\sect{Baruch's A20.}
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" consisting of ${N}$ parallel links which can be
opened from one end (see figure).
\begin{center}
\includegraphics[scale=0.7]{A20.eps}
\end{center}
If the links ${1, 2, 3, ..., p}$ are all open the energy to open
to ${p+1}$ link is $\epsilon$ and if the earlier links are closed
the energy to open the link is infinity. The last link ${p=N}$
cannot be opened. Each open link can assume ${G}$ orientations
corresponding to the rotational freedom about the bond.
\begin{itemize}
\item [(a)]
Construct the canonical partition function. Find then the average
number of open links ${\langle p\rangle}$ as function of $x=G e^{
-\varepsilon/k_{B}T}$. Show that
\[ \langle p
\rangle=\half N[1+\frac{1}{6}N(x-1)+O(N^{3}(x-1)^{3})]\] so that
the slope $d\langle p\rangle /dx\sim N^2$ at $x=1$. Plot ${\langle
p\rangle/N}$ schematically as function of ${x}$ for large but
finite ${N}$ and for ${N\rightarrow\infty}$ .
\item [(b)]
Derive
the entropy ${S}$ and the heat capacity ${C_{V}}$ at ${x=1}$ for
large but finite ${N}$ and plot ${S\left(x\right)}$ and ${C_{V}
\left(x\right)}$ for ${N\rightarrow\infty}$ . What is the order of
the phase transition?\\
\end{itemize}
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