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\sect{Ising spins with interaction that is mediated by atoms}

Consider a one dimensional Ising model of spins ${\sigma_{i}=±1}$,
where ${i=1,2,3,...,N}$ and ${\sigma_{N+1}=\sigma_{1}}$. Between
each two spins there is a site for an additional atom, which if
present changes the coupling ${J}$ to ${J\left(1-\lambda\right)}$.

The Hamiltonian is then ${H = -J \Sigma_{i}\sigma_{i}+1
\left(1-\lambda n_{i}\right)}$,  where ${n_{i}=0}$ or ${1}$.
There are
${N' = \Sigma n_{i}}$  atoms, so that  ${N' < N}$.   Evaluate the
partition sum by allowing all configurations of spins and of atoms.
If the atoms are stationary impurities one needs to evaluate the
free energy ${F}$ for some given random configuration of the atoms:
Then one can average ${F}$ over all configurations. Evaluate the
averaged ${F}$. Find the entropy difference between the two results
and explain its origin.



