  The nanoscopic structure and the stationary propagation velocity of
(1+1)-dimensional solid-on-solid interfaces in an Ising lattice-gas model,
which are driven far from equilibrium by an applied force, such as a magnetic
field or a difference in (electro)chemical potential, are studied by an
analytic nonlinear-response approximation together with kinetic Monte Carlo
simulations. Here we consider the case that the system is coupled to a
two-dimensional phonon bath. In the resulting dynamic, transitions that
conserve the system energy are forbidden, and the effects of the applied force
and the interaction energies do not factorize (a so-called hard dynamic). In
full agreement with previous general theoretical results we find that the local
interface width changes dramatically with the applied force. However, in
contrast with other hard dynamics, this change is nonmonotonic in the driving
force. However, significant differences between theory and simulation are found
near two special values of the driving force, where certain transitions allowed
by the solid-on-solid model become forbidden by the phonon-assisted dynamic.
Our results represent a significant step toward providing a solid physical
foundation for kinetic Monte Carlo simulations.
