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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.

Urb
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Quantumwhisp
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    Implementation: An insulated system with a piston that ensures constant pressure (i.e. with a weight on top). – Sebastian Riese Jan 02 '16 at 22:06
  • @SebastianRiese: That's the answer, isn't it? – CuriousOne Jan 03 '16 at 03:24
  • wouldn't the two systems interchange energy anytime they would interchange volume? – Quantumwhisp Jan 03 '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 03 '16 at 14:42
  • I mean interchange in the way when you look at the microstates of the whole system. – Quantumwhisp Jan 06 '16 at 13:50
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    Connecting two vessels with a movable wall is called 'the adiabatic piston problem' which somehow is still somewhat controversial https://physics.stackexchange.com/questions/257815/thermodynamic-equilibrium-will-the-piston-move/258592 – lalala Jul 30 '17 at 15:54
  • Microcanonical ensemble has constant energy. – Roger V. Sep 24 '21 at 05:02
  • @SebastianRiese, why do you say that $E$ can't be a natural variable? It is definitely a natural variable for the microcanonical ensemble. – SmallPieceOfBread Jan 24 '22 at 07:47
  • No it's not. $E(S, V, N)$ is the thermodynamic potential of the microcanonical ensemble, the natural variables are $S$, $V$, and $N$. – Sebastian Riese Jan 24 '22 at 11:56
  • @SebastianRiese kann you give the conditions the "natural variables" do have to satisfie? I can easily think of a way to set up a system with fixed energy, but a system with fixed entropy? Bugs me. – Quantumwhisp Jan 24 '22 at 13:16
  • Yes, that is certainly the case, it boils down to definitions (not really physical content). It is quite possible that other nomenclature is also common. The definition I am familiar with is that the natural variables are the arguments of the thermodynamic potentials that can be constructed by Legendre trafos from the free energy $F$. ... – Sebastian Riese Jan 24 '22 at 19:55
  • So yes, you prepare the microcanonical ensemble with a fixed energy, but when you then take $S$ as thermodynamic potential you end up with the "wrong" variables E.g. $\partial_E S = 1/T$, but $\partial_S E = T$.. So you define $E(S,V,N)$ as the thermodynamic potential of the ensemble and the natural variables to make the connection to the canonical ensemble (and classical thermodynamics) – Sebastian Riese Jan 24 '22 at 19:57

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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. – Quantumwhisp Aug 02 '17 at 05:43
  • Are you asking about systems of few particles? –  Aug 02 '17 at 12:49