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The Bogoliubov-Born-Green-Kirkwood-Yvon or Time-Dependent Density Matrix(TDDM) hierarchy of equations for higher density matrices is truncated at thethree body level in approximating the three body correlation function by aquadratic form of two body ones, closing the equations in this way. Theprocedure is discussed in detail and it is shown in non-trivial model casesthat the approximate inclusion of three body correlation functions is veryimportant to obtain precise results. A small amplitude approximation of thistime dependent nonlinear equation for the two body correlation function isperformed (STDDM*-b) and it is shown that the one body sector of thisgeneralised non-linear second RPA equation is equivalent to the Self-ConsistentRPA (SCRPA) approach which had been derived previously by different techniques.It is discussed in which way SCRPA also contains the three body correlations.TDDM and SCRPA are tested versus exactly solvable model cases.

physical review. b./physical review. b

Progress in Many Body Theory with the Equation of Motion method. Time  dependent Density Matrix meets Self-Consistent RPA. Applications to solvable  Models