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# Mesoscopic limit for non-isothermal phase transition

 Dirr, Nicolas and Luckhaus, Stephen 2001. Mesoscopic limit for non-isothermal phase transition. Markov processes and related fields 7 (3) , pp. 355-381.

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## 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 ${\log(\gamma^{-1})}.$

Item Type: Article Publication Published Mathematics Q Science > QA Mathematics Non-isothermal phase change; Kac potential; Random time change; Microscopic model for phase field equations Polymat 1024-2953 http://mech.math.msu.su/~malyshev/mprf.h... 04 Jun 2017 02:52 http://orca-mwe.cf.ac.uk/id/eprint/13080