Statistical ensembles: equivalence in the thermodynami limit and connections

Question

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?

Answer 1

Since you used the mathematical-physics tag, I'll assume that you're after rigorous results.

The strongest form of equivalence of ensembles is at the level of probability measures. What is usually done is to consider the ensemble with less constraints as a part of a bigger ensemble in which additional constraints are enforced. I'll discuss the equivalence between microcanonical and grand canonical, but other cases are treated similarly.

In this case, you consider a system in a large volume $V$ with fixed total energy $E$ and total number of particles $N$; the corresponding probability measure $\mu^{\rm micro}_V$ is uniform (microcanonical). You then consider a part $\Delta\subset V$ of this large system; let us denote the marginal of $\mu^{\rm micro}_V$ in $\Delta$ by $\mu_{V,\Delta}$. You then let $V$ increase to infinity while keeping both the energy density $e=E/V$ and particle density $\rho=N/V$ constant. Let $\mu_\Delta^{e,\rho} = \lim_{V\to\infty} \mu_{V,\Delta}$ be the limiting marginal ($\Delta$ is kept fixed). We then take the limit as $\Delta$ tends to infinity. The limiting measure $\mu_\Delta^{\rm grand} = \lim_{\Delta\to\infty} \mu_\Delta^{e,\rho}$ can then be shown to be the grand canonical (Gibbs) measure at temperature $T$ and chemical potential $\mu$, where the latter are the thermodynamic quantities conjugate to $e$ and $\rho$.

Such results hold rather generally, provided that the system is not at a phase transition (otherwise they are generally false, and at least require a more specific formulation).

Note that equivalence of ensembles also fails for models with very long-range interactions, such as mean field models. Actually, thermodynamic potential usually fail to be convex in such cases, so that even equivalence of ensemble at the level of thermodynamic potentials does not hold.

There are many places where you can read about this. For example, this paper is nice and accessible (for the mathematically inclined). There are other good papers aimed at physicists discussing these issues; for example this one. This older extremely influential paper also offers a beautiful discussion.