2.3. Grand canonical partition function Consider a one-dimensional lattice gas on M sites. The variable ti has values 0 and 1, according whether the relative site is occupied or not. The Hamiltonian is given by H = −J M



i=1 ti ti+1 with ti+M = ti . Moreover, we assume that M



i=1 ti = N

a. Compute the grand canonical partition function Z(z,M,T) of such a system.

b. Show that for J > 0 the zeros of Z(z,M,T) are along the circle | z |= e−J of the complex plane of the variable z, where J = βJ.