===== Particle number expectation value =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $ w $ ... grand canonical weight |
| @#FFBB00: definiendum | @#FFBB00: $ \langle\hat N\rangle(\beta,\mu) := \sum_{N=0}^\infty w_N(\beta,\mu)\cdot N $  |

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=== 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 ===
  * Given the [[grand potential]] $\Omega(\beta,\mu)$, we find

^ $  \langle\hat N\rangle = - \frac{\partial}{\partial\mu}\Omega $ ^

  * For the deviation of the particle number, we find

^ $\frac{1}{\beta}\frac{\partial}{\partial\mu}\langle\hat N\rangle = \langle {\hat N}^2\rangle-\langle\hat N\rangle^2$ ^

  * The entry [[grand canonical partition function]] shortly discusses free bosons and fermions. 

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=== Context ===
[[Grand canonical weight]]