= Phase separation
A system can split into macroscopic regions with distinct local <order parameters>, densities or compositions. In a short-range system at <phase coexistence>, regions of two phases with means $M_1,M_2$ and volume fractions $q,1-q$ have spatial mean $M=qM_1+(1-q)M_2$. Their interface contribution is subextensive, so their thermodynamic <free-energy density> is the corresponding convex combination of the phase densities. This can lie below the homogeneous-branch <Landau free energy> at the same $M$. The equilibrium <constrained order-parameter free energy> is consequently convexified, with a straight coexistence segment described by the <Maxwell construction>. A nonconvex local potential remains useful for homogeneous branches, metastability and interfaces, but must not be identified with the full equilibrium constrained density throughout a coexistence interval.
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