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Fourier law, phase transitions and the stationary Stefan problem

journal contribution
posted on 2023-06-08, 15:46 authored by Anna De Masi, Errico Presutti, Dimitrios Tsagkarogiannis
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.

History

Publication status

  • Published

Journal

Archive for Rational Mechanics and Analysis

ISSN

0003-9527

Publisher

Springer Verlag

Issue

2

Volume

201

Page range

681-725

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2013-09-17

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