The equivalence of ensembles for classical systems of particles

For systems of particles in classical phase space with standard Hamiltonian, we consider (spatially averaged) microcanonical Gibbs distributions in finite boxes. We show that infinite-volume limits along suitable subsequences exist and are grand canonical Gibbs measures. On the way, we establish a variational formula for the thermodynamic entropy density, as well as a variational characterization of grand canonical Gibbs measures.