Exact macroscopic description of phase segregation in model alloys with long range interactions.

We derive an exact nonlinear nonlocal macroscopic equation for the time evolution of the conserved order parameter $\ensuremath{\rho}(\mathbf{r},t)$ of a microscopic model binary alloy undergoing phase segregation: a d-dimensional lattice gas evolving via Kawasaki exchange dynamics, satisfying detailed balance for a Hamiltonian with a long range pair potential ${\ensuremath{\gamma}}^{d}J(\ensuremath{\gamma}|x|)$. The macroscopic evolution is on the spatial scale ${\ensuremath{\gamma}}^{\ensuremath{-}1}$ and time scale ${\ensuremath{\gamma}}^{\ensuremath{-}2}$, in the limit $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. The domain coarsening, described by interface motion, is similar to that obtained from the Cahn-Hilliard equation.