Fourier Law, Phase Transitions and the Stationary Stefan Problem

@article{DeMasi2010FourierLP,
  title={Fourier Law, Phase Transitions and the Stationary Stefan Problem},
  author={Anna De Masi and Errico Presutti and Dimitrios K. Tsagkarogiannis},
  journal={Archive for Rational Mechanics and Analysis},
  year={2010},
  volume={201},
  pages={681-725}
}
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… 
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