Effective mobility and diffusivity in coarsening processes

We suggest that coarsening dynamics can be described in terms of a generalized random walk, with the dynamics of the growing length L(t) controlled by a drift term, ?(L), and a diffusive one, D(L). We apply this interpretation to the one-dimensional Ising model with a ferromagnetic coupling constant decreasing exponentially on the scale R. In the case of nonconserved (Glauber) dynamics, both terms are present and their balance depends on the interplay between L(t) and R. In the case of conserved (Kawasaki) dynamics, drift is negligible, but D(L) is strongly dependent on L. The main pre-asymptotic regime displays a speeding of coarsening for Glauber dynamics and a slowdown for Kawasaki dynamics. We reason that a similar behaviour can be found in two dimensions.