Hierarchy Bloch equations for the reduced statistical density operators in canonical and grand canonical ensembles

Starting from Bloch equation for a canonical ensemble, we deduce a set of hierarchy equations for the reduced statistical density operator for an identical many-body system with two-body interaction. They provide a law according to which the reduced density operator varies in temperature. By definition of the reduced density operator in Fock space for a grand canonical ensemble, we also obtain the analogous Bloch equation and the corresponding hierarchy reduced equations for the identical and interacting many-body system. We discuss their possible solutions and applications.