# Statistical ensembles: equivalence in the thermodynami limit and connections

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?

• @Qmechanic : The OP did use the mathematical-physics tag. Any good reason for removing it? (Plus the OP is a math undergraduate, so it's actually likely that it is what he wants...). Jun 27, 2019 at 16:55
• @YvanVelenik: The MP tag is the most misused tag on Phys.SE. Put it back in if you think it is warranted. Jun 27, 2019 at 17:01
• @Francesco Bilotta, " any time we let a new state variable vary we transform the distribution... What is the meaning of these transforms?" We use the method of uncertain Lagrange multipliers: we are looking for the maximum entropy, fixing the average value of this state variable. Naturally, when this (extensive) state variable tends to infinity (with an increase in the number of particles), the average coincides with the total value. I think this is an intuitive meaning of the equivalence Jun 29, 2019 at 14:27

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.

• Thanks for the thorough answer and reference! I have some background in measure theory so I will indeed read on this. And physically, what interpretation and intuition one can attach to this equivalence? Jun 28, 2019 at 7:00
• I am not sure what you mean by physical intuition. On the one hand, this is just the statistical mechanics counterpart of the fact that, in thermodynamics, you have many equivalent ways of choosing the set of thermodynamic parameters that you use to describe the state of the system. On the other hand, in statistical mechanics, equivalence of ensembles (at the level of thermodynamic potentials) is just a manifestation of the fact that different ensembles are related via Laplace transforms, since this implies that the associated thermodynamic potentials are related via Legendre transforms. Jun 28, 2019 at 7:44
• @FrancescoBilotta If you wish to understand why the ensembles are given by measures related via Laplace transforms, then the easiest way is through their derivation using maximum entropy. A (hopefully) pedagogical introduction to this can be found in Section 1.2 of our book, while a discussion of equivalence of ensembles (at the level of thermodynamic potentials) can be found, for the lattice gas, in Section 4.4. Jun 28, 2019 at 7:51