apply as well to other nonlinear Schr\"odinger-type models which purport to provide an approximate wave function to describe physical reaction processes.NUCLEAR REACTIONS Structural analysis of scattering theory for time-dependent Hartree-Fock single determinantal wave functions; scattering matrix defined as a time average; channels labeled by time-averaged properties of periodic solutions; time-averaged channel orthogonality; classical and quantal time-dependent Hartree-Fock droplets.
The variational derivation of the time-dependent Hartree-Fock equation is reviewed. When norm-violating variations are included, a unique time-dependent Hartree-Fock Hamiltonian, which differs from that customarily used in time-dependent Hartree-Fock analyses, is implied. This variationally "true" Hartree-Fock Hamiltonian has the same expectation value as the exact Hamiltonian, equal to the average energy of the system. Since this quantity remains constant under time-dependent Hartree-Fock time evolution, we suggest the label "constant
〈H〉" for this form of time-dependent Hartree-Fock theory.NUCLEAR REACTIONS Unique time-dependent Hartree-Fock Hamiltonian obtained from variational principle by norm-violating variations.