Phase coexistence

Phase coexistence occurs when two or more thermodynamic phases have equal free energy under the same external conditions. A first-order transition crosses a coexistence locus, where the competing phases are simultaneously stable.

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 $M=q{M}_1+(1-q)M_2$. 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.

A pure thermodynamic phase is an extremal equilibrium Gibbs state rather than a probabilistic mixture of distinct macroscopic phases. On an ordered Ising branch, taking the thermodynamic limit before removing a selecting conjugate field gives a nonzero magnetization. A symmetric mixture can instead have zero mean but a nondecaying two-point contribution. The connected correlation function and correlation length describing ordinary critical fluctuations should be evaluated in a chosen pure phase when this distinction matters.

For a bulk free-energy density $f(\varphi )$, two compositions coexist across a flat interface whenThe first condition equates the chemical potentials, and the second equates the pressure. Equivalently the same straight line of slope $\mu$ is tangent to the graph of $f$ at both compositions. A stable coexistence pair uses a supporting tangent, so the resulting mixture lowers the bulk free energy.

Adding $d\varphi$ to a free-energy density adds a constant at fixed total compositional order parameter. In the common-tangent construction for phase coexistence, it shifts the tangent slope by $d$ while preserving its points of contact. Thus coexistence compositions are unchanged, whereas their chemical potential shifts. This does not assert unchanged equilibrium composition at a fixed externally imposed chemical potential.