Landau theory offers a powerful framework for understanding phase transitions in various systems. It introduces the concept of an order parameter to describe the transition between ordered and disordered states, allowing us to analyze the behavior of materials near critical points.

By expanding the free energy in terms of the order parameter, Landau theory provides insights into both continuous and discontinuous phase transitions. This approach helps us predict and explain phenomena like ferromagnetism, superconductivity, and liquid-gas transitions in a unified manner.

Landau Theory of Phase Transitions

Fundamentals of Landau theory

• Phenomenological approach describes phase transitions based on the concept of an order parameter characterizing the transition
• Assumes free energy can be expanded in powers of the order parameter near the critical point when the order parameter is small
• Assumes free energy is analytic and can be expanded in a Taylor series with the system in thermodynamic equilibrium
• Applicable to continuous (second-order) phase transitions (ferromagnetic and superconducting transitions)

Construction of Landau free energy

• Landau free energy functional $F$ depends on the order parameter $\phi$
  • $F(\phi) = F_0 + \alpha \phi^2 + \beta \phi^4 + \gamma (\nabla \phi)^2 + ...$
  • $F_0$ free energy of the disordered phase
  • $\alpha$, $\beta$, and $\gamma$ coefficients depend on temperature and other system parameters
• Second-order phase transition free energy expansion includes even powers of $\phi$
  • $F(\phi) = F_0 + \alpha \phi^2 + \beta \phi^4$
  • $\alpha$ changes sign at the critical temperature $T_c$, while $\beta > 0$
• First-order phase transition free energy expansion includes odd powers of $\phi$
  • $F(\phi) = F_0 + \alpha \phi^2 + \beta \phi^4 + \gamma \phi^6$
  • $\alpha$ and $\beta$ change signs at different temperatures, leading to a discontinuous transition

Order parameters in phase transitions

• Order parameter $\phi$ physical quantity distinguishes between ordered and disordered phases
  • Zero in the disordered phase and non-zero in the ordered phase
• Examples of order parameters for different systems
  • Magnetization (ferromagnetic transitions)
  • Superconducting gap (superconducting transitions)
  • Density difference between liquid and gas (liquid-gas transitions)
• Equilibrium value of the order parameter minimizes the Landau free energy
  • Found by solving $\frac{\partial F}{\partial \phi} = 0$
• Behavior of the order parameter near the critical point characterizes the nature of the phase transition
  1. Continuous change for second-order transitions
  2. Discontinuous jump for first-order transitions

Stability analysis near critical points

• Stability of phases determined by the sign of the second derivative of the free energy
  • $\frac{\partial^2 F}{\partial \phi^2} > 0$ indicates a stable phase
  • $\frac{\partial^2 F}{\partial \phi^2} < 0$ indicates an unstable phase
• Second-order transition disordered phase becomes unstable below $T_c$ and the ordered phase emerges as the stable phase
• First-order transition both phases can coexist and be metastable near the transition temperature
  • Metastable phase corresponds to a local minimum of the free energy
  • Stable phase corresponds to the global minimum of the free energy
• Metastable phase can persist until a spinodal point is reached
  1. At the spinodal point, the metastable phase becomes unstable
  2. The system undergoes a rapid transition to the stable phase