  In this article, we present a concise and self-contained introduction to
nonequilibrium statistical mechanics with quantum field theory by considering
an ensemble of interacting identical bosons or fermions as an example. Readers
are assumed to be familiar with the Matsubara formalism of equilibrium
statistical mechanics such as Feynman diagrams, the proper self-energy, and
Dyson's equation. The aims are threefold: (i) to explain the fundamentals of
nonequilibrium quantum field theory as simple as possible on the basis of the
knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in
describing nonequilibrium systems from Dyson's equation on the Keldysh contour
to the Navier-Stokes equation in fluid mechanics via quantum transport
equations and the Boltzmann equation; (iii) to derive an expression of
nonequilibrium entropy that evolves with time. In stage (i), we introduce
nonequilibrium Green's function and the self-energy uniquely on the round-trip
Keldysh contour, thereby avoiding possible confusions that may arise from
defining multiple Green's functions at the very beginning. We try to present
the Feynman rules for the perturbation expansion as simple as possible. In
particular, we focus on the self-consistent perturbation expansion with the
Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable
approximation, which has a crucial property for nonequilibrium systems of
obeying various conservation laws automatically. We also show how the
two-particle correlations can be calculated within the Phi-derivable
approximation, i.e., an issue of how to handle the
"Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".
