Fourier law, phase transitions and the Stefan problem

Abstract.

When hydrodynamic or thermodynamic limits are performed in systems which are in the phase transitions regime we may observe perfectly smooth profiles develop singularities with the appearance of sharp interfaces.

I will discuss the phenomenon in stationary non equilibrium states which carry non zero steady currents. The general context is the one where the Fourier law applies, but here it is complemented by a free boundary problem due to the presence of interfaces.

I will specifically consider an Ising system with Kac potentials which evolves under the stochastic Kawasaki dynamics. In a continuum limit the evolution is described by an integro-differential equation, as proved by Giacomin and Lebowitz in [1], see also [2]. I will then study its stationary solutions with a non zero current (produced by suitable boundary conditions) and derive, in the infinite volume limit, macroscopic profiles with an interface proving that the profiles satisfy a stationary Stefan problem and obey the Fourier law.

[1] G.B. Giacomin, J.L. Lebowitz: "Phase segragation dynamics in particle systems with long range interactions. I. Macroscopic limits". J. Stat. Phys. 87, 37 (1997); and "ibid II. Interface motion". SIAM. J. Appl. Math. 58, 1707 (1998)

[2] G.B. Giacomin, J.L. Lebowitz and R. Marra: "Macroscopic Evolution of Particle Systems with Short and Long Range Interactions". Nonlinearity 13, 2143 (2000)