I am self-studying Statistical Mechanics from J. Sethna's book Entropy, Order parameters, complexity. In one of the exercises (page 206), the Landau theory for the Ising Model is derived.

Starting with a general, local free energy density of this form:

$$F^{Ising}\{m,T\} = F(\textbf{x},m,\partial _j m, \partial _j \partial _k m, ...) .$$

we are supposed to "Taylor expand in gradients", keeping terms with up to two gradients of m, to get:

$$F^{Ising}\{m,T\} = A(m,T) + V_i(m,T) \partial _i m + B_{ij}(m,T) \partial _i \partial_j m + C_{ij}(m,T)( \partial_i m )(\partial_j m).$$

I do not understand what "Taylor expanding in gradients" means, nor how to get to the second equation (what do $A$, $V$, $B$ and $C$ represent?). It is not explained in the book, and I could not find anything related to this online.

Note that I am not a physicist by training and I am unfamiliar with much of the jargon (I do know what gradients and Taylor expansions are, though).


Suppose we have an expression for the free energy $F(\vec{x},T,m,\partial_j m,\partial_i \partial_j m,...)$, but we also assume that $m$ is approximately constant. This means that higher-order gradients (and large products of lower-order gradients) of $m$ are very small, so we can approximate $F$ as an expression involving only up to, say, two gradients of $m$ in any given term.

What are the possible forms an expression can take under these circumstances? Well, first of all, it could be independent of any gradient, and just be a function of $T$ and $m$ itself. Assuming that term is present in our expression for $F$, we'll call it $A(m,T)$. An expression could also involve just one gradient of $m$, and it might affect different directions differently; assuming all of these possible expressions are present, we'll label them $V_i(m,T)\partial_i m$. Similarly, we'll call those expressions which involve a second-order gradient $B_{ij}(m,T)\partial_i\partial_j m$, and those expressions with a product of two gradients will be labeled $C_{ij}(m,T)(\partial_i m)(\partial_j m)$. These are all of the possible expressions involving two or fewer instances of $\partial$. Adding all four possible terms together, we get the expression you have above.

If it seems like we didn't really do anything in the last paragraph, you're half-right: this is just setting up a general expression that will be elaborated upon in the rest of the book. Applying this expression to a physical situation will impose constraints on the values of $A,V_i,B_{ij},$ and $C_{ij}$; maybe $A(m,T)=0$ in some situations, or $V_i(m,T)=Ke^{-m/T}$, where $K$ is a constant dependent on the lattice spacing or something. The point is, the values for the parameters in your expression will be determined by the physical situation you're considering.

  • $\begingroup$ Indeed, later in the problem it does exactly what I said: various symmetries are imposed to better determine the form of the functions above. $\endgroup$ – probably_someone Jun 19 '17 at 18:40
  • $\begingroup$ Thank you, this very much clarifies what is done in the book. I still don't see how this is a Taylor expansion though. $\endgroup$ – enricosandro Jun 20 '17 at 18:32
  • $\begingroup$ It's a generalization of what you're used to seeing. This might help: math.ubc.ca/~feldman/m200/taylor2dSlides.pdf $\endgroup$ – probably_someone Jun 20 '17 at 18:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.