Super Grand Canonical Ensemble

[qbslug]

Can we take the grand canonical ensemble and then switch the roles of the thermodynamic conjugate variable pair (P, V) making P (pressure) the parameter and V (volume) the variable and allowing it to fluctuate in the system. The macrostate would then be defined by the pressure temperature and chemical potential allowing the variables of energy, number of particles and volume to fluctuate. We can call this the "Super Grand Canonical Ensemble".

Why is there no super grand canonical ensemble? Is it because in the grand canonical ensemble the volume is an imaginary boundary (and thus user defined anyway). Or perhaps the super grand canonical ensemble gives no new information or has no thermodynamic potential to work with? Why no concern for the switching the PV pair?

 

[DrDu]

This ensemble exists, but for some reason, it hasn't got a sticky name. I think it is often abbreviated as J or Omega. It's differential yields the Gibbs Duhem relation, i.e. it vanishes. Abrikosov, Gorkov, Dzyaloshinskii, Quantum field theoretical methods in statistical physics (2ed., Pergamon, 1965) make some use of it and I have it seen discussed also in other texts on statistical mechanics, however I don't remember where.

 

[|squeezed>]

I think you 'll find it in Mandl's "Statistical Physics". Landau also uses Ω=-PV while Pathria sticks to PV.

 

[DrDu]

I think I have been wrong. The potential Omega I had in mind is simply the grand potential. I believe the super-grand-potential doesn't exist for the following reason: It depends only on intensive quantities and thus cannot be an extensive quantity. In fact it vanishes identically. E.g. $$G=\sum_i \mu_i N_i$$ and forming $$G-\sum_i \mu_i N_i=0$$.

 

[pmacias]

I would like to address the concept of the "Super Grand Canonical Ensemble" proposed in the content. While it may seem like an interesting idea to switch the roles of the thermodynamic conjugate variable pair (P, V) in the grand canonical ensemble, there are several issues that need to be considered.

Firstly, the grand canonical ensemble is a well-established and widely used ensemble in statistical mechanics, which allows for fluctuations in the number of particles, energy, and volume while keeping temperature and chemical potential constant. Switching the roles of the P and V variables would essentially create a new ensemble, which may not have a well-defined physical meaning. It is important to note that ensembles in statistical mechanics are not arbitrary choices, but rather have a physical significance and are derived from fundamental principles.

Moreover, the grand canonical ensemble already takes into account fluctuations in the volume through the use of an imaginary boundary. This boundary is not user-defined, but rather a mathematical construct that allows for the inclusion of volume fluctuations in the ensemble. Therefore, the proposed "Super Grand Canonical Ensemble" would not provide any new information or insights that are not already accounted for in the grand canonical ensemble.

Furthermore, in thermodynamics, the choice of the thermodynamic variables is determined by the thermodynamic potentials, such as the Helmholtz free energy or the Gibbs free energy. In the grand canonical ensemble, the choice of variables (N, V, E) is consistent with the Helmholtz free energy, while in the proposed "Super Grand Canonical Ensemble", the choice of variables (N, P, E) would not correspond to any known thermodynamic potential. This lack of a thermodynamic potential would make it difficult to derive meaningful relationships and equations for this ensemble.

In conclusion, while the idea of a "Super Grand Canonical Ensemble" may seem intriguing, there are fundamental issues that need to be addressed before considering its validity. The grand canonical ensemble is a well-established and physically meaningful ensemble, and there is no need to switch the roles of the P and V variables.