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Exact and approximate many-body dynamics with stochastic one-body density matrix evolution
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, , where each state evolves according to the Stochastic
Schr\"odinger Equation (SSE) given in ref. \cite{Jul02}. 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.Comment: 12 pages, 1 figur