Understanding the changing minima picture of phase transitions

Question

Fix a simple $U,V,N$ system which is in contact with a temperature and pressure reservoir, so that the minimization of $G$ is the relevant criterion for equilibrium. This question is somewhat related to this one here, though I don't think the answer is fully captured.

In his textbook, Callen gives diagrams of the following sort to describe phase transitions. We note that throughout the discussion $N$ is assumed constant; it is a tacit spectator variable.

In this particular question, I am not interested in the critical point but rather an arbitrary position on the coexistence curve; say, the middle inset. In this diagram, Callen seems to be showing that at fixed $T,P,N$, $V$ is variable! But that is of course absurd since there are only 3 DOF.

What I think perhaps is going on is that our system/substance has underlying phases which each have separate nominal fundamental equations (i.e. each phase $i$ has a $G_i(T,P,N)$), but the actual fundamental equation is the minimum over all such phases at the given $T,P,N$ ($G(T,P,N) = \min_i G_i(T,P,N)$). The phase transition happens when we cross over from one $i$ to some other $j$ in this minimum. However, this description does not comport with Callen drawing $V$ as a continuous variable. My picture seems to suggest that there should be discrete points $V_i$ (since $T,P,N$ fix $V_i$ for the given phase) corresponding to the minimums given in the inset.

Since Callen's description in terms of a continuous variable is very much different than mine, it seems I must be missing something?

Answer 1

You have hit a point where Callen's explanation of what is going on is not as clear as other parts of his textbook. In Section 9.1, he tries to make the puzzling dependence of the Gibbs free energy on the volume understandable, mentioning fluctuations. Still, there is no explicit argument, and I never found that part convincing.

However, it is possible to give a sound argument based on thermodynamics.

We know that at constant pressure and temperature (and number of molecules), the stable thermodynamic equilibrium corresponds to the absolute minimum of Gibbs' free energy with respect to every variable controlling a possible internal constraint in the system.

Now, among all the microstates of the system corresponding to the $p,T,N$ macrostate, we can find several systems whose volume is $V_1$, another number whose volume is $V_2$, and so on. We can think of each of such sets as a constrained system with fixed $p,T,N$ and a constraint variable $V$ varying over a continuous interval. Notice that such an approach through constrained systems can be easily implemented in a numerical simulation. The redundant variable $V$ should be considered the variable expressing the constraint. The actual thermodynamic equilibrium state corresponds to the minimum of Gibbs' free energy with respect to such a constraint variable.

Notice that the same interpretation applies to every thermodynamic system. This observation may help to understand why the $N$ variable of a simple system does not play any role: In a magnetic system, the Gibbs free energy is a function of temperature and external field. Again, the description of the role of the possible phases is encoded in a function $A(T,H,M)$, where $M$ is the magnetization density. The same interpretation as a function describing a constrained system applies, while the variable $N$ does not play any role for such systems.

A final comment about fluctuations: I do not find an argument based on them a convincing way to introduce a Gibbs free energy depending on the volume. However, their presence helps to make it understandable why we can think of the state at constant $p,T,N$ as corresponding to systems with different volumes.