  Entropy in nonequilibrium statistical mechanics is investigated theoretically
so as to extend the well-established equilibrium framework to open
nonequilibrium systems. We first derive a microscopic expression of
nonequilibrium entropy for an assembly of identical bosons/fermions interacting
via a two-body potential. This is performed by starting from the Dyson equation
on the Keldysh contour and following closely the procedure of Ivanov, Knoll and
Voskresensky [Nucl. Phys. A {\bf 672} (2000) 313]. The obtained expression is
identical in form with an exact expression of equilibrium entropy and obeys an
equation of motion which satisfies the $H$-theorem in a limiting case. Thus,
entropy can be defined unambiguously in nonequilibrium systems so as to embrace
equilibrium statistical mechanics. This expression, however, differs from the
one obtained by Ivanov {\em et al}., and we show explicitly that their ``memory
corrections'' are not necessary. Based on our expression of nonequilibrium
entropy, we then propose the following principle of maximum entropy for
nonequilibrium steady states: ``The state which is realized most probably among
possible steady states without time evolution is the one that makes entropy
maximum as a function of mechanical variables, such as the total particle
number, energy, momentum, energy flux, etc.'' During the course of the study,
we also develop a compact real-time perturbation expansion in terms of the
matrix Keldysh Green's function.
