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\sect{Fluctuations in the grand canonical ensemble}

A fluid in a volume ${V}$ is held (by a huge reservoir) at a
temperature ${T}$ and chemical potential ${\mu}$. Do not assume an
ideal gas. Find the relation between ${\langle \left(E-\langle
E\rangle\right)^{3}\rangle}$ and the heat capacity
${C_{V}\left(T,z\right)}$ at constant fugacity ${z}$. Find the
relation between ${\langle\left(N-\langle
N\rangle\right)^{3}\rangle}$ and the isothermal compressibility
${\chi T\left(V,\mu \right)= -\left(\partial v/ \partial \mu
\right)|_{V,T}}$ where ${v=V/\langle N\rangle}$. [Hint: Evaluate 3rd
derivatives of the grand canonical partition function.] Find
explicitly results in case of a classical ideal gas.







