Ginzburg-Landau model for superconductivity 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?
 A: 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.
