Statement of the problem:
Consider a polymer chain consisting of $N$ linked monomer units with $N\gg 1$. Every monomer can be in state $\alpha$ or $\beta$ with energies $E_\alpha$ and $E_\beta$. In configuration $\alpha$ a monomer has length $l-a$ and length $l+a$ in configuration $\beta$. The system is in a thermal reservoir with temperature $T$. One end of the chain is fixed at location $x=0$ and the other and is attached to a particle attached to a spring with potential energy $U=\frac{1}{2}\kappa (x-L_0)^2$. Find the length of a chain in an equilibrium $L(T)$ using the canonical ensemble.
HINT: Use the thermodynamical limit to calculate the partition function as an integral instead of a sum. Enumerate the integral using the saddle point method.
My attempt at solution: Let m be the number of monomers in a state a. Hamiltonian of the system is $$H=mE_\alpha + (N-m)E_\beta+\kappa L(L-L_0).$$ You cannot just split the partition sum into a product of partition sums of each monomer because the energy of each monomer depends on the whole configuration and hence they are not independent. So im thinking about maybe using the length L of the whole chain as a continuous variable in the thermodynamical limit as H depends only on L. However I am stuck at this point a I have no idea how to actually do it. When Im trying to find the density of states in terms of L theres a factor of ${N}\choose{m}$ (theres this many possible states for each set Length) causes a problem. Thank you for your help.
