Hi all. The blog has been very quiet (I hope exams are going well for you all).
Anyhow, I decided I'd sit down and figure out what on earth the transfer matrix method was all about (this is given on pg 360 of Nelson). I had previously glossed over this portion of the text because I didn't understand at all what the transfer matrix did (or why it's even called the transfer matrix) and hadn't the time to ponder it deeply.
So... here's my take on things, with
some assistance.
Suppose I have a single link in my polymer chain (not a very exciting polymer!). By Nelson's notation, my partition function is
Now, let's write this as
just for the fun of it (this will be useful in a second!).
Suppose I add a second monomer, so my `polymer' is now a dimer. There are four ways this can happen, depending on the states of the existing monomer and the new one (call them 'up' and 'down', or
u and
d). Thus I can have
uu, ud, du and
dd. Each of these options contributes to the partition function of the dimer. Since two options go with having the first monomer 'up' (
uu and
ud), and two with 'down' (
du and
dd)
I can write a matrix equation, which for now I'll represent as
.
This needs filling in. Let's do the
uu term. If the first monomer is 'up' and I add another 'up' then the term
needs to be multiplied by
. Repeating the process for the other combinations yields
.
We call the matrix in the middle the `transfer matrix',
T, but it would probably be more sensible to call it the `extension matrix', since it accounts for all possible ways in which the system may be extended by one monomer.
Now, suppose I take my dimer and add another monomer, to make a trimer. Since the interactions are local (they only depend on the nearest-neighbours) there are again only `four' ways in which I can do this (the rest of the possible configurations are already counted in the partition function of the dimer). Thus I simply get
.
It's now quite simple to see that for a polymer of
N residues the partition function is
,
which is a really neat result.
To summarise: the transfer (extension) matrix finds the terms which must be added to the partition function every time a new monomer is added, so the polymer partition function may be built up by repeated application of
T. Enjoy!
By the way, the equations here were rendered with an
online LaTeX interpreter which outputs images: very useful and fast!