In the Landau theory of phase transitions, one typically considers a "free energy" $F$ as a function of the temperature $T$ and the "order parameter" $\psi$: $F(T, \psi)$.
For the sake of clarity, let's consider the liquid-gas transition, in which the order parameter is usually taken to be the difference of densities between the ordered phase and the disordered phase. In this case, it looks that the order parameter $\psi$ is closely related to the the volume $V$, a global variable of state.
What is the correct interpretation of $\psi$ as a thermodynamic variable?
I consider several logical possibilities, but all interpretations look troublesome to me:
$\psi$ is a variable of state replacing the volume. In this case I have a few related concerns:
Is the "free energy" $F$ the Helmholtz or the Gibbs free energy?
- If $F$ is the Gibbs free energy, $F$ should depend on the temperature $T$ and the pressure $p$ (the conjugate variable of the volume), so it shouldn't depend on $\psi$.
- If it is the Helmholtz free energy, it's ok that it depends on $\psi$, but the experiments are usually carried out at constant pressure, not constant volume.
Usually, state variables are something one can select at will [for instance, think of the phase diagram $(p, T)$]. However, the order parameter $\psi$ is not selected at will, but it is selected by the system as a result of the minimization process according to the Landau theory.
$\psi$ is yet another variable of state in addition to volume and temperature.
Why the pressure is not usually taken into account as a variable of state, and only the temperature is considered?
What is the conjugate variable to $\psi$?
Can really $\psi$ and $V$ be taken as independent?
$\psi$ is not a variable of state, is something else.
- What else can $\psi$ be from the thermodynamic point of view?
- Same question as above, why is pressure not usually taken into account in Landau theory?