Statistical field theory

Statistical field theory assigns probability distributions to spatial field configurations, often obtained by coarse graining an interacting many-particle system. A local order parameter $\varphi (\mathbf{x})$ can encode a slowly varying magnetization or a density difference. In thermal equilibrium, a typical effective description useswhere $\mathcal{F}$ 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.^[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.