Derivation: Maximum entropy implies minimum energy

Question

Callen claims in his book (chapter 5 in my copy at least) that the condition of minimum energy for fixed entropy is exactly equivalent to the condition of maximum entropy for fixed energy. I have seen this claim restated multiple times, and there are a bunch of answers in this site, like this one, which directly cite Callen's claim and derviation.

I have some problems with the way he derives this equivalency. Callen starts his derivation by stating that in the absence of internal constraints, $dS = 0$ and $d^2S < 0$ since entropy is maximum at this point. He then states that this implies that $$ \frac{\partial^2 S}{\partial X^2} < 0 $$ for all $X$. I understand he is roughly referring to the hessian of $S$, however, if this is the case, what he should probably say is that the Hessian matrix of the entropy $\mathcal{H}(S)$ is negative definite at that point, which means that $v^{T}\mathcal{H}(S) v < 0~\forall v$.

Callen finishes his derivation by stating that, since he can prove from this that

$$ \frac{\partial^2 U}{\partial X^2} > 0 $$ energy is a minimum at this point.

I don't think this implies the existence of a minimum, which, similarly to what was stated before requires the Hessian to be positive definite, which is stronger than all $\partial^2_X U$ being positive.

My question is twofold:

1) Does the partials $\partial^2_X U$ being positive imply that there is a minimum as Callen states?

2) If it doesn't then his derivation that the internal energy has a minimum at that point doesn't hold (as shown here, in wikipedia for instance), what would be a correct derivation of this fact?

Answer 1

We need to be careful about what "the partials $\partial^2_X U$ being positive implies a minimum" means. If we mean that we have a given set of coordinates $X_i$ and that $\partial^2 U / \partial X_i^2 > 0$ for all $i$, then the claim is false. For example, a Hessian matrix like

$$\begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix}$$

has all $\partial^2 U / \partial X_i^2 > 0$ but is not positive definite.

But if we mean that if $X$ can be any coordinate at all, chosen from any coordinate system we want, then the claim is true: taking a unit vector $v$ in the direction of the coordinate $X$ (keeping all other coordinates in our coordinate system fixed), then

$$v^T \mathcal{H}(U) v = \frac{\partial^2 U}{\partial X^2},$$

so that if all second derivatives are positive, the Hessian is positive definite and there is a minimum.

Thankfully, Callen's proof works for any coordinate system and any coordinate, so the claimed principle is true in thermodynamics.