Practical difference between canonical and grand canonical ensembles I'm currently doing some calculations which require evaluating various standard thermal expectation values in the canonical ensemble (both bosons and fermions). Now, in order to make my theoretical machinations easier, I am actually using the grand canonical ensemble, where the chemical potential acts as a Lagrange multiplier enforcing the constraint $\langle \hat{N} \rangle = N$, where $N/V$ is the fixed density of the physical system. The justification for this is that the relative fluctuations in $\langle \hat{N} \rangle$ should vanish in the thermodynamic limit, in which case I expect fixing the average number to be physically equivalent to fixing the number once and for all. (Also, this approach seems to be adopted by a several presumably trustworthy references, see for example Simons & Altland Section 6.3.) This intuition seems reasonable, but I wonder if matters may be more subtle than this argument implies.

Do thermal averages in the thermodynamic limit of the grand canonical and canonical ensembles coincide?

I'm hoping for either a more rigorous justification supporting this procedure, or examples where it can go horribly wrong. Pointers to appropriate references would also be much appreciated.
 A: As you know, the thermodynamic limit requires the system to grow to infinite size while keeping the same density, which lets you neglect surface effects. It also requires the lack of long-range interactions so that distant parts can act independently. So, you need to neglect gravitational interactions, allow the system to be charge neutral, etc.
Even in the thermodynamic limit, the different ensembles can behave differently near phase transitions. For example, if you park the grand canonical ensemble at a liquid-gas boundary, then the system is free to be filled with liquid or gas or a mixture. So, you get a giant and non-negligible fluctuation in the system's energy and particle number. By comparison the phase transition is extended in the canonical ensemble, with a range over which you have a liquid-gas mixture. Another example is the boson condensate: once the condensate has occurred, the total particle number in the grand canonical ensemble has a geometric distribution (!) and so there are giant fluctuations.
There are probably some other issues but I can't think of them off the top of my head. Anyway, just avoid long-range interactions, extreme conditions, and critical phenomena, and your calculations should be fine.
