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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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    $\begingroup$ 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. $\endgroup$ – knzhou Apr 9 '18 at 22:00
  • $\begingroup$ 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). $\endgroup$ – knzhou Apr 9 '18 at 22:01
  • $\begingroup$ Nice article, neural networks are always interesting =) $\endgroup$ – Ignacio Apr 9 '18 at 22:03
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    $\begingroup$ 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? $\endgroup$ – Ignacio Apr 9 '18 at 22:05
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    $\begingroup$ 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 $\endgroup$ – Ignacio Apr 11 '18 at 20:22

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