# $\alpha$ in the Grand Canonical Partition Function

When moving from the Canonical Partition Function to the Grand Canonical Partition function, we relax the restriction on the number of particles N by introducing the sum $$\mathcal{Z} = \sum_{N'=0}^\infty Z(N') e^{-\alpha N'}$$ where alpha is a parameter to 'tune' the number of particles to N (the experimentally controlled value). In the derivation done in my lecture we started with $\alpha > 0$, and showed (using some approximations) that $$\ln(Z(N)) = \alpha N + \ln\mathcal{Z}$$ Now it was determined using the Free Energy F that we can identify $\alpha$ as $$\alpha = -\beta \mu$$ where $\mu$ is the chemical potential and $\beta=\frac{1}{k_BT}$. Now since temperature is positive, I would think that this identification of $\alpha$ would fix the sign of $\mu$. When considering quantum statistical distributions, however, in the Fermi-Dirac case, we see that it is possible for $\mu$ to be positive, in the form of the Fermi Energy. $\textbf{Is this a contradiction?}$ This does not arise in the Bose-Einstein case since $\mu$ is always negative. What am I missing?

• Generally $\mu$ can be positive or negative. Your lecturer seems to have chosen to introduce the grand canonical ensemble in a way that is not the most general possible, but may have been a lot simpler. My guess is that by assuming that $\alpha > 0$ allows him to look at the first sum you wrote down and say "this will probably converge" without thinking too hard about it, although any rigorous treatment would have to consider what happens to the $Z(N')$ terms. – By Symmetry Jun 9 '17 at 15:42
• Could you recommend a place where I could find the more general treatment? – Sreekar Voleti Jun 9 '17 at 15:43
• K. Huang, Statistical Mechanics. Or M. Tuckerman, same title. – valerio Jun 10 '17 at 7:40