Zipper

A zipper has N links; each link has a state in which it is closed with energy 0 and a state in which it is open with energy ε. We require that the zipper only unzip from one side (say from the left) and that the link can only open if all links to the left of it (1,2,..., n-1) are already open. (This model is sometimes used for DNA molecules.)

a) Find the partition function.

b) Find the average number of open links ⟨n⟩ and show that for low temperatures τ << ε, ⟨n⟩ is independent of N.

SOLUTION

a) A partition function may be written as

(1)

where we have used the fact that a state with n open links has an energy ε_n = nε. So

(2)

where η ≡ e^-ε/r.

b) The average number of open links ⟨n⟩ is given by

(3)

If τ << ε then η << 1 and

which does not depend on N. It is also zipped up tight!