Particle number fluctuations

In similarity with Sec. , we find that the average number of particles can be obtained as the derivative of the natural logarithm of the gran partition function w.r.t. the chemical potential, divided by $\beta$:

$$\bar{N} = \sum_N N p(N) = \frac{1}{\mathcal{Z}}\sum_N N \exp \{\beta N\mu\} Z(N) = \frac{1}{\beta}\frac{\partial \ln \mathcal{Z}}{\partial \mu}.$$ (3.52)

We see that $\ln \mathcal{Z}$ is extensive, because $\mu$ is intensive. The average quadratic fluctuations can be obtained from

$$(\delta N)^2 = \bar{N^2} - \bar{N}^2 = \frac{1}{\beta^2}\frac{\partial^2 \ln \mathcal{Z}}{\partial \mu^2},$$ (3.53)

and therefore $(\delta N)^2$ is also extensive, as it is proportional to $\ln \mathcal{Z}$, meaning that $\delta N\propto N^{1/2}$ and so $\delta N/N \propto N^{-1/2}$. For a macroscopic system this fraction is negligibly small and so the system can be regarded as having, effectively, a constant number of particles. For such a system averages taken in the gran canonical ensembles are therefore the same as those taken in the canonical or in the microcanonical ensembles.