Talk:Grand canonical ensemble

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[edit] Dirac brackets

Not a physics expert, but is that formula missing a bracket? Also, which Hamiltonian are we talking about here? The Classical one or quantum one? Any help appreciated. Soo 17:39, 17 August 2005 (UTC)

These (partial brac-kets) are called Dirac brackets. Basically a bra or a ket denotes a quantum mechanical state. And the combination, in 'bra'|'ket' order represents the overlap between two states. Statistical mecahnics aims at summing over all nonoverlapping states. Density matrix is often described as a sum over a series of 'ket'-'bra's. However I personally feel it is quite a convoluted way to define grand canonical ensemble. (It's more like g.c.e. provides a meaning of a mixed state (or an ensemble) described by a density matrix.) It will be a much longer story if I explain Dirac brackets in details, sorry. Please consult some quantum mechanics books. Czhangrice 21:51, 23 May 2007 (UTC)

[edit] re merge tag

i suggest no merge. although no one has found the time to expand this stub, the grand canonical ensemble, like the microcanonical ensemble and canonical ensemble, certainly deserves its own page. Mct mht 02:01, 24 August 2006 (UTC)

I agree with the above comment. --HappyCamper 05:00, 25 August 2006 (UTC)
Similar agreement. - Mostly anonymous student: 128.146.34.189 08:32, 16 November 2006 (UTC)

[edit] Partition sum

My physics book gives the partion sum as

\sum_N\sum_i e^{-\beta(E_i -\mu N)}

where N is the number of particles of the system and runs from 0 to infinity. 82.135.75.113 20:14, 6 April 2007 (UTC)

Your textbook is right. The mentioned formula is wrong.
This problem is fixed. Czhangrice 21:32, 23 May 2007 (UTC)

[edit] Ornstein-Zernike

The link seems to be of no relevance for me. As far as I see - I am not a specialist in it - it deals with some rather special questions concerning th canonical(!?) ensemble. In any case the referenced articles does not improve the understanding of thermodynamical potentials at all.


Grand canonical ensemble is roughly a bag of canonical ensembles. In g.c.e, system exchanges particles and energy with the environment (or other systems in the ensemble); In c.e, system only exchanges energy with the environment. I hope that I have made clear upon these points. I didn't see too much relevance of the reference either. Czhangrice 21:37, 23 May 2007 (UTC)

[edit] Logarithm of grand canonical partition function

Does anyone know if there is an official name to the logarithm of the grand canonical partition function? I ask because the log of other partition functions have specific names. Notably, the log of the canonical partition function is (up to a factor of temperature) the free energy while the log of the density of states(essentially the partition function of the microcanonical ensemble) is the entropy. Moreover, one could imagine defining a statistical ensemble at fixed pressure and temperature. The log of the associated partition function would then be the enthalpy. So does anyone have any idea about this? Joshua Davis 21:19, 5 July 2007 (UTC)

Yes, take a look at characteristic state function. So, for this case (grand canonical ensemble), it's related to pressure-volume work. --HappyCamper 23:46, 5 July 2007 (UTC)
Thanks. I'm afraid this isn't quite what I had in mind, though. In particular, one doesn't define the pressure-volume work as the log of the grand canonical partition function. Rather, for extensive systems there is the identity, E=TS+PV+\mu N \,\; (I may have gotten some signs wrong and I'm assuming only one chemical species for simplicity). So if you do the appropriate Legendre(or Laplace) transforms to get to grand canonical ensemble, then you end up with \log \Xi \sim PV \,\;. But for non-extensive systems(black holes, for example), this doesn't hold, I think. Similarly, the Gibbs free energy for extensive systems satisfies G = \mu N \,\;, but one doesn't define it this way. Joshua Davis 04:22, 6 July 2007 (UTC)