= Landau-Ginzburg theory
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= LG theory
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Landau-Ginzburg theory promotes an <order parameter> to a spatial field and includes an energy cost for its variation. For a scalar field,
$$
F[\phi]=\int d^dx\left[\frac\kappa2|\nabla\phi|^2+\frac a2\phi^2+\frac b4\phi^4-h\phi\right].
$$
The local terms select preferred phases, while the gradient term makes abrupt spatial changes costly. Minimizing this <free-energy functional> can describe a smooth <domain wall> between two ordered regions.
Expanding around a uniform stable minimum gives a quadratic fluctuation kernel and an <Ornstein--Zernike correlation function>. The framework therefore connects an equilibrium phase diagram with measurable spatial correlations and interfaces. Minimization alone is a mean-field treatment; integrating fluctuations can change the predictions and is central to the <renormalization group>. The derivative expansion assumes fields varying slowly compared with microscopic lengths.
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