\sect{Ising in interaction with lattice gas}

Consider a one dimensional Ising model of spins $\sigma_{i}=\pm1$ 
labeled $i=1,2,3,...,M$, with periodic boundary condition. 
Between each two spins there is a site $n_i=0,1$ that can be occupied by an atom. 
If the atom is present the feromagnetic coupling 
is decreased from $J$ to $(1-\lambda)J$. 

\Dn

(1) Evaluate the partition sum assuming that there 
are $N$ atoms in the $M$ sites. Allow all configurations 
of spins and of atoms. Calculate the free energy $F$.

\Dn 

(2) If the atoms are stationary impurities one needs to evaluate the
free energy $F$ for some random configuration of the atoms. 
What is the entropy difference between the results?


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%