  We derive the PDE governing the hydrodynamic limit of a Metropolis rate
crystal surface height process in the "rough scaling" regime introduced by
Marzuola and Weare. The PDE takes the form of a continuity equation, and the
expression for the current involves a numerically computed multiplicative
correction term similar to a mobility. The correction accounts for the fact
that, unusually, the local equilibrium distribution of the process is not a
local Gibbs measure even though the global equilibrium distribution is Gibbs.
We give definitive numerical evidence of this fact, originally suggested in
Gao, et. al., Pure and Applied Analysis (2021). In that paper, an approximate
PDE -- our PDE, but without the correction term -- was derived for the limit of
the Metropolis rate process under the assumption of a local Gibbs distribution.
Our main contribution is to present a numerical method to compute the corrected
macroscopic current, which is given by a function of the third spatial
derivative of the height profile. Our method exploits properties of the local
equilibrium (LE) state of the third order finite difference process. We find
that the LE state of this process is not only useful for deriving the PDE; it
also enjoys nonstandard properties which are interesting in their own right.
Namely, we demonstrate that the LE state is a "rough LE", a novel kind of LE
state discovered in our recent work on an Arrhenius rate crystal surface
process.
