Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 46 / 1 / c

Let $A(M)=V(M)+hM$ be the zero-field constrained order-parameter free energy. At coexistence in a conjugate field $h_c$, two global minima $M_1<M_2$ obeyThus the same line of slope $h_c$ is tangent to $A$ at both endpoints: the common-tangent construction for phase coexistence. Equivalently,This is the Maxwell equal-area construction on the equation-of-state curve $h=A^{\prime }(M)$: the signed areas on either side of the horizontal coexistence line cancel. Equal slope alone finds stationary phases; the integral additionally enforces equal free energy.

If the mean order parameter is constrained between the endpoints, the system can separate into two macroscopic phases. With $\lambda =(M-M_1)/(M_2-M_1)$, their bulk free-energy density is $(1-\lambda )A(M_1)+\lambda A(M_2)$. The interface contributes subextensively in the thermodynamic limit. The resulting constrained order-parameter free energy is the convex envelope of $A$, with a straight coexistence segment. This explains why a nonconvex uniform Landau free energy must be convexified when describing stable constrained equilibrium, and why a spinodal point is not the Maxwell construction boundary.