On using intensive variables to describe the macrostate

Question

I am currently reading through Callen and, though Callen is very thorough, I find myself puzzled at the following question.

In Callen's treatment, the thermodynamic formalism is introduced axiomatically to describe systems which are described by $M+1$ extensive variables, $U, X_1,...,X_M$, where I have used that for any system I have ever seen or heard of, $U$ is one of the relevant extensive parameters (using information from outside of Callen, I think this must be so since the Hamiltonian is a constant of the motion trivially).

Callen then gives as a postulate that there exists some function $S$ of these $M+1$ extensive variables. Indeed, Callen says, for every system to which thermodynamics can be applied we have some configuration space of the variables $S,U,X_1,...X_M$ on which there is some surface which defines the entropy function (and where this surface has certain properties that allow, for example, $U$ to be considered as a function of the others but not in general for $X_i$ to be considered as a function of the others.

Let's now specify to simple $(U,V,N)$ systems for simplicity. At some point, Callen begins giving expressions for entropy in forms like $S = S(T,P,N)$. My question is, how do I know whether this is valid and/or actually defines a function for $S$ in general? For instance, I could imagine that there are two macrostates with the given $(T,P,N)$, and that these two macrostates could have different entropies $S$. Are "functions" (are these parametrizations) like $S = S(T,P,N)$ only valid in some neighbourhood of a given point? I'm looking for the details which seem to be omitted here. This discussion seems relevant to what I'm learning about right now; in Chapter 3.9 all manner of derivatives are defined. For example, the thermal expansion coefficient is defined as $$\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P,N},$$ but it seems to me that even defining this assumes that we can speak about $V$ as a function of $T,P,N$?

Answer 1

The most basic postulate of equilibrium thermostatics as presented by Callen is that given a system in equilibrium it is completely described by its macroscopic parameters and said description is a functional relationship between a thermal parameter called entropy $S$, conserved mechanical parameters $X_k;k=1,2,..M$ and the internal energy is given as $U=f^u(S,X_1,..X_M)$, the so-called caloric equation of state. Note below $^\#$.

In Callen's treatment, the function $U=f^u(X_0, X_1,X_2,...X_M)$ is supposed to be piece-wise smooth, continuous and continuously differentiable function, of the variables ${X_0,X_1,...X_M}$, say, $X_0=S$, $X_1=V, X_2=N, X_3=\text{electric charge}, X_4=\text{electric dipole moment},etc....$

The possibly several smooth, continuous/continuously differentiable pieces, over discrete non-overlapping domains $\mathcal D_m$ of the configuration variables ${X_0,X_1,...X_M}$ are the various phases of the system. The boundaries that separate these neighboring domains $\mathcal D_m$ are described by equilibrium equations formed by these smooth functions similar to the case of $M=2$ resulting in the usual formulation of the Clausius-Clapeyron equation.

These domains, say, $\{X_k\} \in \mathcal D_m$, are the largest neighborhoods of any point around within which the functions of the type $U=f_{m}^u(\{X_k\})$ defined and smooth, one smooth function $f_{m}^u$ for one domain $\mathcal D_m$.

In the Gibbs equation, $$U=TS+\sum_{k=1}^MY_kX_k$$ the internal energy is a function $U=f^u(S,X_1,..,X_M)$, and in this form $Y_j(X_0,\{X_k\})=\frac{\partial f^u}{\partial X_j}$, $j=0,1,..,M$, and $T=Y_0,S=X_0$. The functions $Y_k=y_k(S,X_1,X_2,..X_M)$ are the individual thermal equation of state operational in the domain in which the $U$ is defined and smooth. The caloric and the $M$ thermal equations of state together fully characterize the equilibrium, hence, quasi-static behavior of the phase in the domain where these functions are smooth.

Assuming that $U=f^u$ is 1st order homogeneous, the differential $dU$ satisfies $$dU=TdS+\sum_{k=1}^MY_kdX_k \tag{1}\label{1}$$

Within these domains of "smoothness", the so-called multivariable inverse function theorem ensures that quite freely one can interchange any of the independent variables with the dependent variable, and specifically in your question interchange $S$ with $U$, or $S$ with $T$, and also with a Legendre transform you may replace any number of the variables $X_k$ with its conjugate intensive $Y_k$.

By changing from the variable $S$ to the variable $U$ we can rewrite the differential Gibbs equation to $$dS=\frac{1}{T}dU - \sum_{k=1}^M\frac{Y_k}{T}dX_k\tag{2}\label{2}$$ In $\eqref{2}$ the intensive function ${Y_k}=\frac{\partial f^u}{\partial X_j}$ is function of the explicit variables $S,X_1,..X_M$ that will have to be replaced by the variables $U,X_1,..,X_M$ by substituting for $S$ the function $S=f^s(U,X_1,..,X_M)$.

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$^\#$ Not all "varieties" of thermostatics make the postulate that the equilibrium is completely described by its macroscopic parameters where the external variables, as are defined by Callen, can be externally controlled. One mathematically convenient way of introducing irreversibility into thermostatics is to include so-called concealed or internal variables that maybe observed but cannot be controlled externally. For example, the gain of a field effect transistor (FET) depends on its gate temperature that cannot be controlled directly, but it could be observed by some IR technique to some extent. Such temperature is an example of an internal variable.