Chapter 7.1, Problem 6P

Show that when a system is in thermal and diffusive equilibrium with a reservoir, the average number of particles in the system is $\bar{N}=\frac{kT}{Z}\frac{\partial Z}{\partial \mu }$, where the partial derivative is taken at fixed temperature and volume. Show also that the mean square number of particles is $\overline{N^2}=\frac{{\left(kT\right)}^2}{Z}\frac{{\partial }^{2}Z}{\partial {\mu }^2}$

Use these results to show that the standard deviation of N is ${\sigma }_N=\sqrt{kT\left(\partial \bar{N}\right)/\partial \mu }$, in analogy with Problem 6.18. Finally, apply this formula to an ideal gas, to obtain a simple expression for ${\sigma }_N$ in terms of $\bar{N}$. Discuss your result briefly.