As often stated in books, near a phase transition we may express the free energy density as a power series in the order parameter $\phi(\mathbf{r})$. Up to quartic contributions, we have $$f=f_{0}-h\phi(\mathbf{r})+a_{2}\phi(\mathbf{r})^{2}+\frac{1}{2}|\mathbf{\nabla}\phi(\mathbf{r})|^{2}+a_{3}\phi(\mathbf{r})^{3}+a_{4}\phi(\mathbf{r})^{4}$$ where the second term is a coupling to an external field.
Now, I'm a bit confused about a few things:
The terms with coefficients $a_{i}$ make sense to me - if we do a Taylor series we get a polynomial. However, why can't we include a term $a_{1}\phi(\mathbf{r})$?
The external field term sort of makes sense, but this doesn't really stem from a power series does it? It's just something we put in by hand.
Where does the gradient term come from? Is this also put in by hand - I don't see how this could come from a power series... And, why does it have to have coefficient $\frac{1}{2}$? Why can't we use $|\mathbf{\nabla}\phi(\mathbf{r})|$ and higher powers?
If anyone has any input on these points, or can suggest a good source of explanation, that would be useful :)
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