Landau free-energy functional

A Landau free-energy functional is a phenomenological model for equilibrium behavior expressed in terms of an order parameter (see order parameter ). It captures phase transitions by encoding symmetry and stability constraints in a low-order expansion.

Landau theory for a uniform order parameter

For a scalar order parameter $m$ (e.g., Ising magnetization), the Landau free-energy density near a transition is often written as

• Symmetry: for an Ising-type $\mathbb{Z}_2$ symmetry at $h=0$, only even powers of $m$ appear.
• Temperature dependence: one commonly models $a=a_0\,(T-T_c)$ (or $a$ changing sign at criticality).

Minimizers and spontaneous magnetization

At $h=0$:

• If $a>0$, the unique minimizer is $m=0$ (disordered phase).
• If $a<0$, the minimizers are $m=\pm\sqrt{-a/b}$ (ordered phase), corresponding to symmetry breaking and nonzero magnetization; see spontaneous magnetization and spontaneous symmetry breaking .

Multiple local minima encode metastability (see metastable states ).

Landau–Ginzburg functional for spatially varying order parameter

To model interfaces and spatial fluctuations, one introduces an order-parameter field $\phi(x)$ and writes (schematically) $$ \mathcal{F}[\phi] ={} \int d^d x, \left( \frac{\kappa}{2}|\nabla \phi(x)|^2

• \frac{a}{2}\phi(x)^2
• \frac{b}{4}\phi(x)^4

• h,\phi(x) \right), \qquad b>0,; \kappa>0. $$
• The gradient term penalizes rapid spatial variation and leads to finite interface costs, connecting to surface tension and interfaces .
• The local polynomial term reproduces the uniform Landau picture when $\phi$ is constant.

Correlation length from the quadratic approximation

In the disordered phase ($a>0$), the Gaussian (quadratic) approximation yields a characteristic length scale

which can be interpreted as a correlation length; compare correlation length and the Ornstein–Zernike description Ornstein–Zernike form .

Relationship to mean-field and large deviations

• The uniform Landau function $f(m)$ is the natural outcome of a mean-field approximation when written as an effective free energy in terms of $m$; see mean-field approximation .
• In models where $m$ satisfies an LDP, $f(m)$ often matches (up to constants) the magnetization rate function or its variational representation; see rate function for magnetization .

Landau functionals provide a bridge between microscopic equilibrium theory and macroscopic thermodynamic stability; compare thermodynamic stability .