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I am looking the simplest system that has both discontinous phase transition and a continous phase transition between the same phases (you can change one parameter).

discontinous transition: first order transition where (some) measurable has a discontinuity/non-analyticity as a function of (some) order parameter

continous transition: measurable is continous/analytical for all values of order parameter.

simplest: Requires the smallest number of mathematical symbols and is still defined exactly c.f. ising model.

EDIT: Removed water as an example. Above critical point, water has no phase transition as Alexei pointed out. So, all the transitions of water are first order...

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    $\begingroup$ I don't know why you removed water --- it seems exactly the right example. Below the critical point, you obviously have discontinuous transitions, and if you move along that discontinuity, you get a continuous transition at the critical point. In fact, these two facts are really fundamentally linked. Furthermore, many magnetic models are in the same universality class, so behave the same. $\endgroup$
    – genneth
    Sep 27, 2012 at 10:13
  • $\begingroup$ @genneth, Can you distinguish the two phases above or at the critical point? I think this is the essence of my question (or at least what I want to know). To my knowledge, you should be able to distinguish the different phases regardless of the transition type. Also (now) it seems that in all the examples there exists only one critical point for transition of type a and rest is a transition of type b... $\endgroup$
    – Juha
    Sep 27, 2012 at 11:13
  • $\begingroup$ @Juha "in all the examples there exists only one critical point for transition of type a and rest is a transition of type b" This is not right. It is indeed the case for the so-called isolated Curie point. This is, however, very rarely met in nature, if any. But for the so-called, first order transition close to the second order (the alternative name is the tri-cricital point vicinity) the line of the second order transition becomes the line of the first order one. $\endgroup$
    – Alexei Boulbitch
    Nov 5, 2012 at 13:53
  • $\begingroup$ Why do not you look into some textbook. For example, in the book Toledano, J. C. & Toledano, P. The Landau Theory of Phase Transitions. (World Scientific, Singapore, 1987) you will find lots of examples on this subject. You may also look into another book as well. I only gave this reference, since it was right at hand. $\endgroup$
    – Alexei Boulbitch
    Nov 5, 2012 at 13:55
  • $\begingroup$ In fact your requirement to be able to always distinguish between phases is always achieved, if the phase transition is associated with the symmetry break. This is the case of 99% of phase transitions. If the transition is isostructural, no symmetry break takes place, and you may only distinguish the two phases by the volume of elementary cell in the transition point, if the transition is first order and the volume dependence upon, say, temperature exhibits a discontinuity. $\endgroup$
    – Alexei Boulbitch
    Nov 5, 2012 at 14:03

4 Answers 4

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Let me give a more mathematical perspective on that.

As far as I know, the classical example of a system you are looking for -- is the basic landau theory with a cubic term:

$$F=r\Psi^2+s\Psi^4+\alpha\Psi^3,\quad r=r_0(T-T_c)$$

  • If $\alpha=0$ then you have just a standard theory for 2nd order phase transistion.

  • While at $\alpha\ne0$ you'll have a discontinuity in the order parameter dependence on temperature.

Regarding your comment: if one restricts values of $\Psi$ to be positive -- then one gets exactly what you want. The restriction is quite natural if you, say, have two-component order parameter $\vec{\phi}=(\phi_1,\phi_2)$ and $\Psi=|\vec{\phi}|$.

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  • $\begingroup$ Can you modify this so that there is more values for $\alpha$ by adding higher order terms? E.g. discontinuity when $\alpha' \lt 0$ and second order when $\alpha' \ge 0$. $\endgroup$
    – Juha
    Sep 27, 2012 at 11:17
  • $\begingroup$ @Juha: I think so. See my edits. $\endgroup$
    – Kostya
    Sep 27, 2012 at 11:41
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I think you are wrong with water. Above the critical point there is no transition in water at all. This is also true for any other isostructural transition: as soon as there is no symmetry difference between the phases, in the continuous case you cannot say, if the transition has already taken place. A correct example should be probably, KHP (potassium dihydrogen phosphate, see here: http://en.wikipedia.org/wiki/Monopotassium_phosphate or KDP (Potassium deuterium phosphate). You may look into the books of Tonkov: 1. Tonkov, E. Y. High pressure phase transformations (Gordon and Breach, Philadelphia u.a., 1996). 2. Tonkov, E. Y. High Pressure Phase Transformations: A Handbook: (Gordon and Breach SA, 1992). and may be also in this one: Tonkov, E. Y. & Ponyatovsky, E. G. Phase Transformations of Elements Under High Pressure (Advances in Metallic Alloys) (Crc Press Inc, 2004). You will find there lots of examples of various transitions, and I have also seen there examples of transitions of the second order that come into the first order through a so-called, tri-critical point. It should be the example you are after. To my knowledge, there are, however, only few such examples in structural phase transitions. There should be some more examples among magnetics, but here I am not a specialist.

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  • $\begingroup$ I would be interested in these magnetic systems. Can you perhaps point where to start looking? $\endgroup$
    – Juha
    Sep 26, 2012 at 11:55
  • $\begingroup$ I am not aware much about magnetics. You may go through Tonkov anyway, he did not consider any special type of transition, but just enumerated all phases and transitions that are there in a material under consideration. $\endgroup$
    – Alexei Boulbitch
    Sep 27, 2012 at 7:04
  • $\begingroup$ Just the comment to the answer of Kostya. The question seems to be concerning experimental examples. If one speaks about the theoretical description, then the example with the cubic term has only a so-called, isolated Curie point, which is indeed the second order. Experimentally such case takes place in some liquid crystals. The other theoretical example would be the so-called, first order close to the second with the potential possessing the second, fourth and sixth order terms. These are sometimes observed in solid phases, the KHP is the example. $\endgroup$
    – Alexei Boulbitch
    Oct 4, 2012 at 14:39
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First, make sure to read up on definitions to clarify what you are looking for - classification of phase transitions isn't 100% science, and has a little bit of fussiness to it. Wikipedia's page isn't terrible.

Second, I can't tell you whether it is the simplest or not, but as I understand your question, the Ising model itself satisfies your conditions, as long as you include a magnetic field (and are in dimension d=2 or higher, so that there is a phase transition!).

At zero magnetic field, if we decrease the temperature of the Ising model from $T > T_c$ to $T < T_c$, the magnetization M(T) increases continuously, i.e. we can get an arbitrarily small M by taking T arbitrarily close to Tc. Now, suppose the model is below Tc and we have a nonzero magnetic field, $h > 0$; the magnetization will then be positive. As we decrease h to zero, the magnetization will discontinuously change from being positive for any $h>0$ to being negative for any $h < 0$. So for $T<T_c$, there is a first-order phase transition line. The phase diagram can be seen on this webpage and this is discussed more in most stat mech textbooks. My favorite for this is Nigel Goldenfeld's Lectures On Phase Transitions And The Renormalization Group.

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  • $\begingroup$ Note: the link is now broken $\endgroup$
    – Nihar Karve
    Oct 31, 2021 at 6:43
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Many order-disorder transitions are second order, for example the order-disorder transition in $\beta$-brass. So brass would be an example of a material showing both first and second order transitions.

However even simpler is, as Alexei pointed out, the magnetic transition in iron that happens at the Curie temperature is second order. So iron is probably the simplest system that fits your criteria.

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