Work out the weight factors by consulting [Chapter 10.6 from Jos'

book!](/downloads/Thijssen_Chapter10_Monte_Carlo_Method.pdf). It is essential to

get these right, otherwise your simulation will not work.

To quantitatively investigate the curling of polymers, it is useful t

the end-to-end distance of the polymers. Compare the case with

interactions to the case without interactions between the atoms.

- a polymer is a chain of subunits (monomers)

- the distance between monomers is fixed (1 in dimensionless units)

- the angle between neighboring bonds is fixed to multiples of $90^\circ$. With this, the subunits of the polymer are restricted to the sites of a square lattice.

- One lattice site can only be occupied by one monomer.

This set of rules turns the polymer problem into a *self-avoiding

random walk* on a square lattice.

![](/figures/polymer-selfavoiding.svg)

It would be possible to sample all self-avoiding random walks by starting

from a free random walk, and throwing away all paths with intersections.

However, this quickly becomes too expensive computationally!

In this project we instead employ a Monte Carlo method, to sample

only a finite number of representive polymers. In particular, we

employ the Rosenbluth that builds a polymer step by step: an existing polymer is grown

by adding a new subunit onto an unoccupied lattice site. In this way we sample

only allowed polymers. However, not every polymer then has the same

probability (as desired by us), as the growing process favors certain

configurations more than others!

In accordance with what we have learned for [approximate importance sampling](proj2-monte-carlo.md#importance-sampling),

we need to correct for this by introducing a weight.

This is derived in detail in the [lecture notes for the polymer project](proj2-polymers.md)

In particular, make sure you understand how Eq. (2) is derived there :