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Order parameter is used to describe second order phase transition. It seems that in some papers it is used in the first order phase transitions. Can first order phase transition have an order parameter? If so, how can we define the order parameter in liquid-gas transition (first order)?

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Yes, there may still be some order parameters in the presence of first-order transitions. But much like free energy, the order parameter is discontinuous at the transition point. For the liquid/gas phase transitions, the relevant order parameter is the difference between the densities.

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    $\begingroup$ Dear hlew, the difference between $a$ and $b$ is the expression denoted $a-b$. Does it answer your question? I didn't say that the order parameter behaves in the same way as it does in the 2nd order phase transition and indeed, it doesn't behave in the same way. $\endgroup$ – Luboš Motl Jan 16 '13 at 15:05
  • $\begingroup$ Dear hlew, in the presence of 1st order phase transitions, there is always (at any combination $p,T$) the other phase as well except that it is unstable. Maybe my approach was just a wrong way to deal with the question. Maybe I should have just written that the order parameter is only useful or "routinely used" for 2nd order phase transitions and the right theory for 1st order phase transitions is of course completely different than Landau's theory etc., so any attempt to apply 2nd order phase transition concepts will lead to confusion or chaos. $\endgroup$ – Luboš Motl Jan 17 '13 at 8:13
  • $\begingroup$ Is there a general definition of order parameter? I have only seen examples, but most of the time people are just saying "this is the order parameter", without really saying why they call it that. I know it usually "jumps" at a phase transition.... but many functions could do that. $\endgroup$ – becko Nov 29 '17 at 16:41
  • $\begingroup$ Hi, check e.g. this definition en.wikipedia.org/wiki/Phase_transition#Order_parameters - it's a quantity that is meant to exhibit a relevant un-smoothness at the phase transition point. In most cases, we want the order parameter to be zero above the transition and nonzero below it. $\endgroup$ – Luboš Motl Dec 1 '17 at 12:24
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Your first answer was completely OK. Order parameters can be also used for the description of first order transitions, why not?

Think for example of water to ice transition. There is a jump when you look at the plot of density against temperature (see for example: Density of water (wikimedia)) and this is what we understand (loosely speaking of course!) under the first order phase transition (small subtlety: define the order parameter so, that it remains zero in the disordered liquid phase). This UCI lecture note about phase transitions may also be helpful.

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