Partition Function for Spin-1 One Dimensional Ising Model

[pauladancer]

Homework Statement

Hi everyone,
I'm writing a paper for my statistical mechanics course and require the partition function for the spin-1 Ising model. I've searched for a solution, but can't find one anywhere. I'm hoping to get some help!

Relevant Equations

See below

$$H=-J\sum_{i=1}^{N-1}\sigma_i\sigma_{i+1}$$ There is no external magnetic field, so the Hamiltonian is different than normal, and the spins $\sigma_i$ can be -1, 0, or 1. The boundary conditions are non-periodic (the chain just ends with the Nth spin)
$$Z=e^{-\beta H}$$
$$Z=\sum_{\sigma_1}...\sum_{\sigma_{N-1}}e^{\beta J\sum_{i=1}^{N-2}\sigma_i\sigma_{i+1}}\sum_{\sigma_N}e^{\beta J\sigma_{N-1}\sigma_N}$$
and here's where I get lost, I'm not sure how to evaluate this sum

 

[pauladancer]

Update: It seems that the transfer matrix method is the best way to do this, but I cannot find the correct expression for the partition function in matrix form if the boundary conditions are not periodic.

 

[Dr_Nate]

I don't think I'll be of any real help, but let me try.

• For your paper, are you explicitly not allowed to use periodic BCs?
• Are you allowed to model it numerically?
• Have you tried all the summation identities that seem reasonable? Maybe an index shift would work.

 

[pauladancer]

I am modelling DNA as a one dimensional Ising model, and so I don't think it would be wise to use periodic boundary conditions since DNA isn't circular (in eukaryotes anyway). The main part of my paper will be comparing this to a numerical/computational result :) Digging through the internet I've come to the conclusion that the first N-1 sums will contribute a 2, and the final sum contributes $$e^{-\beta J}+e^{\beta J}+1=2cosh(\beta J) +1$$ so the final result is $$Z=2^{N-1}(2cosh(\beta J)+1)^{N-1}$$ I'm not sure I completely understand where the $$2^{N-1}$$ comes from, so if someone could clarify that I'd really appreciate it!