The short version of this question is:
- Is there, or could there be, a system with a phase transition where adding a small amount of heat causes a discontinuous jump in its temperature?
Below are my reasons for thinking there might be.
At a first-order phase transition there is a discontinuity in the first derivative of $\log Z(\beta)$, where $\log Z$ is essentially the free energy and $\beta=1/k_B T$ is the inverse temperature. As a consequence, its Legendre transform $S(E)$ has a segment of zero second derivative. (Here $S$ is the entropy and $E$ the expected value of the energy. The two functions are related by $S(E) = \log Z(\beta) + \beta E$.) This means that the function $E(\beta)$ has a discontinuity. These basic properties of first-order phase transitions are illustrated below:
The slope of the third plot, $d E/d\beta$, is related to the heat capacity, which becomes infinite when $\beta$ is at the critical value.
However, it seems to me that the opposite phenomenon could also happen, where the discontinuity is in the first derivative of $S(E)$, and consequently $\log Z(\beta)$ has a straight line segment, and $E(\beta)$ has a section of zero rather than infinite slope, like this:
Rather than having an infinite heat capacity for a critical value of $\beta$, such a material would have a zero heat capacity for a critical value of its energy density, meaning that at the critical point, adding a small amount of energy would cause a discontinuous change in temperature.
It seems that it wouldn't be too hard to construct a toy model that exhibits this "dual" type of phase transition. All you really need is a very high density of states at the critical energy value. (However, I have not explicitly constructed such a model yet.)
In a similar way, one could construct the dual of a continuous phase transition. Here the second derivative of $S(E)$ would diverge at the critical point, and the heat capacity would smoothly approach zero around the transition.
I have never seen anyone refer to these types of transition, but I don't know whether this is because (a) they don't happen, (b) they're not considered very interesting, or (c) I just don't know the correct term for this phenomenon. Therefore my questions are
Does this type of transition occur in physical systems? If so, does this transition type have a name, and is there a well-studied example?
If not, is there a fundamental reason why it can't happen? What assumptions are needed to prove that it can't?