\sect{Tension of a rubber band}

The elasticity of a rubber band can be described by a one
dimensional model of a polymer. The polymer cosnists 
of ${N}$ monomers that are arranged along a straight line,  
hence forming a chain. Each unit can be either in a state
${\alpha}$, with length ${a}$ and energy ${E_{a}}$, 
or in a state ${\beta}$, with length ${b}$ and energy ${E_{b}}$.

\Dn

(1) Write down the partition function ${Z_{G}({\beta},f)}$.

(2) Derive the relation between the length ${L}$ of the chain molecule and the tension ${f}$ applied at the ends of the molecule.

(3) Find the compressibility ${\chi_{T}=(\partial L/\partial f)_{T}}$.

(4) Describe the dependence of L and ${\chi_{T}}$ on ${(fa/T)}$. 

(5) Write down the partition function ${Z({\beta},L)}$. 
What is the probability function of $L$ for ${f=0}$ ?

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