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The conserved number operator isSubstitution of $\Psi =\sqrt{\rho }{e}^{-i\theta }$ into the Landau-Ginzburg theory givesThe imaginary term is a total derivative, soThus $\rho$ is the number density and is the canonical momentum conjugate to $\theta$. The canonical commutation relation is $[\theta (\mathbf{x}),\rho (\mathbf{y})]=i{\delta }^{(d)}(\mathbf{x}-\mathbf{y})$. Consequently, for ${\theta }_0=\int d^{d}x\theta$ in volume $V$,Equivalently, the averaged phase $\bar{\theta }={\theta }_0/V$ obeys $\Delta \bar{\theta }\Delta N\ge 1/2$. This number-phase conjugacy means that a state of sharp $N$ has no sharp phase, whereas a phase-selected state exhibiting spontaneous symmetry breaking must superpose different number sectors. Such sectors become effectively degenerate in the thermodynamic limit.

The thermodynamic order parameter is the conserved scalar composition difference $\varphi$ of the binary fluid mixture. The momentum density $\rho \mathbf{v}$ is an additional conserved hydrodynamic mode; it is not a symmetry-breaking order parameter, but it must be retained because momentum relaxes only through spatial transport.

Composition conservation gives a continuity equation,The current $\mathbf{J}=-M\nabla \mu$ is the leading isotropic, dissipative constitutive law: it drives material down gradients of the chemical potential. For the Landau-Ginzburg theorythe functional derivative isThese statements give the advective Cahn--Hilliard equation.

Constant mass density and incompressible flow require $\nabla \cdot \mathbf{v}=0$. Momentum conservation gives the Navier-Stokes equation: material acceleration equals the sum of the Newtonian viscous force $\eta {\nabla }^2\mathbf{v}$, the pressure force $-\nabla P$, and the Korteweg force $-\varphi \nabla \mu$. The pressure is the Lagrange multiplier enforcing incompressibility. Together these equations are Model H dynamics; isothermality removes the need for a separate energy equation.

Landau-Ginzburg theory assumes a local gradient expansion for a slowly varying order parameter. At zero field the microscopic spin-flip symmetry sends $m\mapsto -m$, so only even powers occur. Rotational symmetry makes the leading derivative term $(\nabla m{)}^2$, while locality and analyticity near the transition organize higher powers and derivatives. Stability requires a positive gradient coefficient and, at an ordinary continuous transition, ${\alpha }_4(T_c)>0$; the coefficient ${\alpha }_2(T)$ changes sign at $T_c$.

A nonrelativistic Landau-Ginzburg theory for a complex order parameter with global $U(1)$ symmetry isIn the broken phase $r<0$, write $\psi =\sqrt{n}{e}^{i\theta }$. The first term isup to the sign convention for the phase. Thus the momentum conjugate to $\theta$ is $-n$: this is number-phase conjugacy. Canonical quantization gives the corresponding local commutator, while the spatial integral relates the global phase to total particle number.

Consequently a state with a sharply selected phase, as used to describe spontaneous symmetry breaking, cannot be an eigenstate of particle number. It is a coherent superposition of charge sectors with number fluctuations. Conversely, an exact finite-volume number eigenstate has no definite phase; the phase-selected broken-symmetry states emerge in the thermodynamic limit.

For a binary fluid mixture, the compositional order parameter $\varphi (\mathbf{r})$ may be taken as the local concentration difference between the two species. In a closed systemis fixed by the total amount of each species. This is the sense in which $\varphi$ is conserved; locally it changes through a current and obeys a continuity equation.

For a symmetric mixture, the Landau-Ginzburg theory free energy isA term $h{\int }_V\varphi ,d\mathbf{r}$ is constant on the allowed configurations and therefore changes neither equilibrium probabilities nor the dynamics. Equivalently, it merely shifts the chemical potential by a spatial constant, whose gradient vanishes.

Fourier expansion in volume $V$ diagonalizes the quadratic Landau-Ginzburg theory:whereEach independent real mode contributes a Gaussian integral proportional to $(T/K{)}^{1/2}$. Taking $-T\log Z$, dividing by $V$, and replacing the sum by an integral gives, up to field-independent conventions,

For a Landau-Ginzburg theory with positive gradient stiffness $\kappa$, $r<0$ and quartic coupling $u>0$, the degenerate minima are $\pm M_0$, $M_0^2=|r|/u$. A planar domain wall solves $\kappa {\varphi }^{\prime \prime }=r\varphi +u{\varphi }^3$. Multiplying by ${\varphi }^{\prime }$ and using the limiting minima gives $\kappa ({\varphi }^{\prime }{)}^2/2=u({\varphi }^2-M_0^2{)}^2/4$. Separating this equation yields the displayed profile, and integrating its excess energy gives the positive surface tension $2\sqrt{2\kappa }|r{|}^{3/2}/(3u)$. Its area cost is subextensive compared with bulk volume, permitting phase-separated common-tangent construction states when the average order parameter is constrained.

A statistical Hamiltonian assigns an energy to a microscopic configuration. Its Boltzmann weight defines the canonical ensemble, and summing or integrating that weight gives a partition function. A reduced Hamiltonian absorbs inverse temperature into its coefficients, so the weight is $e^{-H_{\mathrm{reduced}}}$. An Ising model uses discrete Ising spin configurations, while Landau-Ginzburg theory uses a coarse scalar field.