Landau Description of Phase Transitions

The order parameter can be a very complicated quantity (as we will see later). First we suppose that the order parameter is a scalar and the free energy is insensitive to the sign of S. In this case, near to the free energy can be expressed as  

The thermodynamically stable state S of the system is foimd by minimizing F with respect to S. The stability conditions are  

A simple calculation then shows that for T T, S = 0 is the stable state, whereas for T T, 0 values are possible, with the temperature depen- 

Just as the example of the soap film geometries showed, not all transitions are of second order, but the majorities are of first order. To describe phenomenologically a first-order transition, we need to incorporate a third-order term in the Landau free energy. 

Plot of the free energy at the function of order parameter S corresponding to (3.3) at different temperatures with respect to the I-N phase transition T. The positions of the minima determine the equilibrium order parameter S . The temperature dependence of S is plotted in the insert. 

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