# Is there any zero-order phase transition in nature?

Theoretically, a finite jump in the free energy phase diagrams can naturally be called a zeroth-order phase transition according to the Ehrenfest classification. We always hear about the first- and the second-order phase transitions, but I'm curious whether there is any zero-order phase transition in nature? If yes, in which kinds of critical phenomena and if not, why? Is there any experimental proof for it? I searched but found nothing.

Note: I found this (open access) paper about the existence of such a critical phenomenon for a weakly non-ideal Bose gas in the Bogolybov model, which primarily is based on theoretical exploration.

• What do you consider a martinsitic transition to be? Commented Mar 3, 2021 at 20:39
• @ Jon Custer - I'm not familiar with the martensitic transitions but it seems they are known to be strongly first order (not sure yet). For example see this short review (sciencedirect.com/science/article/abs/pii/S0927025613002310) and also this one (inis.iaea.org/collection/NCLCollectionStore/_Public/28/024/…). There are a number of papers which confirm that martensitic transitions are first order.
– SG8
Commented Mar 3, 2021 at 20:53
• I agree, its just that over the years different books have referred to it in different ways. The diffusionless aspect of having the unit cell distort to form the martensite is a bit unusual (at first glance). Commented Mar 3, 2021 at 21:02

A zeroth-order phase transition defined as a finite jump of some free energy or any other fundamental equation is incompatible with the requirement of convexity (concavity) for such state functions. It is a convex analysis theorem that every convex function must be continuous. Moreover, a convex function must have left and right derivatives almost everywhere. The requirement that fundamental equations should have a well definite convexity is equivalent to the stability of thermodynamic systems. Therefore, it is hardly conceivable to get rid of it.

I have read Maslov's paper cited in the question, but I find it difficult to follow the author's argument. However, due to the basic status of the convexity argument, I think that it is to Maslov to explain how his Statistical Mechanics argument could avoid contradiction with basic Thermodynamics.

• Thank you. But, are you sure a zeroth-order phase transition is incompatible with the requirement of convexity (concavity) of the state function of free energy? As far as I know, theoretically, we can a zero-order phase transition which satisfies all global and local criterions of thermal stability. For example, zeroth-order phase transition have been found in mathematical models of black holes and many explorations have revealed that during this transition the thermodynamic stability is satisfied. Of course, the entropy is discontinuous at the zero-order phase transition.
– SG8
Commented Mar 4, 2021 at 8:21
• The following papers are about the zero-order phase transitions in black hole (totally based on theoretical considerations) : [1]journals.aps.org/prd/abstract/10.1103/PhysRevD.88.101502 [2]journals.aps.org/prd/abstract/10.1103/PhysRevD.103.044025 [3]iopscience.iop.org/article/10.1088/0264-9381/33/24/245001/meta
– SG8
Commented Mar 4, 2021 at 8:22
• @SG8 within classical thermodynamics, there is no room for doubts. Zeroth-order phase transitions are incompatible with convexity. However, gravitational systems are not usual thermodynamic systems. One of the fingerprints is that they are not additive nor extensive, and they show the wrong convexity. Commented Mar 4, 2021 at 8:35
• @SG8, the issue is what does it mean to go beyond classical thermodynamics. Theories cannot constrain how the world behaves. However, concepts cannot be stretched beyond some limits. After that, it is probably better to change the concepts. That's the way Science goes on. What I think is not fair is to silently extend the meaning of some concept without any warning on that. That's precisely the case of the so-called black-holes thermodynamics. The problem is that people use the same words as for lab systems even if the coherence is not obvious at all. Commented Mar 4, 2021 at 9:22
• @SG8 I can confirm. I am not aware of any experiment showing indication of a possible zeroth-order phase transition in the usual thermodynamic systems. Commented Mar 4, 2021 at 19:55