## Particle number expectation value

### Set

 context $w$ … grand canonical weight definiendum $\langle\hat N\rangle(\beta,\mu) := \sum_{N=0}^\infty w_N(\beta,\mu)\cdot N$

#### Discussion

The notation “$\langle\hat N\rangle$” is chosen for the function because we can also introduce the sequence of observables $\hat N$ defined to give us the particle number of each canonical ensemble, i.e. $\hat N_N=N$, and then the above coincides with the proper grand canonical expectation value of $\hat N$. Notice that this $\hat N$ is sometimes denoted by $N$, which can get a little confusing.

#### Theorems

$\langle\hat N\rangle = - \frac{\partial}{\partial\mu}\Omega$
• For the deviation of the particle number, we find
$\frac{1}{\beta}\frac{\partial}{\partial\mu}\langle\hat N\rangle = \langle {\hat N}^2\rangle-\langle\hat N\rangle^2$