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I keep bumping into order parameters in scientific papers, reviews, articles, etc, but I can never get a firm grip on them. Order parameters seem terribly subjective to me. Basically the way I understand them is "just choose some function that helps you differentiate between phases, then normalize it so that its value is 0 in one and 1 in the other". But there must be much more to it than that, otherwise they wouldn't be this useful or widespread. Is the variable one chooses unique? Is there always a canonical order parameter choice? If there is one, then how do I know I have chosen the right one to describe my phase transition?

I have seen explanations of Landau theory in which a thermodynamic potential is expanded as series around an "order parameter" $\Psi$, but I have never seen an explanation where it was explicitly stated how one needs to proceed in order to choose or find this variable, if it always exists, etc. How do you do this?

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Ignacio
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    Of course it's not unique, it depends on the problem! The whole point is you use your physical understanding to figure out what the order parameter should be. This is like asking for the canonical, rigorous way to write a poem. – knzhou Apr 09 '18 at 22:00
  • But if you don't want to think about it physically, you may be interested in recent work using machine learning to identify order parameters (e.g. see here). – knzhou Apr 09 '18 at 22:01
  • Nice article, neural networks are always interesting =) – Ignacio Apr 09 '18 at 22:03
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    I mean, I could probably say "the hamiltonian depends on the problem, you have to be artful and find the nicest hamiltonian" but there is a canonical choice of hamiltonian, namely, the one that correctly defines the dynamics of your system. Isn't there such a thing for order parameters? Are they just a variable you choose in order to distinguish stuff? – Ignacio Apr 09 '18 at 22:05
  • No, the order parameter doesn't necessarily exists, for example, just take liquid-gaz transition, there is not an order parameter. I think that for all first order phase transition you can not find an order parameter, that's why Landau theory is just a phenomenological way to explain some cases, and you have not a universal way of explaining phase transitions (you have "toplogical hypothesis" but it's verified for some cases and it's not obvious) – Giuseppe Apr 09 '18 at 22:40
  • @Giuseppe am not sure what you mean by "there is not an order parameter", I thought the order parameter for the liquid-gas phase transition was usually taken to be the density difference, $\Delta \rho$. For instance, see https://physics.stackexchange.com/questions/346119/why-cant-the-density-difference-between-the-liquid-and-solid-be-an-appropriate – Ignacio Apr 09 '18 at 22:58
  • @Ignacio, The point is that Landau noted that phase transitions with vanishing latent heat (so not first order transitions) were accompanied by a symmetry change of the physical states of a system, and he constructed all his theory on this, so this is false to apply Landau theory on liquid-gas transition. – Giuseppe Apr 11 '18 at 12:56
  • So if you want to be rigourous : You have to compare the symmetry groups of your Hamiltonian and your physical state; if the symmetry group of your physical state is included in the symmetry group of your Hamiltonian, then you can use Landau theory. Therefore the order parameter has to vanish in the more symmetric phase and to be different from zero in the less-symmetric phase. – Giuseppe Apr 11 '18 at 12:59
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    If I am not mistaken Landau theory can be used for first order phase transitions if one keeps certain powers of order parameters that would vanish for second order transitions – Ignacio Apr 11 '18 at 20:22
  • @Ignacio Landau theory can be used only for symmetry breaking transitions. I don't think there are first order transitions which have a symmetry breaking, so, from what i know the answer is : No. if you want to have a good review on this and on phase transitions in general (what we know actually), take a look on this book : Marco Pettini Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics. This guy was my teacher and his book is very very interesting for fundamental points like what you're asking – Giuseppe Apr 14 '18 at 19:28
  • @Giuseppe Landau theory can be used even when there is no symmetry breaking! Check out this answer : https://physics.stackexchange.com/questions/313758/whats-the-rigorous-definition-of-phase-and-phase-transition – user140255 Apr 28 '18 at 01:49

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