Macroscopic evolution of particle systems with short- and long-range interactions

We consider a lattice gas with general short-range interactions and a Kac p otential J(gamma)(r) Of range gamma (-1), gamma > 0, evolving via particles hopping to nearest-neighbour empty sites with rates which satisfy detailed balance with respect to the equilibrium measure. Scaling spacelike gamma ( -1) and timelike gamma (-2), we prove that in the limit gamma --> 0 the mac roscopic density profile rho (r, t) satisfies the equation partial derivative/partial derivativet rho (r, t) = del . [sigmas(rho)del d eltaF(rho)/delta rho (r)]. Here sigma (s)(rho) is the mobility of the refer ence system, that with J = 0, and F(rho) = integral [f(s)(rho (r)) -1/2 rho (r) integral J(r -r')rho (r')dr dr'], where f(s)(rho) is the (strictly con vex) free energy density of the reference system. Beside a regularity condi tion on J, the only requirement for this result is that the reference syste m satisfy the hypotheses of the Varadhan-Yau theorem leading to (*) for J = 0. Therefore, (*) also holds if F achieves its minimum on non-constant den sity profiles and this includes the cases in which phase segregation occurs . Using the same techniques we also derive hydrodynamic equations for the d ensities of a two-component A-B mixture with long-range repulsive interacti ons between A and B particles. The equations for the densities rho (A) and rho (B) are of the form (*). They describe, at low temperatures, the demixi ng transition in which segregation takes place via vacancies, i.e. jumps to empty sites. In the limit of very few vacancies the problem becomes simila r to phase segregation in a continuum system in the so-called incompressibl e limit.