Can a Super Grand Canonical Ensemble Exist?

[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 $$G-\sum_i \mu_i N_i=0$$.