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?
- 1$\begingroup$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– knzhou2018-04-09 22:00:20 +00:00CommentedApr 9, 2018 at 22:00
- $\begingroup$Nice article, neural networks are always interesting =)$\endgroup$Ignacio– Ignacio2018-04-09 22:03:27 +00:00CommentedApr 9, 2018 at 22:03
- 1$\begingroup$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– Ignacio2018-04-09 22:05:10 +00:00CommentedApr 9, 2018 at 22:05
- 1$\begingroup$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– Ignacio2018-04-11 20:22:20 +00:00CommentedApr 11, 2018 at 20:22
1 Answer1
Yes and no, the order parameter is subjective in the same way the phase is subjective. We come to understand phases by observing systems and noticing certain features about them that are different, we then look at these features and use them to subjectively define what a phase is, once we have that definition we can look into creating an order parameter. A nice example for this is BCS superconductivity. These superconductors are characterized by a host of properties such as resistivity approaching 0, quantization of magnetic flux through rings of superconducting material, the expulsion of magnetic field from the bulk and much more. The idea here then is that a superconducting system has to satisfy a whole host of requirements that we determined by experimentation. We then decided to call a system that follows all of this a "superconductor" which is in the "superconducting phase". The order parameter describing the phase change from regular material to superconductor must then also reproduce all the properties we observe from a superconducting phase in experiment when it is nonzero. This is how you determine "correctness" in reality, you look at experiments and determine all the properties you associate with a phase and see if you can find a variable that can distinguish between phases that possess all those properties and those that don't. You could then imagine that if you found such a variable it might contain physically important information about the system and so you could assume the free energy of the system depends on this parameter and note that this parameter is small near a phase change (by construction) allowing you to expand to low order and do Landau theory, the fundamental assumption rests upon the fact that you assume you've found a quantity that you think the free energy of system should depend on.
Explore related questions
See similar questions with these tags.