Fluctuation Terms in Landau Theory

[maverick_starstrider]

Alright, here is something that is driving me insane. I feel like I've looked through every stat mech book on the planet and not a one discusses this stuff properly. My difficulty is in the study of phase transitions when one applies Landau's approach of expanding the free energy about the order parameter at the critical point. If one then assume that the order parameter fluctuates in space Landau originally added a term of the form gradient(M) dot gradient(M) and simply said something like "this is the simplest spatial derivative that respects the relevant symmetries". My question is this, what is the GENERAL spatial expansion about the critical point? I've never seen a series expansion that had a term like gradient(M) dot gradient(M) (maybe I missed that day in vector calc), what are the lower order terms? what are the higher order terms? Why do they go to zero/become negligible?

What I would really like is an explanation/discussion of how one in general forms a series expansion in space about the critical point and then a discussion of why all but a gradient(M) dot gradient(M) term can be neglected. And I don't want to really get my hopes up but a discussion of how accurate the spatial expansion is in general and when it is not so valid (when an expansion in space isn't valid, not the ginzburg criterion) would be amazing. Anyway, if someone could point me to a source that actually discusses in detail this non-trivial series expansion (or just explain it here) rather than plopping down gradient(M) dot gradient(M) and saying "well here's what Landau used and he knew his stuff" that would be amazing. Thanks in advance.

P.S. I've already been through Reichl, Huang, Pathria, Cardy and Kadanoff (and Landau obviously who started the whole thing by plopping down a term with a sentences justification). Not a one does it.

 

[nbo10]

Have you tried Landau and Lifgarbagez? It's been a while but, The expansion is performed about M(r) because M is small at the critical point. I think the grad M(r) dot grad M(r) term is the first term that is invariant wrt M(r).

 

[maverick_starstrider]

Have you tried Landau and Lifgarbagez? It's been a while but, The expansion is performed about M(r) because M is small at the critical point. I think the grad M(r) dot grad M(r) term is the first term that is invariant wrt M(r).

? that's exactly what I'm asking about. In Landau and Lifgarbagez they introduce the grad(r) dot grad(r) term by just handwavingly saying it is the FIRST term that matches symmetries or whatever. The question I'm asking is what are the OTHER possible terms (whether they are invariant or not) and what kind of series expansion produces that term to begin with. And for that matter what is it invariant with respect to? It's this maddening situation where every textbook in existence seems to slam down that term and basically says "It's here because Landau and Lifgarbagez and *mumble* *mumble* symmetry". I'd just like to see how one creates a general expansion of the order parameter about T and r and then some explanation of why the other r terms are physically unallowable.

 

[maverick_starstrider]

This is one of the fundamental concepts in advanced statistical mechanics. It directly leads to the Ginzburg criterion and such. Does no one know where it comes from?

 

[Catalyst3146]

There are a few things to address here. I'll try to give you some rough ideas, then point you to some more advanced resources at the end.

First, for the Landau-Ginzburg functional to represent your physical problem, it must have the same symmetries as the underlying system. This is why the expansion only contains symmetry allowed terms. For standard case (ferromagnet) this mean translational and rotational symmetries in space and
some kind of spin rotational (or spin-flip for an Ising type magnet) in order parameter space. Stability also constrains the expansion as well, as the energy must be bounded below.

Second, the gradient term. One way to think of this is as an expansion about small magnitude and uniformity. Specifically, in Fourier space imagine doing a double expansion in both the order parameter M and the coefficent functions in wave vector k about M=0 and k=0 (again only with symmetry allowed terms). That is write :
$$F[M] = F[0] + \sum_{k} a_1(k) M(k)M(-k) + \sum_{k_1 k_2 k_3} a_2(k_1,k_2,k_3) M(k_1)M(k_2)M(k_3)M(-k_1-k_2-k_3) + \cdots$$
Then expand the functions $$a_1$$ and $$a_2$$ as:
$$a_1(k) = a_1(0) + a_{1} k^2 +\cdots$$
$$a_2(k_1,k_2,k_3) = a_2(0,0,0) + \cdots$$
This again has to be motivated physically, that is the problem should suggest a uniform state is a good place to expand about.

Here is a more advanced view which I'll leave to you to explore at your discretion. In technical terms one can start with a microscopic Hamiltonian for your problem of interest and derive the effective action for a given order parameter using a gradient expansion. This gives automatically gives you the symmetry constraints. Another reason the series is usually truncated at this order is due to ideas from the renormalization group.

 

[maverick_starstrider]

Thanks for your post. My concern is how one MATHEMATICALLY arrives at the sum you presented above or the "gradient expansion" you mention. I understand how, once one has a general expansion, certain terms can be disqualified on physical grounds. However, I cannot for the life of me find any reference or proof or derivation of a series expansion of the form

$$f(x)=A \nabla f(x) + B \nabla^2 f(x) + C \vert \nabla f(x) \vert^2 + \ldots$$

Specifically the C term (which is the only term that is kept) is simply bizarre to me. This is where my trouble lies. MATHEMATICALLY how does one expand "about small magnitude and uniformity". What's the general mathematical expression and how is it derived. I suspect it has something to do with the calculus of variations but I've scoured the internet to find it and haven't had any luck. For example, say I wanted to include higher order fluctuation terms, what are they? How do I find them out? Do I just randomly mash together gradients and exponent until I trial and error another term that obeys symmetries? Surely not, there must be some GENERAL expansion that I am simply unfamiliar with. I feel like it's understood that this is the first symmetry obeying term of some super secret Taylor type expansion that I've never heard of. I would like to know what that expansion is. Without any symmetry or physical considerations what are all its terms. If the term was truly just "guessed" then why that term. Why not something bizarre like $$m^2 \vert \nabla m \vert^2$$?

 

[Catalyst3146]

Your concerns are perfectly valid. The Landau approach is a guess, as really one has no idea what the coefficients are and which ones are the largest. This depends completely on the underlying physical system. A priori all we are assuming about the system are symmetries which is why we only include the terms with the proper symmetry and again why we include all of them, as a priori there is no reason for them to be absent. One can justify (partially) why only a few of the terms are kept using the renormalization group (at least in the vicinity of the critical point).

In the example you've shown the first term usually absent due to a reflection symmetry in the order parameter, and the next two are equivalent once integrate (up to boundary terms). The bizarre term you mention is at fourth order in m and 2nd in k, and if you expect uniformity then you'd expect the coefficient here to be small. This is not necessary and again depends on the full details of the system in question.

Mathematically, expansion of a function of a given symmetry is most generally done using the representation theory of the symmetry group (see any book on representation theory for physics, specifically the section on basis functions). The gradient expansion is discussed in many advanced condensed matter books as well as many quantum field theory books (the terminology might be a bit different).

 

[welshtill]

GL's equation was first obtained from phenomenological consideration plus some symmetry constraints. Microscopic alculation can be performed by path-integral method in a more clearer way. Details are given in A.Altland and B.Simons book:"condensed matter field theory" on page286.