# 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.

• Related StackExchange questions: this, this, this and possibly this. – Mark Mitchison Oct 8 '13 at 15:57
• For a specified problem , at fixed temperature, $\langle \hat{N} \rangle$ depends on the chemical potential. For perfect fermion gas, perfect boson gas, at fixed temperature, one find that $\langle \hat{N} \rangle/V$ depends on the chemical potential (except in the case of photons, where the chemical potential is zero) – Trimok Oct 8 '13 at 16:53
• @Trimok Right, sorry, should have been clearer that I want the density to be fixed. Ultimately $N$ will always appear in the combination $N/V$ anyway. Have edited accordingly. The reason I phrased it thus is that experimentally these systems will obviously have finite $N$ and $V$. It's simply that these quantities will be large enough that I expect the thermodynamic limit to predict the correct behaviour. – Mark Mitchison Oct 8 '13 at 16:56
• @Trimok Also, I don't really understand the purpose of your comment. Are you saying what I'm doing does/doesn't make sense? Or just making an observation? – Mark Mitchison Oct 8 '13 at 17:16
• I just noted that, at fixed temperature, fixing $\langle \hat{N} \rangle/V$ means fixing the chemical potential, but it is not in contradiction with the disparition of relative fluctuations for large $N$ or large $V$. – Trimok Oct 8 '13 at 17:30

• Thanks. I understand the issue with critical phenomena. Regarding long-range interactions I understand why the thermodynamic limit does not formally exist due to instability. However I don't see why specifically long-range interactions would affect the fundamental issue: the number fluctuations. Is it not the case that when $N$ is extremely large (but not formally infinite) expectation values calculated in the two ensembles should converge? For example, how about large finite-temperature systems that are not charge neutral, but appropriately confined so they don't just explode? – Mark Mitchison Oct 8 '13 at 16:51