Mesoscopic limit for non-isothermal phase transition

Dirr, Nicolas ORCID: https://orcid.org/0000-0003-3634-7367 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
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	Non-isothermal phase change; Kac potential; Random time change; Microscopic model for phase field equations
Publisher:	Polymat
ISSN:	1024-2953
Related URLs:	http://mech.math.msu.su/~malyshev/mprf.h...
Last Modified:	18 Oct 2022 13:13
URI:	https://orca.cardiff.ac.uk/id/eprint/13080

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