  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.
