  We consider the problem of heat transport by vibrational modes (conduction)
between Langevin thermostats connected by a central device. The latter is
anharmonic and can be subject to large temperature differences and thus be out
of equilibrium. We develop a self-consistent Green's function formalism to
describe high-temperature and non-equilibrium transport, and derive a formula
for the heat current for up to quartic anharmonicity (4th-order in the
potential energy). We show the importance of including quartic terms in the
anharmonic potential in order to properly describe thermal expansion and
temperature dependence to leading order in anharmonicity. This formalism paves
the way for accurate and efficient modeling of thermal transport in highly
non-equilibrium situations beyond perturbation theory.
