Section 3.1.2.2.2. Basic assumptions and strategy

Our aim is to determine the above displacement as a function of temperature. Landau's strategy is to determine by a variational method. One considers an arbitrary displacement d of the ion. For given temperature T and pressure p (or volume V), and specified values of the components of d, there is, in principle, a definite value for the free energy F of the system. This function is a variational free energy since it is calculated for an arbitrary displacement. The equilibrium displacement is defined as the displacement that minimizes the variational free energy F. The equilibrium free energy of the system is . Note that, strictly speaking, in the case of a given pressure, one would have to consider a variational Gibbs function () in order to determine the equilibrium of the system. We will respect the current use in the framework of Landau's theory of denoting this function F and call it a free energy, though this function might actually be a Gibbs potential.

The former strategy is not very useful as long as one does not know the form of the variational free energy as a function of the components of the displacement. The second step of Landau's theory is to show that, given general assumptions, one is able to determine simply the form of in the required range of values of the functions' arguments.

The basic assumption is that of continuity of the phase transition. It is in fact a dual assumption. On the one hand, one assumes that the equilibrium displacement has components varying continuously across the transition at . On the other hand, one assumes that F is a continuous and derivable function of , which can be expanded in the form of a Taylor expansion as function of these arguments.

Invoking the continuity leads to the observation that, on either side of , is small, and that, accordingly, one can restrict the determination of the functional form of to small values of and of . F will then be equal to the sum of the first relevant terms of a Taylor series in the preceding variables.