Mesoscopic limit for non-isothermal phase transition
Nicolas Dirr and Stephan Luckhaus
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Submission date: 06. Nov. 2000
published in: Markov processes and related fields, 7 (2001) 3, p. 355-381
MSC-Numbers: 60K35, 80A22, 82C22
Keywords and phrases: non-isothermal phase change, kac-potential, random time change, microscopic model for phase field equations
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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 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, while time is not rescaled. By writing 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 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