
\sect{Fluctuations in the grand canonical ensemble}

%Baruch's  A21

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.
\begin{itemize}
\item [(a)]

Find the relation between $\langle (E -\langle E\rangle)^3\rangle$
and the heat capacity $C_V(T,\zeta)$ at constant fugacity $\zeta$.

\item [(b)]
Find the relation between $\langle(N- \langle N\rangle)^3\rangle$
and the isothermal compressibility
$k_{T}(V,\mu)=-(\partial v/\partial\mu)_{V,T}$ where $v=V/\langle
N\rangle$

[Hint: Evaluate 3rd derivatives of the grand canonical partition function.]

\item [(c)]

Find (a) and (b) explicitly for a classical ideal gas.\\

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
