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\sect{Fluctuations of $N$ in the grand canonical ensemble}

Show that
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\[\langle\triangle
N^{2}\rangle=\left(\frac{1}{\beta}\,\frac{\partial}{\partial\mu}\right)^{2}\ln
Z=T\left(\frac{\partial N}{\partial\mu}\right)_{TV}\]
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From that prove the equation
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\[\frac{\langle\triangle N^{2}\rangle}{\langle
N\rangle^{2}}=-\frac{T}{V^{2}}\left(\frac{\partial V}{\partial
{p}}\right)_{N,T}\]
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The last step demands manipulation of equations in thermodynamics

Hints:
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\[\left(\frac{\partial N}{\partial
\mu}\right)_{T,V}=\left(\frac{\partial \mu}{\partial
N}\right)_{TV}^{-1}\]
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\[d\mu=\nu d{p}+ SdT, \,\nu=\frac{V}{N},\, S=\frac{S}{N}\]
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\[\frac{\partial\mu}{\partial\nu}|_{T}=\nu\frac{\partial
{p}}{\partial\nu}|_{T}
\]