# Understanding the changing minima picture of phase transitions

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?

• You again ask a thing I need to go back and dig out my Callen to check. But in the meantime, I think, because pV=NkT, fixed p and T, then V is a surrogate for N, and any one of them extensive variables is the last DoF. i.e. N is not kept constant in this picture. In fact, it cannot even be v=V/N that is being meant. Now, at fixed p and T, the inserts seem to say that the analytical form of G varies like that when V changes. I think there will be Maxwell's construction for phase change, flattening the bottom as a mixture of phases. I'll check this when I get back home. Commented Jul 25, 2023 at 4:03

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.

• Thank you for this answer as always Giorgio. I’ve thought about it over the last day but am not sure I find myself fully satisfied. I understand what you are saying in general about internal constraints, but I do not believe that $V$ is a variable expressing an internal constraint here. It is not the volume of some simple subsystem, but the volume of the overall system here right? As I think about it more, I find myself more and more convinced that Callen is being sloppy in letting $V$ be a continuous variable here. I think it’s likely he is doing so in the sense that there are…
– EE18
Commented Jul 27, 2023 at 2:24
• …nonequilibrium states at this $T,P,N$ which are such as to have different $V$. The sloppiness is in assigning a value of $G$ to such no equilibrium states. The only points which should have values assigned are the two minima, with the volumes corresponding to the two different phases. I think the interpretation I’ve given above comports with Callen’s discussion about fluctuations (nonequilibrium) in Ch 9.1 — what do you think?
– EE18
Commented Jul 27, 2023 at 2:27
• @EE18 I think this is one of the few cases where thinking in terms of Statistical Mechanics may help understand a purely Thermodynamic problem. It allows us to understand the meaning of a system at fixed $P$ and $V$ (or $H$ and $M$): in a numerical simulation at fixed $P$, we generate configurations with different $V$ (the fluctuations). If we collect all the configurations with the same $V$ in different classes, we have many systems corresponding to the same $P$ and different $V$. We get a set of values from the corresponding partition functions we can consider a sampling of $G(T,P,V)$.... Commented Jul 27, 2023 at 3:44
• @EE18 ... of course these "constrained states" are not equilibrium states, then the existence as macroscopic states can only be understood in terms of constraints. The correct macroscopic equilibrium $V$ is obtained by looking for the volume minimizing such function values. Notice that this is precisely what we do in Mean Field approximations. Commented Jul 27, 2023 at 3:55
• I guess the confusing part for me is how we can even assign a $G$ to these nonequilibrium states at all? Certainly we can’t within the framework of classical thermodynamics, right?
– EE18
Commented Jul 27, 2023 at 4:52