# Does this type of phase transition exist?

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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?

This post imported from StackExchange Physics at 2014-04-08 05:10 (UCT), posted by SE-user Nathaniel
There are quite general results guaranteeing strict convexity of the pressure in $\beta$ (and any other parameters appearing linearly in the Hamiltonian), see projecteuclid.org/euclid.cmp/1103857626 (and more recent works citing the latter). Of course, there are known counterexamples, when their assumptions are violated, see for example link.springer.com/article/10.1007/BF01877543.

This post imported from StackExchange Physics at 2014-04-08 05:10 (UCT), posted by SE-user Yvan Velenik
@YvanVelenik thanks, I'll check those out. At first thought it seems that pressure isn't really relevant, since my question was mostly concerned with the canonical rather than grand canonical ensemble. But on the other hand I'm not sure if they're using "pressure" to mean the energy density, rather than $\partial E/\partial V$. Do you have any insight about that?

This post imported from StackExchange Physics at 2014-04-08 05:10 (UCT), posted by SE-user Nathaniel
I don't think it matters that much for what you're interested in: you can always Legendre-transform to your favorite set of variables... For the proofs I mentioned, however, the choice of ensemble plays an important role.

This post imported from StackExchange Physics at 2014-04-08 05:10 (UCT), posted by SE-user Yvan Velenik

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Addressing parts of the question out of order:

...the heat capacity would smoothly approach zero around the transition. I have never seen anyone refer to these types of transition...

The heat capacity of all substances smoothly approaches zero at absolute zero.

$S(E)$ has a segment of zero first derivative

no, the first derivative is a constant, the second derivative is zero.

$logZ(\beta)$ has a straight line segment

For a phase transition, the Gibbs free energy of the two phases are equal. It seems unrealistic for this to occur at an infinite number of Gibbs free energy values along the line segment.

This post imported from StackExchange Physics at 2014-04-08 05:10 (UCT), posted by SE-user DavePhD
answered Mar 7, 2014 by (65 points)
I don't see your point in your first comment. Of course it does, but what does that have to do with anything? I've corrected the trivial error you pointed out in your second comment.

This post imported from StackExchange Physics at 2014-04-08 05:10 (UCT), posted by SE-user Nathaniel

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