\sect{Tension of a stretched chain}

%Baruch's  A05, revised by DC.

A rubber band is modeled as a single chain of $N\gg 1$ massless non-interacting links,
each of fixed length $a$. Consider a one-dimensional model where the links are restricted 
to point parallel or anti-parallel to a given axis, while the endpoints are constraint 
to have a distance $X=(2n-N)a$, where $n$ is an integer. Later you are requested to use 
approximations that allow to regard $X$ as a continuous variable. Note that the body 
of the chain may extend beyond the length $X$, only its endpoints are fixed. 
In items (c,d) a spring is pushed between the two endpoints, such that the additional 
potential energy $-KX^2$ favors large $X$, and the system is released (i.e. $X$ is free to fluctuate).

\begin{center}
\includegraphics[scale=0.35,angle=0]{A05}
\end{center}

\begin{itemize}

\item[(a)] Calculate the partition function $Z(X)$.
Write the exact combinatorial expression. Explain 
how and why it is related trivially to the entropy $S(X)$. 

\item[(b)] Calculate the force $F(X)$ that the chain applies 
on the endpoints. Use the Stirling approximation 
for the derivatives of the factorials. 
  
\item[(c)] Determine the temperature $T_c$ below which 
the ${X=0}$ equilibrium state becomes unstable.

\item[(d)] For ${T<T_c}$ write an equation for the 
stable equilibrium distance $X(T)$. Find an explicit 
solution by expanding $F(X)$ in leading order.  

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%