A one-dimensional polymer (a chain), made of (N + 1) monomers, is diffusing on top of a one-dimensional lattice having a lattice constant a = 1. The i-th monomer (i = 0, 1, 2,..., N) is located at a coordinate $x_i$ and has energy of $$ U(x_i)=V(x_i)+\sum_{n.n,j}{f(|x_i-x_j|)} $$ where $V(x_i)$ is some function, the sum is over nearest neighbors and: $$ f(x)=\begin{cases} 0, &x=1\\ \infty, &\text{otherwise} \end{cases} $$
I am trying to use the transfer matrix to obtain the canonical partition function of the polymer, $Z_{N+1}$.
I thought of two ways to go about this:
(1) either I denote each point of the lattice, starting with the location of the 0-th monomer, with $y_j$ such that $y_j=1$ if the node is occupied and 0 otherwise, and then use: $$ Z=\sum_{\{x_0\}}\sum_{\{x_N\}}\sum_{\{y_i\}}\prod_{j=0}^{M}\exp\left[-\beta V(x_0+j)δ_{y_i,y_{i-1}}δ_{y_i,y_{i+1}}δ_{y_i,1}\right] $$ To get some kind of transfer matrix, but this seems problematic since each multiplication has to take the two nearest neighbors into account, or:
(2) I can maybe use $z_i=|x_i-x_{i-1}|$ to get the partition function, but then I have: $$ Z_{N+1}=\sum_{\{x_0\}}\sum_{\{y_i\}}\prod_{i=1}^N \exp\left[-\beta V(x_0+\sum_{n=1}^i y_n)\delta_{y_i,1}\delta_{y_{i+1},1}\right] $$
But this seems very complex, since the V function has a sum in its argument. Am I going about this in the wrong way, or am I missing something?
