= Solution
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 example
$$
\mathcal A[m]=A_{\rm reg}+\int d^Dx\left\{\frac\kappa2(\nabla m)^2+\frac r2m^2+\frac u4m^4+\frac v6m^6-hm+\cdots\right\},\qquad\kappa>0.
$$
At $h=0$, 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. \b[Spontaneous order is defined by taking the <thermodynamic limit> before removing a selecting field], for example $m_0=\lim_{h\downarrow0}\lim_{N\to\infty}\langle m\rangle$. This distinction also matters for the connected correlations used below.
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