\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). \begin{center} \includegraphics[scale=0.7]{A20.eps} \end{center} 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? %%begin{figure} \putgraph{Ex234} %%\caption{}\label{} %%end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%