We consider a lattice gas with general short-range interactions and a Kac potential Jγ (r) of range γ-1, γ > 0, evolving via particles hopping to nearest-neighbour empty sites with rates which satisfy detailed balance with respect to the equilibrium measure. Scaling spacelike γ-1 and timelike γ-2, we prove that in the limit γ → 0 the macroscopic density profile ρ(r, t) satisfies the equation (formula presented) (*) Here σs(ρ) is the mobility of the reference system, that with J ≡ 0, and ℱ(ρ) = ∫[fs(ρ(r)) -1/2ρ(r) ∫ J(r - r′)ρ(r′)dr dr′], where fs(ρ) is the (strictly convex) free energy density of the reference system. 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 (*) for J ≡ 0. Therefore, (*) also holds if ℱ achieves its minimum on non-constant density profiles and this includes the cases in which 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 ρA and ρB are of the form (*). 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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics