@@ -22,7 +22,7 @@ where $\mathbf{r}_\text{cm} = \frac{1}{L+1} \sum_{i=0}^L \mathbf{r}_i$ is the ce

We will now give a complete description of the polymer toy model. We will continue

to describe a polymer by an ordered set of points $\{\mathbf{r}_0, \mathbf{r}_1, \dots, \mathbf{r}_L\}$. In a polymer, the distance between subunits is determined by the length of the chemical bonds, and we will thus assume a fixed bond length $a$ between subunits $\left|\mathbf{r}_{i+1} - \mathbf{r}_i\right| = a$. In contrast, for many polymers the angle between bonds is far less restricted, and we use this as the degree of freedom that determines the shape of the polymer.

To simplify the model even further, we will assume that the angle between different bonds is restricted to be a multiple of $90\degree$. In doing so, we restrict the polymer points $\mathbf{r}_i$ to be on a square lattice with lattice constant $a$. Without restricting generality, we can set $a=1$.

To simplify the model even further, we will assume that the angle between different bonds is restricted to be a multiple of $90^{\circ}$. In doing so, we restrict the polymer points $\mathbf{r}_i$ to be on a square lattice with lattice constant $a$. Without restricting generality, we can set $a=1$.

In the above example, you may already see the last ingredient for our toy model: Since a polymer has a finite extent in space, two subunits cannot come too close. In our lattice model we implement this by demanding that one lattice point cannot be occupied by two different subunits. In other words: $\mathbf{r}_i \neq \mathbf{r}_j$ for all $i \neq j$.