On the validity of an einstein relation in models of interface dynamics

We consider models of interface dynamics derived from Ising systems with Kac interactions and we prove the validity of the Einstein relation=, where is the proportionality coefficient in the motion by curvature, is the interface mobility, and is the surface tension.     

Interface motion in a planar spin-flip model derived from exclusion on the line

We consider a two-dimensional spin-flip model, which can be interpreted as the limit of the Ising model at low temperature and a small nonzero external field. In the hydrodynamic limit and for a special class of initial conditions, the motion of the interface is governed by a nonlinear partial differential equation with a lattice-distorted curvature term. In our proofs we use results about the hydrodynamic behavior of the weakly asymmetric exclusion process on the integers and also on the nonnegative integers with a trap at the boundary.