# Renormalisation group in statistical mechanics

## Main Question or Discussion Point

I am currently trying to get my head round RG in the context of statistical mechanics and am not succeeding! I would be grateful for any help. I have a specific question, but any clarification of RG in general would be useful. Here is my understanding of the main ideas:

1. Define microscopic system with some Hamiltonian H.
2. Coarse grain this system in some (arbitrary) fashion, which gives rise to a system with a new H in terms of the coarse grained variables.
3. Now we use some input from experiment: near the critical point the system appears to be self similar at all scales, so we impose the requirement that H is of the same form in the new system so that the coarse grained system behaves as it should; the new H has some different set of constants in general given by a transformation.
4. We can then go on to find critical exponents and the critical coupling.

The problem comes when I read or hear statements such as in Leo Kadanoff's book where he says that "each phase of the system can be described by a special set of couplings, K*, which are invariant under the RG transform". The procedure outlined above (if i've got it right!) only applies around the critical point from step 3 onwards, as elsewhere the system certainly doesn't look the same on all scales. Why does the recursion relation we previously obtained have any meaning away from the critical point?

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atyy
I understand it something like this:
(i) Start from a microscopic Hamiltonian
(ii) Coarse grain and rescale
(iii) At the critical point, there should be self similarity, ie. there will be a fixed point of step (ii).
(iv) Search for fixed points of step (ii).
(v) If a fixed point of step (ii) is related to critical phenomena, it should have some properties eg. existing only for T=Tc.

atyy
I understand it something like this:
(i) Start from a microscopic Hamiltonian
(ii) Coarse grain and rescale
(iii) At the critical point, there should be self similarity, ie. there will be a fixed point of step (ii).
(iv) Search for fixed points of step (ii).
(v) If a fixed point of step (ii) is related to critical phenomena, it should have some properties eg. existing only for T=Tc.
Right, I forgot the rescaling but was nearly there! It seems to me though that having the same Hamiltonian or close enough so that we can find fixed points corresponds to a rather special situation.

I'll take a look at that book though, hopefully that will clear things up.