In Chapter 5 of his famous textbook on thermodynamics, Callen argues for the "equivalence" of the maximum entropy (Max-Ent) principle and the minimum energy (Min-En) principles. I quote from Callen first:
Entropy Maximum Principle. The equilibrium value of any unconstrained internal parameter is such as to maximize the entropy for the given value of the total internal energy.
Energy Minimum Principle. The equilibrium value of any unconstrained internal parameter is such as to minimize the energy for the given value of the total entropy.
As far as I know (though Callen never makes this explicit in what, I think, represents an error or at least an oversight) the Max-Ent principle applies if and only if the (composite) system under investigation is isolated. On physical grounds, an isolated system is always characterized by fixed energy $U$. It's with that background that I ask the following.
Callen first gives a "qualitative" proof (in one direction of the equivalence) which goes as follows:
Assume, then, that the system is in equilibrium but that the energy does not have its smallest possible value consistent with the given entropy. We could then withdraw energy from the system (in the form of work) maintaining the entropy constant, and we could thereafter return this energy to the system in the form of heat. The entropy of the system would increase ($dQ = TdS$), and the system would be restored to its original energy but with an increased entropy. This is inconsistent with the principle that the initial equilibrium state is the state of maximum entropy! Hence we are forced to conclude that the original equilibrium state must have had minimum energy consistent with the prescribed entropy.
Aside from the fact that one seems to be applying the Max-Ent principle to a non-isolated system in this "argument", I have an even bigger objection to the first line, wherein we are supposing that we are at some fixed/constrained entropy!
The "mathematical argument" seems to suffer (at least it seems to me) from that same shortcoming. See here for the very nice answer from Chemomechanics which reproduces Callen's derivation. Translating the result of that proof into words, we seem to have shown that (assuming a simple $U,V,N$ system for simplicity) if on the surface $\psi(S,U,V,N) = 0$ we are at a point such that $S$ is locally maximum at the given $U$, then that point is also locally a $U$ minimum for that given $S$ (i.e. the $S$ corresponding to the max entropy).
My problem with the above is that Callen goes on (or seems to go on) to claim that the two principles are in every way equivalent, but this seems absurd given that each principle (on my reading of things) is applicable in completely different contexts. Callen for example analyzes the diathermal piston problem using both principles, which makes no sense to me given that the composite system of two gases is either at fixed energy or at fixed entropy (though I'm not sure how one would arrange the latter) -- but not both!
To adapt a clever (if controversial) phrase, this question can be summed up as "don't the Max-Ent and Min-En principles have nonoverlapping magisteria and, if so, how does it makes sense to talk about their being equivalent?" I am hoping someone can elucidate exactly in what sense these are equivalent. There have been many questions asked on this site to that effect but none which suitably address this last point, I think.
Edit: Please consider the following. Perhaps if someone can point out the error in this deduction it will point out where my misunderstanding arises from.
Suppose I give you an isolated system at some $U_i$ and tell you that some internal constraint has been released. By Max Ent, you can then search in configuration space (as restricted to the $U_i$ constraint) for the state which maximizes $S$ on this plane. Suppose I find that this final state has $S_f>S_i$ But if I use Min En as the method as applied to the same system, how do I proceed? The initial state has some entropy $S_i$ -- do I search on this plane of $S = S_i$ for the minimum energy state (that seems to be what Min En says)? And, if so, since this minimum energy state will have $S = S_i < S_f$ (and so cannot be the same state as I just found with the maximum entropy principle), surely (though of course I have erred somewhere since all of you, Callen, and Gibbs disagree with me!) I have just shown that the minimum energy principle has led me to a different conclusion than the maximum entropy principle?