Exact and approximate many-body dynamics with stochastic one-body density matrix evolution

Abstract

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 ab = |Φa Φ b |, where each state evolves according to the Stochastic Schrödinger Equation (SSE) given in ref. [1]. A stochastic Liouville-von Neumann equation is derived as well as the associated Bogolyubov-Born-Green-Kirwood-Yvon (BBGKY) 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 (Extended TDHF) 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.

Stochastic quantum dynamics beyond mean field

“…As a consequence, the number of trajectories necessary to properly describe the problem increases very fast and prevent from using such a technique. Specific methods, that explicitly use the QMC flexibility, can however be proposed to reduce statistical fluctuations [104]. Second, implementation of QMC requires to solve non-linear stochastic equations.…”

“…If we denote by {|β i } i=1,N and {|α i } i=1,N the set of N single-particle states, we assume in addition that for each couples of SD, associated singles-particle wave-functions verify β j | α i = δ ij . Accordingly, the one-body density matrix associated to a given D reads [116,104,88] …”

“…Since the evolution is exact, any one-, two-or k-body observable Q estimated through Q ≡ T r(D(t)Q) will follow the exact dynamics [104].…”

Unitary evolution with fluctuations and dissipation

“…The solution of a Lindblad equation, even after a reduction to a Monte Carlo wave function approach, in the case of nuclear fission becomes easily prohibitive numerically. We should mention here that over the years many extensions of the time-dependent meanfield approaches have been suggested in nuclear physics, in order to incorporate fluctuations in a fully quantum treatment [25][26][27][28][29][30][31], with implementations illustrated for very small quantum systems or suffering from fundamental flaws [32,33], c.f. [34].…”

Comment on the recent article "Fission dynamics of from saddle to scission and beyond" by Bulgac et al., published in Phys. Rev. {\bf C 100}, 034615 (2019)

“…In general, methods like auxiliary fields methods applied in real-time evolution, due to the necessity to have a term proportional to √ i∆t in the singleparticle evolution, the evolution is non-unitary and the one-body density is non-hermitian (see for instance [3,23,24]). Here also, it would not make sense to interpret a single trajectory.…”