Equivalence of Gibbs ensembles for classical lattice systems

For lattice systems with many-particle absolutely summable interaction it is shown for all $\beta>0$ and $1>\rho>0$that the limiting generating functionals of the canonical and grand canonical ensembles satisfy the Bogolyubov equation and in this sense the ensembles are equivalent. For systems with binary interaction, it is shown that the Bogolyubov equation has several solutions for the parameters $(z,\beta)$ for which the one-to-one correspondence with the parameters $(\rho,\beta)$ is broken.