# Do ensemble averages correspond to the most probable state for interacting particles?

Most introductory statistical mechanics books state that in the thermodynamic limit, ensemble averages go towards the value that corresponds to the most probable state.

They justify this statement using an example of a non-interacting system where the spin/energy of each molecule/particle/spin are statistically identical and independent. The probability distribution for the sping/energy in question goes towards a Gaussian distribution according to the Law of Large Numbers (LLN) and the Central Limit Theorem (CLM).

Example source:

In most thermodynamic systems, molecules/particles/spins interact and are not statistically independent.

For systems with interacting molecules/spins, when can we still expect the ensemble average to correspond to the most probable state in the thermodynamic limit?

In the thermodynamic limit, macroscopic observables take on deterministic values under any extremal Gibbs state: both a suitable law of large numbers and suitable large deviation principles (with rate functions given by the relevant thermodynamic potential) hold generally. This is, for instance, discussed (for classical systems on a lattice) in Chapter 6 of our book and, in much greater generality, in Georgii's book (in particular, in Chapters 7 and 15).

Concerning typical fluctuations, they are usually of CLT type, although there are counterexamples, in particular at a critical point.

• Infinite-volume Gibbs measures is quite complicated so it will take me some time to get through your chapter. I might come back with a question or two. Dec 6, 2021 at 14:10
• If you find the setting of Chapter 6 to be too abstract, you might have a look at the more concrete Chapter 4 (about the lattice gas), in which we analyze, among other things, the typical particle density in the grand canonical ensemble and show that, away from phase transitions, the probability of observing a large deviation (that is, a macroscopic fluctuation away from the average value) is exponentially small in the size of the system, with rate essentially given by the free energy density (see Section 4.6). At a phase transition, things are in general more complicated(see Section 4.12.1). Dec 6, 2021 at 14:29
• The discussion in Chapter 4 is only for finite systems (sometimes taking the thermodynamic limit at the end) and limited to very specific systems and quantities, but the arguments are much closer to those routinely used by physicists (although more care is is taken with mathematical rigor). The setting of Chapter 6 allows a much more elegant treatment of infinite systems, without having to pass to the limit, and makes it possible to prove very general results. But this comes at the cost of having a much more abstract framework (and statements). Dec 6, 2021 at 14:35

I think that your statistical mechanics books hints at the 2. law of thermodynamics. "Entropy tends to increase".

A system probably changes towards more probable overall (macroscopic) states. The underlying assumption is that all microstates have the same probability. Entropy for an overall state is higher the more microstates we have in the system for that. The concepts are intended for large systems - like Avogadro's number or so.

In the light of this one would say that the peaks of your gaussian will have more underlying microstates.