Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits

We present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter p(r, t) of a binary alloy undergoing phase segregation. Our model is a d-dimensional lattice gas evolving via Kawasaki exchange dynamics, i.e., a (Poisson) nearest neighbor exchange process, reversible with respect to the Gibbs measure for a Hamiltonian which includes both short-range (local) and long-range (nonlocal) interactions. The nonlocal part is given by a pair potential γ^dJ(γ |x - y|), γ > 0, x and y in ℤ^d, in the limit γ → 0. The macroscopic evolution is observed on the spatial scale γ^-1 and time scale γ^-2, i.e., the density p(r, t) is the empirical average of the occupation numbers over a small macroscopic volume element centered at r = γx. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (Part II) we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.