Density Fluctuations

The goal of this section is to show that the fluctuations in the number of particles in the system at constant V and T can be computed from the grand canonical partition function and that the size of the fluctuations in the number of particles $\sqrt{\overline{(\delta N)^2}}$ decays as $1/\sqrt{\overline{N}}$. The ensemble average square fluctuation in the number of particles

$$\overline{(\delta N)^2} = \overline{(N - \overline{N})^2} = \overline{N^2} - \overline{N}^2,$$ (86)

(where the overline indicates an ensemble average), can be computed as follows

$$\overline{(\delta N)^2} = \sum_j p_j N_j^2 - (\sum_j p_j N_j)^2.$$ (87)

Substituting Eq. (80) into Eq. (87) we obtain

$$\overline{(\delta N)^2} = \Xi^{-1} \sum_j N_j^2 e^{-\beta E_j + \beta \mu N_j} - \left(\Xi^{-1} \sum N_j e^{-\beta E_j + \beta \mu N_j} \right)^2.$$ (88)

Therefore,

$$\overline{(\delta N)^2} = \frac{\partial^2 \text{ln} \Xi}{\partia... ...u)^2} \Bigg )_V = \frac{\partial \overline{N}}{\partial (\beta \mu)} \Bigg )_V.$$ (89)