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 rho(r, t) of a microscopic model binary alloy undergoing phase segregation: a d-dimensional lattice gas evolving via Kawasaki exchange dynamics, satisfying derailed balance for a Hamiltonian with a long range pair potential gamma(d)J(gamma/chi/). The macroscopic evolution is on the spatial scale gamma(-1) and rime scale gamma(-2), in the limit gamma --> 0. The domain coarsening, described by interface motion, is similar to that obtained from the Cahn-Hilliard equation.