The phenomena of interest in phase kinetics can be viewed on many different space and time scales. On a very large scale, observed over long times, phase interfaces appear very sharp, and the evolution is specified in terms of equations for the motion of the sharp interface. On shorter “mesoscopic” length scales over which the the interface between the phases is not sharp, one has different equations, typically of Cahn-Hilliard type, for the phase densities. It is of interest to relate the solutions of the equations of motion for the sharp interface to solutions of the mesoscopic equations. A typical way to approach such a problem is to use matched asymptotic expansions.

A similar problem arises in the kinetic theory of dilute gasses in which one seek to relate solutions of the Euler equation to solutions of the Boltzmann equation. A different set of methods developed by Caflisch has been successful *this context, which is based on a controlled Chapman-Enskog expansion. **Recent joint work with Carvalho and Orlandi has developed this approach **in the phase kinetics setting. In these lectures, we shall explain this approach, **and develop some of tools need to apply it, with special emphasis on potential **applications involving surface diffusion.*