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I am puzzled by the definition of continuous and first-order phase transitions and the use of Landau theory. I think I am lost in technicality.

In the phase diagram below the critical point, we have coexistence lines. The macroscopic properties of matter change abruptly moving from one phase to another. This change manifests itself mathematically as discontinuous molar entropy/molar volume.

Above the critical point, the two phases can not be distinguished where we have continuous phase transition.

If this is true for all cases, the whole point of the Landau theory is to determine if a phase transition (at criticality) is continuous or not.

Or can we have a case in which below the critical point where we have coexistence lines there are continuous phase transitions?

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    $\begingroup$ It seems to me as a rather confused description. I suggest reading the Goldenfeld's book chapter on the Landau theory (although short descriptions of the Landau theory can be found in many other places). $\endgroup$ Aug 24, 2021 at 14:23
  • $\begingroup$ I will check that. thank you $\endgroup$
    – GGphys
    Aug 24, 2021 at 14:46

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There are two different ideas that are mixed up in this question. Let me try to separate them.

The first has to do with the form of a thermodynamic phase diagram. Consider, for example, the phase diagram of a magnet in the plane of temperature T and external magnetic field H. This is a 2-dimensional space. As GGphys points out, there is a line in this space across which the magnetization changes discontinuously. For a symmetric magnet, this line is on the $H=0$ axis, from $T=0$ to some higher temperature $T=T_c$. If we cross this line, say from $H >0$ to $H<0$, there is a discontinuity in $M$, and we call this a first-order phase transition. At very high temperature, $M = 0$ when $H = 0$. In practice, on the line of discontinuity, $M$ decreases and goes to zero at $T = T_c$. Above $T_c$, there is no discontinuity. We call the behavior at $T_c$ a second-order phase transition, and we call the point $H=0, T = T_c$ the "critical point".

The vicinity of the point $H=0$, $T=0$ is a special place with long-range correlations of the magnetization and, singularities in thermodynamic functions. For example, the magnetic susceptibility $\partial M/\partial H$ goes to infinity at $H=0$, $T=0$. So a special, more sophisticated, theory is needed to describe this region.

This is all for a 2-dimensional phase diagram. If we add more thermodynamic variables, all of this extends to higher dimensions. An example is an antiferromagnet in an external magnetic field or a liquid-gas system with a varying density of a solute in the liquid. Maybe the example of the antiferromagnet is most clear. We can imagine a "staggered" magnetic field $H_s$ that turns on the antiferromagnetic order. The variables are then $H_s$, a uniform field $H$, and the temperature. This is a 3-dimensional thermodynamic space. The line of discontinuities becomes a plane of discontinuities in the plane $H_s$ = 0, and the second-order phase transition becomes a line of critical points at $H_s = 0$, $T = T_c(H)$. There can also be other features. If you make the uniform $H$ field very large, all of the spins will line up uniformly. So if you start from the antiferromagnetic phase and raise $H$, eventually you will find a transition (usually, discontinuous or first-order) above which there is a uniform magnetization. It is hard to make a staggered magnetic field in the lab, so when we measure the phases of an antiferromagnet we are restricted to the $H_s = 0$ plane. This has a (vertical) line of second-order phase transitions at $T = T_c(H)$ and a (horizontal) line of first-order phase transitions at $H = H_f(T)$ that meet and end at an even more special point called the "tricritical" point. As you add more thermodynamic variables, more complicated behaviors are possible.

So far, I have not talked about Landau theory. Landau theory is an approximation to the free energy in which you assume that $M$ is small and write the Gibbs free energy as a polynomial in $M$. This gives an approximate description in the vicinity of the critical point. Even away from the critical point, Landau theory gives a qualitative description of the phase diagram. For example, for the antiferromagnet above, the Landau theory would have a polynomial in $M$ and the staggered magnetization $M_s$, and it would show the various phase transitions that I have described.

Landau theory is then useful to get a qualitative picture of the phase diagram in however many dimensions you have. It turns out that it is also useful as a starting point for a quantitative theory of the thermodynamics near a critical point.

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