  A BBGKY-like hierarchy is derived from the non-equilibrium Redfield equation.
Two further approximations are introduced and each can be used to truncate and
solve the hierarchy. In the first approximation such a truncation is performed
by replacing two-particle Green's functions (GFs) in the hierarchy by their
values at equilibrium. The second method is developed based on the cluster
expansion, which constructs two-particle GFs from one-particle GFs and neglects
the correlation part. A non-equilibrium Wick's Theorem is proved to provide a
basis for this non-equilibrium cluster expansion. Using those two
approximations, our method of solving the Redfield equation, for instance, of
an N-site chain of interacting spinless fermions, involves an eigenvalue
problem with dimension $2^{N}$ and a linear system with dimension $N^2$ in the
first case, and a nonlinear equation with dimension $N^2$ in the second case,
which can be solved iteratively via a sequence of $N^2$ linear systems. Other
currently available direct methods correspond to a linear system or an
eigenvalue system with dimension $4^N$ plus an eigenvalue system with dimension
$2^N$. As a test of the methods, for small systems with size N=4, results are
found to be consistent with results made available by other direct methods.
Although not discussed here, extending both methods to their next levels is
straightforward. This indicates a promising potential for this BBGKY-like
approach of non-equilibrium kinetic equations.
