  To take initial correlations into account, a method, based on the
time-independent projection operator technique, that allows converting the
conventional linear inhomogeneous (containing a source caused by initial
correlations) time-convolution generalized master equation (TC-GME) and
time-convolutionless GME (TCL-GME) into the homogeneous form exactly, is
proposed. This approach results in the exact linear time-convolution and
time-convolutionless homogeneous generalized master equations (TC-HGME and
TCL-HGME) which take the dynamics of initial correlations into account via
modified memory kernels governing the evolution of the relevant part of a
distribution function of a many-particle system. However, to derive the desired
nonlinear equations (the Boltzmann equation in particular) from this actually
linear equation, we should make an additional approximation neglecting the
time-retardation of a one-particle distribution function which restricts the
time scale by times much smaller than the relaxation time for a one-particle
distribution function. To obtain the actually nonlinear evolution equations and
avoid any restrictions on the time scales, we develop a new method, based on
using a time-dependent operator converting a distribution function of a total
system into the relevant form, that allows deriving the new exact nonlinear
generalized master equations. To include the initial correlations into
consideration, we convert the obtained inhomogeneous nonlinear GME into the
homogeneous form by the method which we used for conventional linear GMEs. The
obtained exact homogeneous nonlinear GME describes all evolution stages of the
system of interest and treats initial correlations on an equal footing with
collisions by means of the modified memory kernel.
