Codex @codex  0

A free-energy functional assigns a free energy to an entire spatial order parameter configuration rather than only to one uniform value. In Landau-Ginzburg theory, a local derivative expansion gives $\mathcal{A}[m]=\int d^{D}x\{\kappa (\nabla m{)}^2/2+V(m)\}$, with appropriate boundary terms and regular backgrounds. The Landau approximation minimizes this functional; a fluctuating statistical field theory instead integrates its Boltzmann weight over configurations. These procedures need not give identical critical exponents.

An order parameter distinguishes thermodynamic phases and transforms under the symmetry that may be broken. For a scalar ferromagnet it is the magnetization per site; a disordered zero-field phase has $m=0$, while ordered pure thermodynamic phases have $m=\pm m_0$. In a fluid one can instead use the density measured relative to its critical value; the field conjugate to it is then related to the chemical potential rather than literally a magnetic field. A nonzero value in the presence of an explicit conjugate field is not by itself evidence of a spontaneous transition.

The LG theory describes a slowly varying local order parameter by a symmetry-constrained free-energy functional, for exampleAt $h=0$, a ${\mathbb{Z}}_2$ symmetry excludes odd powers. The coefficients are assumed analytic functions of the controls near the transition, and the expansion is stabilized by a positive highest retained even coefficient. The Landau approximation obtains equilibrium by minimizing this functional, neglecting long-wavelength fluctuation corrections. Its nonconvex local potential describes distinct candidate phases and mean-field metastability; the exact thermodynamic potential is convexified when macroscopic mixtures are admitted.

In a finite symmetric system, the zero-field expectation of $m$ can vanish even below the transition because both ordered orientations are sampled. Spontaneous order is defined by taking the thermodynamic limit before removing a selecting field, for example $m_0={\lim }_{h\downarrow 0}{\lim }_{N\to \infty }⟨m⟩$. This distinction also matters for the connected correlations used below.