Fourier law, phase transitions and the stationary Stefan problem

De Masi, Anna, Presutti, Errico and Tsagkarogiannis, Dimitrios (2011) Fourier law, phase transitions and the stationary Stefan problem. Archive for Rational Mechanics and Analysis, 201 (2). pp. 681-725. ISSN 0003-9527

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Abstract

We study the one-dimensional stationary solutions of the integro-differential equation which, as proved in Giacomin and Lebowitz (J Stat Phys 87:37–61, 1997; SIAM J Appl Math 58:1707–1729, 1998), describes the limit behavior of the Kawasaki dynamics in Ising systems with Kac potentials. We construct stationary solutions with non-zero current and prove the validity of the Fourier law in the thermodynamic limit showing that below the critical temperature the limit equilibrium profile has a discontinuity (which defines the position of the interface) and satisfies a stationary free boundary Stefan problem. Under-cooling and over-heating effects are also studied: we show that if metastable values are imposed at the boundaries then the mesoscopic stationary profile is no longer monotone and therefore the Fourier law is not satisfied. It regains its validity however in the thermodynamic limit where the limit profile is again monotone away from the interface.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
Depositing User: Dimitrios Tsagkarogiannis
Date Deposited: 17 Sep 2013 11:35
Last Modified: 17 Sep 2013 11:35
URI: http://sro.sussex.ac.uk/id/eprint/46291
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