Landau free energy and first-order transitions

For a scalar order parameter with reversal symmetry and no conjugate field, the local expansion contains only even powers. A field adds a linear term that explicitly breaks that symmetry. Analyticity of this coarse-grained local function near small order parameter is an assumption; the final thermodynamic free energy can become nonanalytic after minimization and the infinite-system limit. One must not assume the exact critical free energy itself has an ordinary Taylor series everywhere.

A positive quartic coefficient gives the familiar continuous mean-field onset when the quadratic coefficient changes sign. A negative quartic term requires a positive higher-order stabilizer, for which the sextic term is the simplest choice. This truncated polynomial is useful only over amplitudes where omitted terms remain small or where it is explicitly adopted as an illustrative model. Symmetry alone does not guarantee a continuous transition, fix a critical temperature, or determine the numerical coefficients.

At zero field, the disordered stationary point is $m=0$. Nonzero stationary points obey a quadratic equation for $q=m^2$. For negative u, a pair of ordered minima can coexist locally with the central minimum while the quadratic coefficient is still positive. The first-order coexistence condition requires the nonzero stationary point to have the same free energy as the origin, not merely that the two states both exist.

Combining stationarity with equality of free energies gives a finite jump in squared order parameter and a positive coexistence value of r. In a sample with a conserved spatial average, the equilibrium state between coexisting values may separate into domains, adding interfacial costs at finite size. The resulting equilibrium free energy respects convexity in its proper constrained variable. The nonconvex local polynomial should therefore be read as a coarse-grained landscape rather than the fully convex thermodynamic answer for arbitrary constrained mixtures.

A local minimum with higher free energy is metastable. Reaching the equilibrium phase requires an appropriately sized fluctuation that creates a nucleus and overcomes an interfacial barrier. Its lifetime depends on temperature, sample volume, defects, and dynamics, none of which is determined by uniform minimization alone. Sweeping a control parameter too rapidly can therefore follow different metastable branches and produce hysteresis even though equilibrium comparison selects a unique coexistence condition.

A spinodal solves stationarity and vanishing curvature together: a minimum merges with an intervening maximum. The central minimum loses curvature at r=0, while the ordered extrema first appear at the discriminant threshold above coexistence. In short-range finite-dimensional systems, nucleation can end metastable persistence before this ideal endpoint, and a sharp thermodynamic spinodal need not exist. A smooth experimental fit with a negative quartic coefficient is consequently a model inference requiring checks, not a stand-alone proof of a phase boundary.