= Solution
Choose a coarse-grained <magnetization> $m(\mathbf x)$ as the <order parameter>: it vanishes in the <paramagnetism> phase and develops a spontaneous nonzero value in the <ferromagnetism> phase. In zero applied <magnetic field>, <spin> reversal sends $m\mapsto-m$, so the local <Landau free energy> contains only even powers. A minimal spatially inhomogeneous <free-energy functional> is
$$
\mathcal F[m]=F_{\mathrm{reg}}(T)+\int d^dx\left[\frac{a(T)}2m^2+\frac b4m^4+\frac c6m^6
+\frac\kappa2|\nabla m|^2-hm+\cdots\right],\qquad\kappa>0.
$$
The positive <gradient> coefficient penalizes rapid spatial variations. Near an ordinary transition take $a(T)=a_0(T-T_0)$, $a_0>0$, and positive $b$; retain the sextic term when $b$ can approach or cross zero. The source $h$ is conjugate to the <magnetization> and explicitly breaks <spin> reversal.
Determine equilibrium by minimizing this <free-energy functional>, imposing the appropriate <boundary conditions>. Its <Euler-Lagrange equation> is
$$
\boxed{-\kappa\nabla^2m+a(T)m+bm^3+cm^5-h=0.}
$$
For a uniform system at $h=0$, compare the values of all stable stationary solutions, rather than accepting every solution of this equation. The selected <global minimum> gives the equilibrium <magnetization>; derivatives of its <free energy> give <entropy>, <heat capacity> and <magnetic susceptibility>. At an ordinary transition the two ordered minima emerge continuously from zero. Spatial dependence also permits <domain walls> and an estimate of the <correlation length>.
This is a <mean-field approximation>: it selects a minimizing profile and neglects statistical fluctuations around it. The <symmetry> constrains the expansion, but its numerical coefficients are phenomenological. Close enough to a transition, <thermal fluctuations> near a <critical point> can invalidate the resulting <mean-field critical exponents>.<Statistical Field Theory by David Tong>[Chapter 1, the Landau-Ginzburg free-energy functional, spatial gradients and mean-field phase transitions]{ref}
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