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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