Equivalence of ensembles

Consider a system of particles at a given energy, isolated from the environment, and a second identical system at thermal equilibrium with the environment. If only the energy is considered to be relevant, the maximum entropy principle provides us with two probability distributions on phase space for these two models. The former is called the micro-canonical ensemble, the latter the canonical ensemble. The problem of equivalence of ensembles is whether in the thermodynamic limit small portions of the two ensembles become indistinguishable, provided that the temperature is tuned appropriately. I will show how relative entropy is used to simplify the question, both from the mathematical and physical perspective. Examples will be offered.