# Existence of statistical ensemble with fixed energy but varying volume

To me, every statistical ensemble in statistical physics was introduced beginning with the microcanonical ensemble, in which every microstate is equally probable. A canonical ensemble is described by combining two ensembles, who together shall form a microcanonical ensemble. The microstates in system 1 shall then form the canonical ensemble, system 2 is said to be large compared to system 1, so that it's temperature $$T = \frac{\partial S}{\partial E}$$ doesn't change when the two systems interchange energy. If one still requires every possible microstate of the whole system to be equally probable, then the probabilities for microstates $$\Gamma$$ in the small system scale with a factor $$e^{-\frac{H(\Gamma)}{k_b T}}$$. The canonical ensemble is no longer described by its energy, but by its temperature ( and volume V, particle number N ....)

My question: Is there also a way to describe an ensemble that has a fixed Energy, E, but varying Volume, that means, a EpN Ensemble, or a SpN Ensemble? That would mean that I look at two systems that can interchange Volume, but don't interchange energy, in the same way I described it above for the canonical ensemble.

I am asking because the Enthalpie H(S,p,N) exists, and is a thermodynamical potential that depends on S,p,N, which suggests that such an ensemble exists.

• Implementation: An insulated system with a piston that ensures constant pressure (i.e. with a weight on top). Jan 2, 2016 at 22:06
• @SebastianRiese: That's the answer, isn't it? Jan 3, 2016 at 3:24
• wouldn't the two systems interchange energy anytime they would interchange volume? Jan 3, 2016 at 11:27
• Well, technically, it is an $S, p, N$ ensemble, not an $E, p, N$ ensemble, and the entropy remains constant (in both systems). $E$ can't be a natural variable of a thermodynamic ensemble. Jan 3, 2016 at 14:42
• Connecting two vessels with a movable wall is called 'the adiabatic piston problem' which somehow is still somewhat controversial physics.stackexchange.com/questions/257815/… Jul 30, 2017 at 15:54

## 1 Answer

Yes. It is called the Isoenthalpic-isobaric Ensemble. See my answer for more details, but remember that moving between different ensembles is as simple as a Laplace transform or a Legendre transform (that answer will show more details). So you can really construct as many "ensembles" as you have Thermodynamic variables to do so.

• "Moving between different ensembles is as simple as a Laplace transform or Legendre transform". This is only true for the thermodynamic limit. Aug 2, 2017 at 5:43
• Are you asking about systems of few particles? Aug 2, 2017 at 12:49