In the landau theory we assume order parameter that is equal to zero at $T>T_c$ and none zero at $T<T_c$ wich is valid only for order to disorder phase transition according to my understanding. So that is mean that I can't use Landau theory on Liquid - Gas phase transition?


Away from the critical point, the liquid-gas phase transition is a first order phase transition, so you must use a theoretical description that treats it as one. The standard formulation of the Landau theory does not treat first order phase transitions, but it can be modified so that it does.

The liquid-gas phase transition through the critical point is second order, and the Landau theory can be applied to it.

  • $\begingroup$ but from what I understandood this theory describes only order to disorder phase transition, wich is not the case in liquid -gas. I would like for an example, or a link to a source that show how to treat the liquid gas with landau theory. In addition, landau theory is handle with first order phase transition, you just need to write your Landau functional a bit different. $\endgroup$
    – Sagigever
    May 27 '21 at 16:19
  • $\begingroup$ @Sagigever the order parameter for vapour-liquid transition through critical point is density. $\endgroup$ May 27 '21 at 16:54
  • $\begingroup$ @ Andrew Steane, density difference, strictly speaking $\endgroup$ May 27 '21 at 17:30

The Landau paradigm is getting a bit outdated in my opinion (not only) . There are plenty of different phase transitions which cannot really be categorised as first or second order. Enough to look at lattice models of quantum gravity or the standard model, but in analytic models also the study of Kibir-Żurek mechanism shows that. Many phases such as transition to spin glass, cannot be categorised according to order/disorder... Other aspects like long range entanglement can come into the picture.

The Landau description is a model related to the 20th century , when with 20th century technology we were able to create 20th century materials. Most of them could be analysed fairly easily in the labs. Nowadays we need to trap ions with lasers, excite composite systems, create a 2d lattice to give rise to anyons, create well designed lattices. One should look at phase transitions from a different perspective, because the old categorisation can easily break in many cases.

The Landau theory is not only valid to order-disorder but this is the most typical type of distinction, when your order parameter is zero or nonzero on the two sides.

  • $\begingroup$ Surely if one adopts mean field approach (which is natural simplest assumption) then one gets the Landau model. So this is not going to be outdated; it remains the model which is obtained when a mean field approach is taken. $\endgroup$ May 27 '21 at 9:11
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    $\begingroup$ Well, but the mean field approximation is misleading, it cannot really give (or rarely) higher order phase transitions. If you analyze a higher order phase transition with mean field approximation, you may find that its first order (mistakenly). This is what I meant, but there are sparks of truths in your comment too! $\endgroup$
    – Kregnach
    May 27 '21 at 11:16
  • $\begingroup$ Agreed. But it is nice that one gets critical exponents, and their values are in the right ball-park. $\endgroup$ May 27 '21 at 15:28
  • $\begingroup$ @Kregnach 7, "it cannot really give..higher order phase transitions." According to Ehrenfest's classification phase transitions were labeled by the lowest derivative of the free energy that is discontinuous at the transition. It seems to me that if we take $m^4$ and $m^6$ instead of $m^2$ and $m^4$ (where $m$ - order parameter) in the Landau expansion of free energy we formally obtain according to this classification the third order phase transition and so on... $\endgroup$ May 27 '21 at 18:28

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