MiS Preprint Repository

Motivated by the problem of modelling nucleation in non-isothermal systems, we consider the stochastic evolution of a coupled system of a lattice spin variable $\sigma$ and a continuous variable e (corresponding to the phase and the energy density of a continuum system). The spin variables flip with rates depending both on a Kac-potential type interaction with the spins and on an intercation with the e-field, which plays the role of the external field in ferromagnetics but evolves by a diffusion equation with a forcing depending on the spins.
We analyse the mesoscopic limit, where space scales like the diverging interaction range of the Kac potential, ${\gamma }^{-1}$ while time is not rescaled. By writing $\sigma$ as random time change of a family of independent spins, and thus reducing the problem to investigating integral equations parametrised by independent random variables, we show that as $\gamma \to 0$ the average of the spins over small cubes and the field e converge in probability to the solution of a system of nonlocal evolution equations which is similar to the phase field equations. In some cases the convergence holds until times of order $\log ({\gamma }^{-1})$