I am studying statistical ensembles (microcanonical, canonical and grand canonical) and I would like to better understand how are linked.
It is clear to me that the microcanonical ensemble is somehow the fundamental seed from which we can derive the other two. Often in my course, though, we spoke of equivalence between these three ensembles, in the thermodynamic limit, i.e. for a number of particles tendig to infinity. As a justification we noticed how they indeed generate the same Sackur-Tetrode equation for entropy in the case of an ideal gas, for instance. However I would like to know more about this equivalence and what, physically, motivates it. It seems to me that the three ensembles describe very different systems (isolated, closed, particle interchanging)
Another thing I noticed is that the probability distribution of the canonical is obtained as a Laplace transform of the microcanonical one with respect to energy and that the grand canionical as a Z-transform of the canonical one, with respect to the number of particles. In some sense, any time we let a new state variable vary we transform the distribution. Is there more to this idea? What is the physical meaning of these transforms?