Monday 25 June 2018

condensed matter - Landau Theory of Phase Transitions


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:




  1. 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 $a_{1}\phi(\mathbf{r})$?




  2. 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.





  3. 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 :)




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...