LandauTheorie

Landau theory, named after Lev Landau, is a phenomenological framework for describing phase transitions by introducing an order parameter that measures the degree of symmetry breaking. The central idea is that near a phase transition the free energy can be written as an analytic expansion in this order parameter, reflecting the symmetries of the high-temperature phase. The coefficients depend on temperature, and the sign of the quadratic term signals the onset of order. For a system with symmetry φ → −φ, odd powers of φ are absent.

A typical form is F(φ) = F0 + a(T) φ^2 + b φ^4 + c φ^6 + … with a(T) crossing zero at Tc, for example a(T) = a0 (T − Tc). Minimizing F yields the stable phase: for T > Tc, a > 0 and φ = 0; for T < Tc, a < 0 and a nonzero φ minimizes F. If b > 0, the transition is second order with φ ∝ sqrt(Tc − T). If higher-order terms or a negative b appear, first-order behavior can arise.

Landau theory provides a framework for classifying phase transitions by symmetry and order-parameter behavior, and it yields mean-field critical exponents (for simple cases β = 1/2, γ = 1, α = 0), neglecting fluctuations. Extensions include Landau-Ginzburg theory, which adds a gradient term (∇φ)^2 to describe spatial variations and leads to a characteristic coherence length; this forms the basis of the theory of superconductivity, as well as other continuous transitions.

Limitations include the neglect of fluctuations, which are essential near critical points in low dimensions. Renormalization-group analysis shows deviations from mean-field predictions in such cases. Landau theory remains a foundational, widely used tool for understanding phase transitions and symmetry breaking.