  We analytically and numerically study a fourth order PDE modeling rough
crystal surface diffusion on the macroscopic level. We discuss existence of
solutions globally in time and long time dynamics for the PDE model. The PDE,
originally derived by the second author, is the continuum limit of a
microscopic model of the surface dynamics, given by a Markov jump process with
Metropolis type transition rates. We outline the convergence argument, which
depends on a simplifying assumption on the local equilibrium measure that is
valid in the high temperature regime. We provide numerical evidence for the
convergence of the microscopic model to the PDE in this regime.
