Fourier law, phase transitions and the stationary Stefan problem

TL;DR

This paper investigates stationary solutions of an integro-differential equation related to Kawasaki dynamics in Ising systems, demonstrating the validity of Fourier's law in the thermodynamic limit and analyzing phase transition effects and boundary conditions.

Contribution

It constructs non-zero current stationary solutions, proves Fourier law validity in the thermodynamic limit, and explores effects of boundary conditions and metastability on the profile.

Findings

Fourier law holds in the thermodynamic limit below critical temperature.

Discontinuities in the equilibrium profile define the interface in phase transitions.

Metastable boundary conditions disrupt monotonicity and Fourier law validity at finite size.

Abstract

We study the one-dimensional stationary solutions of an integro-differential equation derived by Giacomin and Lebowitz from Kawasaki dynamics in Ising systems with Kac potentials, \cite{GiacominLebowitz}. 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 however its validity in the thermodynamic limit where the limit profile is again monotone away from the interface.