A particle model for spinodal decomposition

We study a one-dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a "small" drift toward regions of higher concentration. We prove convergence in the continuum limit to a nonlinear parabolic equation whenever the initial density profile satisfies suitable conditions which depend on the strength a of the drift. There is a critical value a_c of a. For a<a_c, the density values are unrestricted, while for a≥a_c, they should all be to the right or to the left of a given interval ℐ(a). The diffusion coefficient of the limiting equation can be continued analytically to ℐ(a), and, in the interior of ℐ(a), it has negative values which should correspond to particle aggregation phenomena. We also show that the dynamics can be obtained as a limit of a Kawasaki evolution associated to a Kac potential. The coefficient a plays the role of the inverse temperature β. The critical value of a coincides with the critical inverse temperature in the van der Waals limit and ℐ(a) with the spinodal region. It is finally seen that in a scaling intermediate between the microscopic and the hydrodynamic, the system evolves according to an integrodifferential equation. The instanton solutions of this equation, as studied by Dal Passo and De Mottoni, are then related to the phase transition region in the thermodynamic phase diagram; analogies with the Cahn-Hilliard equations are also discussed.