  The existence and uniqueness of a steady state for nonequilibrium systems
(NESS) is a fundamental subject and a main theme of research in statistical
mechanics for decades. For Gaussian systems, such as a chain of harmonic
oscillators connected at each end to a heat bath, and for anharmonic
oscillators under specified conditions, definitive answers exist in the form of
proven theorems. Answering this question for quantum many-body systems poses a
challenge for the present. In this work we address this issue by deriving the
stochastic equations for the reduced system with self-consistent backaction
from the two baths, calculating the energy flow from one bath to the chain to
the other bath, and exhibiting a power balance relation in the total (chain +
baths) system which testifies to the existence of a NESS in this system at late
times. Its insensitivity to the initial conditions of the chain corroborates to
its uniqueness. The functional method we adopt here entails the use of the
influence functional, the coarse-grained and stochastic effective actions, from
which one can derive the stochastic equations and calculate the average values
of physical variables in open quantum systems. This involves both taking the
expectation values of quantum operators of the system and the distributional
averages of stochastic variables stemming from the coarse-grained environment.
This method though formal in appearance is compact and complete. It can also
easily accommodate perturbative techniques and diagrammatic methods from field
theory. Taken all together it provides a solid platform for carrying out
systematic investigations into the nonequilibrium dynamics of open quantum
systems and quantum thermodynamics.
