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[QUE/SM-06001] SM-PROBLEM

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{Consider an open system of non-interacting particles obeying Maxwell-Boltzmann statistics. The system is at temperature $T$ and chemical potential $\mu$. Let $\sigma^2_N$ denote the variance of the number of particles in the system. Show that \begin{eqnarray*} \sigma^2_N &=& k_BT\left( \frac{\partial\langle N\rangle}{\partial\mu}\right)_{T,V}, \end{eqnarray*} where $\langle N\rangle$ is the mean number of particles. The angular brackets denote averaging is carried out over a grand canonical ensemble

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