The "geometry" of thermodynamics

Question

I have a vague notion that thermodynamics is best captured in some language like differential geometry or something of the sort, but I am unfamiliar with said language. That being said, it seems to come up over and over again and, unfrortunately, my lack of knowledge hurts me here. In particular, I am studying Callen's derivation (Chapter 2) of the so-called "entropic intensive parameters" (simply the partial derivatives of the entropy with respect to its natural variables $U,V,X_k$ -- we restrict to a simple system for simplicity) and how these relate to the "energetic intensive parameters" (simply the partial derivatives of the entropy with respect to its natural variables $S,V,X_k$). One finds, says Callen, that \begin{equation} \left(\frac{\partial S}{\partial U} \right)_{X_k} \stackrel{(1)}{=} \frac{1}{\left(\frac{\partial U}{\partial S} \right)_{X_k}} = \frac{1}{T} \end{equation} and \begin{equation} \left(\frac{\partial S}{\partial X_i} \right)_{U,X_k; \, k\neq i} \stackrel{(1)}{=} \frac{-\left(\frac{\partial U}{\partial X_i} \right)_{S,X_k; \, k\neq i}}{\left(\frac{\partial U}{\partial S} \right)_{X_i,X_k; \, k\neq i}} =: \frac{-P_i}{T} \end{equation} where in both cases the (1) has used that all of the thermodynamic variables are related as some $\psi(S,U,N,X_k) = const$. That is, I believe the development above uses/assumes that system is described on such a $\psi(S,U,N,X_k) = const$ surface (so that the so-called reciprocal and reciprocity relations can be used). Is this correct? It also seems tacit in Callen's development that this $\psi$ should be such that we can always get one as a function (and not some relation?) of all the others...is this understanding correct too?

Answer 1

The surface and its differential geometry that pops up in the Caratheodory-style thermostatics and one you are alluding to is related to either of two functions $S=S_1(T,X_1,X_2,...,X_N)$ or the $S=S_2(U,X_1,X_2,...,X_N)$. For either function one can define an isentropic surface $S=const$ with the property that the whole of the available $N+1$ dimensional thermostatic space when parametrized by the configuration variables $X_1,...X_N$ and with either $T$ or $U$ can be filled by these surfaces without them having common points; the mathematicians call this a foliation. This is essentially the differential geometric content and it is not related to the partial differential derivatives in your question. Despite its elegance Caratheodory's mathematics does not really add much physical insight.