= Statistical field theory
{wiki}
<Statistical field theory> assigns <probability distributions> to spatial field configurations, often obtained by coarse graining an interacting many-particle system. A local <order parameter> $\phi(\mathbf x)$ can encode a slowly varying <magnetization> or a density difference. In <thermal equilibrium>, a typical effective description uses
$$
Z=\int\mathcal D\phi\,\exp[-\mathcal F[\phi]/(k_BT)],
$$
where $\mathcal F$ is an effective <free energy> and the <functional measure> specifies the configurations being counted.
Minimizing $\mathcal F$ gives a <mean-field theory>, while integrating over neighbouring configurations includes <thermal fluctuations>. <Landau-Ginzburg theory> adds a gradient cost, expressing the energetic penalty for rapidly varying the <order parameter>.<Statistical Field Theory by David Tong>[§1.3.1, effective free-energy functional and partition function; §1.3.2, saddle-point interpretation of mean-field theory]{ref} The field description makes <correlation length> and <critical phenomenon> accessible without tracking each microscopic particle. It still requires a microscopic cutoff or other definition of its short-distance degrees of freedom; a continuum integral is not automatically a complete microscopic model.
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