Section 16. Free energy and convex analysis

On the differentiability of thermodynamic potentials

 The most fundamental (or basic) thermodynamic potential is internal energy. Other thermodynamic potentials are derived from this. Since the most detailed description of a thermodynamic state is in terms of thermodynamic coordinates, internal energy is the most detailed thermodynamic potential (however, do not forget that if we know one potential the rest may be recovered uniquely; see Section 36 for the meaning of `the most detailed’).

 Since we can always write down the Gibbs relation, it is clear that internal energy is not only continuous but once differentiable. The intensive quantities are continuous. Therefore, internal energy is a C1 function (continuously differentiable function) of the thermodynamic coordinate variables.

 Why are intensive variables continuous? From the convexity of  E , we can conclude that intensive variables are almost everywhere continuous. 

 Let  f be a finite convex function on an interval  I . Then,  f is continuously differentiable except at most countably 

 many points in I. (Rockafellar, Theorem 25.3 p244)

I believe thermodynamics cannot conclude anything stronger without further (extra-thermodynamic) input. 

 When can an intensive quantity discontinuous as a function of thermodynamic variables? First, notice that there is no discontinuous example known. If this could happen, it must be at a phase transition point. M E Fisher discussed the possibility statistical-mechanically (see around p43 of his ``Phase and phase diagram’’ in Proceeding of the Gibbs symposium, Yale University May 15-17, 1989 (AMS and AIP, 1989)), saying ``Let us now venture out from the world as it is into the world as it might have been.’‘ As you can read there, if the contribution of multibody interactions is sufficiently contained, then discontinuity in intensive variables is not realized.

 In the usual thermodynamics,  it is assume that intensive quantities are continuous with respect to thermodynamic coordinates .

 What can we say about the Helmholtz free energy  A ? It is derived from  E by a Legendre transformation with respect to entropy  S , so 

A is continuous for all it natural independent variables, but its differentiability with respect to temperature is not guaranteed, although as guaranteed by convex analysis, its left and right derivatives are well-defined.

 Thus, as is clear from this example, the derivatives or thermodynamic potentials can be discontinuous.

 What is we look at internal energy as a function of other variables than the thermodynamic coordinates? For example, if entropy is known as a function of  V and  T , we could consider  E as a function of  T and  V . In this case  S may not be continuous, so even though  E is a C1 function of  S , the continuity of  E is not guaranteed.