[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

## Question: Zipper problem ([1] pr 3.7)

A zipper has links; each link has a state in which it is closed with energy and a state in which it is open with energy . we require, however, that the zipper can only unzip from the left end, and that the link number can only open if all links to the left are already open. Find (and sum) the partition function. In the low temperature limit , find the average number of open links. The model is a very simplified model of the unwinding of two-stranded DNA molecules.

## Answer

The system is depicted in fig. 1.1, in the and states.

Fig 1.1: Zipper molecule model in first two states

The left opening only constraint simplifies the combinatorics, since this restricts the available energies for the complete molecule to .

The probability of finding the molecule with links open is then

with

We can sum this geometric series immediately

The expectation value for the number of links is

Let’s write

and make a change of variables

so that

The average number of links is thus

or

In the very low temperature limit where (small , big ), we have

showing that on average no links are open at such low temperatures. An exact plot of for a few small values is in fig. 1.2.

Fig 1.2: Average number of open links

# References

[1] C. Kittel and H. Kroemer. *Thermal physics*. WH Freeman, 1980.