\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. \Dn (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. \Dn (6) Treating $L$ as a continuous variable, find the probability distribution $P(L)$, assuming $E_a=E_b$ and ${f=0}$. \Dn (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. \Dn (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. \Dn (9) Find what would be the results for $L(T,f)$ in the latter case. \Dn {\em Note:} Above a "linear function" means ${y=Ax+B}$. \\ Please express all results using $(N,a,b,E_a,E_b,f,T,L)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%