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Could someone kindly elaborate more on the Simple Interpretation section from this Wikipedia Article? I refer to the part on the natures of $\alpha , \beta$. Why can one assume that $\alpha(T)=\alpha_0(T-T_c)$, for example?

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I don't have the time to write up an answer right now, but keep in mind that the Ginzburg-Landau model is a phenomenological theory, so the $T-T_c$ dependence is just a natural shift in variables to incorporate the change in behaviour above or below $T=T_c$. And I think the omission of higher order terms in the expansion of $\alpha$ is part of the spirit of the model to keep things simple: only the lowest non-trivial order is taken into account and the model should then work fairly well close to the $T=T_c$ point. I'm not sure about that second part though, I should revisit my notes on it. –  Wouter May 10 '13 at 0:53
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Actually, this is an assumption of the Landau theory: the simplest field model exhibiting a phase transition is analogous to a $\phi^4$ theory, which has the lagrangian density $$ {\mathcal L} = \partial_\mu \phi \partial^\mu \phi - \frac{m^2}{2} \phi^2 - \frac{\lambda}{4} \phi^4\,. $$

For $m^2 > 0$ the potential in the above lagrangian has a single minimum at $\phi=0$, but if $m^2<0$ then it has minima at $\phi = \pm \frac{m}{\sqrt{\lambda}}$.

The assumption that $\alpha = \alpha(T-T_c)$, which has the role of $m^2$, is such that at $T=T_c$ the system undergoes a phase transition from going to a two minima state to a single minimum one (or vice-versa). Taking it to be proportional to $T-T_c$ is, in my view, just the simplest choice and doesn't change the results qualitatively.

In the G-L theory the order parameter $\psi$ near the critical point, where the theory is valid, is taken to be small and so only the first two even terms of the Taylor expansion of the free energy are needed. The constant $\beta$ is of phenomenological nature.

Since $\phi$ is taken to be complex-valued this justifies, in a way, the choice to keep only the even powers. On the other hand, if it was real-valued one could argue, on phenomenological grounds, that the theory is symmetric under $\psi \to -\psi$.

A good reference for this subject is Quantum and Statistical Field Theory, by M. Le Bellac and G. Barton.

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Thank you, Felipe! :) –  valentina spirova May 10 '13 at 10:12
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