Archive for Rational Mechanics and Analysis

, Volume 201, Issue 2, pp 681–725 | Cite as

Fourier Law, Phase Transitions and the Stationary Stefan Problem

  • 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.


Thermodynamic Limit Iterative Scheme Stefan Problem Ising System Magnetization Density 
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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Anna De Masi
    • 1
  • Errico Presutti
    • 2
  • Dimitrios Tsagkarogiannis
    • 2
  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità di L’AquilaL’AquilaItaly
  2. 2.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly

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