# 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). – Sebastian Riese Jan 2 '16 at 22:06
• @SebastianRiese: That's the answer, isn't it? – CuriousOne Jan 3 '16 at 3:24
• wouldn't the two systems interchange energy anytime they would interchange volume? – Quantumwhisp Jan 3 '16 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. – Sebastian Riese Jan 3 '16 at 14:42
• I mean interchange in the way when you look at the microstates of the whole system. – Quantumwhisp Jan 6 '16 at 13:50