Preprint 71/2000

Mesoscopic limit for non-isothermal phase transition

Nicolas Dirr and Stephan Luckhaus

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Submission date: 06. Nov. 2000
Pages: 21
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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Abstract:
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 tex2html_wrap_inline10 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, tex2html_wrap_inline16 while time is not rescaled. By writing tex2html_wrap_inline10 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 tex2html_wrap_inline20 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 tex2html_wrap_inline24

21.02.2013, 01:40