= Solution
An <order parameter> distinguishes thermodynamic phases. For a ferromagnet it is the <magnetization>; for a fluid near its <liquid-gas critical point> it can be the density relative to its critical value. In a system with a scalar <order parameter>, introduce a slowly varying real <scalar field> $\phi(x)$ whose spatial average represents that <order parameter>. The <Landau-Ginzburg theory> organizes the effective <free energy> as a local expansion consistent with the symmetry:
$$
\mathcal F[\phi]=\int d^dx\left\{\frac K2(\nabla\phi)^2+\frac r2\phi^2+\frac u4\phi^4+\frac v6\phi^6-H\phi\right\}.
$$
Here $K>0$ is the stiffness in the <gradient energy>, and $H$ is the <field conjugate to an order parameter>. The even powers assume a symmetry $\phi\mapsto-\phi$ at $H=0$. Without that symmetry further terms, including a cubic term, may be allowed. The coefficients are smooth functions of temperature and other controls; near a mean-field transition one usually writes $r=a(T-T_0)$ with $a>0$. A positive highest retained even coefficient makes the local potential bounded below.
The <partition function> is obtained by weighting configurations with $\exp[-\mathcal F/(k_BT)]$. The <Landau approximation> replaces this functional average by minimization, taking a uniform $\phi=M$ in a homogeneous phase. In a symmetric finite system the zero-field average can vanish even below the transition because it averages both signs of $M$. The spontaneous <order parameter> selects a <pure thermodynamic phase> by taking the <thermodynamic limit> before removing the conjugate field:
$$
M=\lim_{H\downarrow0}\lim_{V\to\infty}
\left\langle V^{-1}\int_V\phi(x)\,d^dx\right\rangle_H.
$$
Thus \b[the order parameter measures the ordering of a selected thermodynamic phase], rather than the average over all symmetry-related phases.
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