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A free-energy functional assigns a free energy to an entire spatial order parameter configuration rather than only to one uniform value. In Landau-Ginzburg theory, a local derivative expansion gives $\mathcal{A}[m]=\int d^{D}x\{\kappa (\nabla m{)}^2/2+V(m)\}$, with appropriate boundary terms and regular backgrounds. The Landau approximation minimizes this functional; a fluctuating statistical field theory instead integrates its Boltzmann weight over configurations. These procedures need not give identical critical exponents.

Landau-Ginzburg theory promotes an order parameter to a spatial field and includes an energy cost for its variation. For a scalar field,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.

Landau-Ginzburg theory describes a slowly varying order parameter by a local free-energy functional consistent with the microscopic symmetries. For a representative scalar system with spin-inversion symmetry, take the dimensionless functionalThe gradient term penalizes spatial variation, while the even local potential implements the zero-field symmetry $\varphi \mapsto -\varphi$. The positive quartic coefficient stabilizes a continuous transition. A negative quartic coefficient requires higher powers and can instead produce a first-order phase transition. A vector order parameter gives the corresponding rotationally invariant model with powers of $|\mathbf{\varphi }{|}^2$.

In the mean-field approximation, replace the field by a constant $M$ and minimize the local potential. Its equation of state and stable zero-field solutions areBelow the transition these two minima exhibit spontaneous symmetry breaking. Differentiating the equation of state gives $\chi =(r+3u{M}^2{)}^{-1}$: it is $1/r$ above the transition and $1/(2|r|)$ within either ordered phase. At $r=0$, $M=(h/u{)}^{1/3}$. The minimized zero-field free-energy density is zero above and $-r^2/(4u)$ below, apart from an analytic background; its second temperature derivative has a finite jump. Consequently

To include fluctuations, write $\varphi =M+\psi$. The quadratic part isA Gaussian functional integral therefore gives $⟨\psi (q)\psi (-q)⟩=(c{q}^2+r+3u{M}^2{)}^{-1}$, with the momentum-volume delta function understood. The Ornstein--Zernike correlation function has correlation length $\xi =\sqrt{c/(r+3u{M}^2)}$. Thus ${\nu }_{\mathrm{MF}}=1/2$, and at criticality the Gaussian covariance is proportional to $q^{-2}$, giving ${\eta }_{\mathrm{MF}}=0$. Continuous-symmetry ordered phases additionally have transverse Goldstone modes, for which the restoring mass vanishes.

The Gaussian integration contributes $\frac{1}{2}\int d^{d}q(2\pi {)}^{-d}\log (c{q}^2+r)$ to the disordered-phase free-energy density. Differentiating twice with respect to $r$ gives a critical contribution proportional to $\int d^{d}q(c{q}^2+r{)}^{-2}$; for $d<4$ its singular part grows as $r^{(d-4)/2}$ and at four dimensions it is logarithmic. Interacting fluctuations thus become important close to the critical point below four dimensions. The Ginzburg criterion tests this failure of the mean-field approximation; the renormalization group then explains modified exponents and universality. Mean-field theory finds the competing minima, while fluctuations determine whether its predicted critical behavior survives at long distance.

An order parameter distinguishes equilibrium thermodynamic phases. For a scalar magnetic system, a slowly varying coarse-grained scalar field $\varphi (x)$ represents local magnetization, and its uniform expectation $M=⟨\varphi ⟩$ is the order parameter. Under spin inversion symmetry, $\varphi \mapsto -\varphi$ and $h\mapsto -h$. The disordered phase has $M=0$ at $h=0$, while a pure ordered phase has $M\ne 0$. In a finite symmetric system the average still vanishes at $h=0$; spontaneous symmetry breaking means taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$ to select a pure thermodynamic phase.

Landau-Ginzburg theory assigns a local free-energy functionalSpin inversion symmetry forbids odd powers at zero conjugate field; locality and slow variation motivate the gradient expansion. The gradient energy penalizes interfaces and determines long-wavelength response. Integrating $e^{-\mathcal{F}/(k_{B}T)}$ over fields includes fluctuations, whereas the Landau approximation replaces that integral by its stable stationary configuration. For a uniform phase this reduces to minimizing $V(M)$. If $f$ is the equilibrium free-energy density, then $M=-\partial f/\partial h$, so the order parameter is also the thermodynamic response to its conjugate field.

An order parameter distinguishes phases by a macroscopic quantity that changes at the transition. For an Ising model ferromagnet it is the magnetization per Ising spin, $M=N^{-1}{\sum }_n⟨{\sigma }_n⟩$. At zero magnetic field the statistical Hamiltonian is invariant under spin inversion symmetry. The disordered phase preserves this symmetry, whereas a selected ordered phase has $M\ne 0$ and a spin-reversed partner with $-M$. At finite volume with symmetric boundary conditions the expected value is zero even below the transition; spontaneous magnetization means taking the thermodynamic limit before sending a symmetry-breaking field to zero.

Landau-Ginzburg theory introduces a coarse-grained local order parameter $\varphi (x)$ and a symmetry-constrained free-energy functional. For a short-range system with scalar spin inversion symmetry, a useful expansion isThe coefficients vary smoothly with microscopic control parameters at the level of the Landau approximation; normally $r=A(T-T_c)$ with $A>0$. The gradient term penalizes spatial variation. The magnetic field is conjugate to the order parameter, and odd powers at $h=0$ are excluded by spin inversion symmetry. With $u>0$, the sextic term may be omitted locally; with $u<0$, a stabilizing positive $v$ is essential. The equilibrium uniform order parameter minimizes the local potential $V(M)$, givingThis equation of state follows from minimization, rather than being imposed as an independent assumption. Vector or tensor order parameters and their allowed invariants describe other symmetry classes.

An order parameter distinguishes thermodynamic phases and transforms under the symmetry that may be broken. For a scalar ferromagnet it is the magnetization per site; a disordered zero-field phase has $m=0$, while ordered pure thermodynamic phases have $m=\pm m_0$. In a fluid one can instead use the density measured relative to its critical value; the field conjugate to it is then related to the chemical potential rather than literally a magnetic field. A nonzero value in the presence of an explicit conjugate field is not by itself evidence of a spontaneous transition.

The LG theory describes a slowly varying local order parameter by a symmetry-constrained free-energy functional, for exampleAt $h=0$, a ${\mathbb{Z}}_2$ symmetry excludes odd powers. The coefficients are assumed analytic functions of the controls near the transition, and the expansion is stabilized by a positive highest retained even coefficient. The Landau approximation obtains equilibrium by minimizing this functional, neglecting long-wavelength fluctuation corrections. Its nonconvex local potential describes distinct candidate phases and mean-field metastability; the exact thermodynamic potential is convexified when macroscopic mixtures are admitted.

In a finite symmetric system, the zero-field expectation of $m$ can vanish even below the transition because both ordered orientations are sampled. Spontaneous order is defined by taking the thermodynamic limit before removing a selecting field, for example $m_0={\lim }_{h\downarrow 0}{\lim }_{N\to \infty }⟨m⟩$. This distinction also matters for the connected correlations used below.

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 isThe 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 isFor 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.^[ref]