First order phase transitions and the thermodynamic limit

We consider simple mean field continuum models for first order liquid–liquid demixing and solid–liquid phase transitions and show how the Maxwell construction at phase coexistence emerges on going from finite-size closed systems to the thermodynamic limit. The theories considered are the Cahn–Hilliard model of phase separation, which is also a model for the liquid-gas transition, and the phase field crystal model of the solid–liquid transition. Our results show that states comprising the Maxwell line depend strongly on the mean density with spatially localized structures playing a key role in the approach to the thermodynamic limit