A phase transition of order k is mathematically characterized by a loss of regularity of free energy f: f is k-1 differentiable but not k differentiable. There are many examples of first and second order phase transitions in experiments and in models. There are also cases where f is C^{\infty} but not analytic (Griffith singularities).
But are their known example of phase transition of order k, k>2 ?
A third order phase transition would mean that quantities like susceptibility or heat capacity are not differentiable with respect to parameters variations. But I have no idea of what this means physically.
Dear Bruno,
you might have a look in the paper by Bollobas, Janson and Riordan :"THE PHASE TRANSITION IN INHOMOGENEOUS RANDOM GRAPHS" especially pages 117-121. They construct random graph models where the giant component has phase transitions of arbitrary exponents larger equal one. Of course It is only weakly related to physical systems in statistical mechanics.
Hope you and the family are well. I'm meanwhile permanently at the Technical University in Wroclaw. In May or June I will visit Serge in Marseille. Maybe we can arrange for a meeting. Would be happy to see you. We could also think of some French Polish travel grants applications. What are your prime topics of interest right now.
My official mail is [email protected]
Many greetings
Tyll
Dear Bruno,
To identify such order you need to work with solvable models. Kazakov and Boulatov (Nucl. Phys. B (Proc. Suppl.) 4 (1988) 93, and references therein) have worked with spin models on dynamical planar random lattices. They calculate critical exponents and show the third order critical behavior. See also Fukazawa et al. (Mod. Phys. Lett. A 5 (1990) 2431), where the authors study the phase diagram of the 3-matrix model, which can describe the BEG model on a random lattice.
Hi Bruno,
I know a couple of references that might be useful for you:
The first one is a paper published in 1983 in Comm. Math. Phys.
http://projecteuclid.org/download/pdf_1/euclid.cmp/1103940230
and the other one is of more applied nature
http://scitation.aip.org/content/aip/journal/adva/1/4/10.1063/1.3650703
Hope they might be useful.
Miguel
The Bose-Einstein is such an example. The entropy and heat capacity are both continuous, whereas the temperature derivative of the latter is discontinuous, hence a third oder phase transition.
Dear all,
Sorry for the late reply, I don't connect so often to research gate. Many thanks for your interesting answers.
Bruno
hello guys, I asked a question 2 months ago that maybe is relevant here:
In the videos bellow, how many phase transitions you see in the water dissipation of mechanical energy and what type of phase transition you see?
https://www.youtube.com/watch?v=9al397N6Tzs&t=23sin here you see how the videos were obtainedhttps://www.youtube.com/watch?v=uMK3OVBjx2Q
Dear Prof. Bruno Cessac, in addition to all the interesting answers in this thread, there is a paper that explains historically the evolution of the Ehrenfest classification of the phase transitions, it might be good to add it since it talks about the Pippard extension of the classification when there are singular points in the specific heat at the Tc as in the ferromagnetic/antiferromagnetic-to-paramagnetic transitions in Ni (ferrom.), MnO (antiferrom.) and other crystals.
Article The Ehrenfest Classification of Phase Transitions: Introduct...
The Bose-Einstein is such an example. The entropy and heat capacity are both continuous, whereas the temperature derivative of the latter is discontinuous, hence a third oder phase transition.
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