\sect{Tension of a rubber band, instability criterion}

The elasticity of a rubber band can be described by a one
dimensional model of a polymer. The polymer consists 
of ${N}$ monomers that are arranged along a straight line,  
hence forming a chain. Each unit can be either in 
a state of length ${a}$ with energy ${E_{a}}$, 
or in a state of length ${b}$ with energy ${E_{b}}$.
We define $f$ as the tension, i.e. the force that is 
applied while holding the polymer in equilibrium. 


\Dn

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

\Dn

(2) For very high temperatures $F_G(T,f) \approx F_G^{(\infty)}(T,f)$, 
where $F_G^{(\infty)}(T,f)$ is a linear function of $T$. 
Write the explicit expression for $F_G^{(\infty)}(T,f)$. 

\Dn

(3) Write the expression for ${F_G(T,f) - F_G^{(\infty)}(T,f)}$.
Hint: this expression is quite simple -  
within this expression $f$ should appear 
only once in a linear combination with other parameters.

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(4) Derive an expression for the length ${L}$ of the polymer 
at thermal equilibrium, given the tension ${f}$. 
Write two separate expressions: one for the infinite temperature 
result $L(\infty,f)$ and one for the difference $L(T,f)-L(\infty,f)$. 

\Dn 

(5) Assuming ${E_a=E_b}$, write a linear approximation for 
the function ${L(T,f)}$ in the limit of weak tension. 

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(6) Treating $L$ as a continuous variable, 
find the probability distribution $P(L)$, 
assuming $E_a=E_b$ and ${f=0}$. 

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(7) Write an expression that relates the function $f(L)$ 
to the probability distribution $P(L)$.
Write also the result that you get from this expression.

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(8) Find what would be the results for $Z_G(\beta,f)$ 
if the monomer could have any length ${\in [a,b]}$. 
Assume that the energy of the monomer is independent of its length.

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(9) Find what would be the results for $L(T,f)$ in the latter case. 

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{\em Note:} Above a "linear function" means ${y=Ax+B}$. \\
Please express all results using $(N,a,b,E_a,E_b,f,T,L)$.


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