Particle number expectation value


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


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.


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


Link to graph
Log In
Improvements of the human condition