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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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For the second order liquid/gas phase transitions, the relevant order parameter is the difference between the densities of liquid and gas. However, for the first order liquid/gas phase transitions (below the critical point), the high temperature phase is gas and the low temperature phase is liquid. What does "the difference between the densities" mean? –  hlew Jan 16 '13 at 13:52
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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. –  Luboš Motl Jan 16 '13 at 15:05
    
I'm still confused. When it is in the low temperature phase i.e. liquid phase, let $a$ be the density of liquid, then what is $b$? Thank you ! –  hlew Jan 17 '13 at 2:55
    
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. –  Luboš Motl Jan 17 '13 at 8:13
    
For the liquid/gas first order phase transitions, the relevant order parameter is just the densities, I think. So at the transition point, there is a jump in the order parameter. Do you think so? –  hlew May 13 '13 at 14:04
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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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Do you mean that the order parameter is defined to be zero in liquid phase and to be the difference between the densities of liquid and ice in the ice phase? –  hlew May 13 '13 at 14:13
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