Macroscopic Evolution of Particle Systems with Short and Long Range Interactions

We consider a lattice gas with general short range interactions and a Kac potential $J_\gamma({r})$ of range $\gamma^{-1}$, $\gamma>0$, evolving via particles hopping to nearest neighbor empty sites with rates which satisfy detailed balance with respect to the equilibrium measure. Scaling space like $\gamma^{-1}$ and time like $\gamma^{-2}$, we prove that in the limit $\gamma \to 0$ the macroscopic density profile $\rho({r},t)$ satisfies a integro-differential equation which is in the form of the gradient flux of the energy functional $\cal F$, with a mobility given by the Einstein relation Beside a regularity condition on J, the only requirement for this result is that the reference system satisfy the hypotheses of the Varadhan--Yau Theorem leading to the equation for $J\equiv 0$. Therefore the equation holds also if $\cal F$ achieves its minimum on non constant density profiles and this includes the cases in which {\sl phase segregation} occurs. Using the same techniques we also derive hydrodynamic equations for the densities of a two component A-B mixture with long range repulsive interactions between A and B particles. The equations for the densities $\rho_A$ and $\rho_B$ are again in the form of the gradient flux. They describe, at low temperatures, the demixing transition in which segregation takes place via vacancies, i.e. jumps to empty sites. In the limit of very few vacancies the problem becomes similar to phase segregation in a continuum system in the so called incompressible limit.