\sect{The zipper model for DNA molecule}

The DNA molecule forms a double stranded helix with hydrogen
bonds stabilizing the double helix. Under certain conditions the
two strands get separated resulting in a sharp "phase transition"
in the thermodynamic limit. As a model for this unwinding, 
use the "zipper model" where the DNA is modeled as a polymer 
with ${N}$ parallel links that can be opened from one end (see figure).

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The energy cost of an open link is $\varepsilon$.
A possible state of the DNA is having links ${1, 2, 3, ..., p}$ open, 
and the rest are closed. The last link cannot be opened. 
Each open link can have ${g}$ orientations, 
corresponding to the rotational freedom about the bond.
Assume a large number of links $N$.

(1) Define $x=ge^{-\varepsilon/T}$ and find the canonical partition function $Z(\beta,x)$.

(2) Find the average number of open links $\langle p \rangle $ as a function of $x$.

(3) Find the linear approximation for $\langle p \rangle $.

(4) Approximate $\frac{\langle p \rangle }{N}$ for large $x$.

(5) Describe the dependence of $\frac{\langle p \rangle }{N}$ on $x$.

(6) Find expressions for the entropy $S(x)$ and the heat capacity $C(x)$ at $x=1$.

(7) What is the order of the phase transition?

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