Subdynamics in a many-body system of quantum particles

For a system of N ≫ 1 spinless quantum particles, the projection operator is introduced, which, surprisingly, exactly transforms the inhomogeneous (with irrelevant initial correlations term) Nakajima-Zwanzig generalized master equation (GME) into completely closed (homogeneous) GME with initial correlations included into the kernel governing its evolution. The obtained equation is equivalent to the closed linear evolution equation for an S -particle ( S < N ) statistical operator, which holds for any initial condition and all timescales. Thus, the introduced projector implements the idea of the subdynamics in the subspace of S -particle complex. It also allows for precise treatment of the initial correlations on an equal footing with collisions. No “molecular chaos” -type approximation is used. The obtained homogeneous equation is further specialized in the linear approximation in the particles density n for the kernel and considered in detail for the one-particle statistical operator. The influence of both the quantum correlations, caused by the particles' statistics, and initial correlations, related to the interparticle interaction, is considered in detail and at different timescales. Connection of the linear equation for a one-particle statistical operator to the nonlinear quantum Boltzmann equation is shown.