Mesoscopic Limit for Non-Isothermal Phase Transition

#### N. Dirr, S. Luckhaus

2001, v.7, Issue 3, 355-381

ABSTRACT

Motivated by the problem of modeling 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 interaction 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 analyze 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 parametrized 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 ${\log(\gamma^{-1})}.$

Keywords: non-isothermal phase change,Kac potential,random time change,microscopic model for phase field equations