  We discuss the origin of the microscopic description of correlations in
quantum many-particle systems obeying Fermi-Dirac and Bose-Einstein statistics.
For correlation operators that give the alternative description of the quantum
state evolution of Bose and Fermi particles, we deduce the von Neumann
hierarchy of nonlinear equations and construct the solution of its
initial-value problem in the corresponding spaces of sequences of trace class
operators. The links of constructed solution both with the solution of the
quantum BBGKY hierarchy and with the nonlinear BBGKY hierarchy for marginal
correlation operators are discussed. The solutions of the Cauchy problems of
these hierarchies are constructed for initial data satisfying a chaos property.
