The (NTp) ensemble
\vspace{15mm}
\noindent
The phase-points are $(\vec{r},\vec{p},V)$.
\begin{eqnarray}
\sum \theta_i & = & 1 \nonumber\\
\sum H_i \theta_i & = & \langle E \rangle\nonumber\\
\sum V_i \theta_i & = & \langle V \rangle \nonumber
\end{eqnarray}
The entropy is maximized:
\begin{displaymath} \ln( \theta ) = 1 + \alpha + \beta E + \gamma V =0
\end{displaymath}
where
$\alpha, \beta$, and $\gamma$ are Lagrange multipliers.
\noindent
By standard arguments
\begin{eqnarray}
Q & = & e^{1+\alpha} \nonumber\\
\beta & = & \frac{1}{k_{B}T} \nonumber\\
\gamma & = & \frac{p}{k_{B}T} \nonumber
\end{eqnarray}
\noindent
The partition function is
\begin{displaymath}
Q(N,T,p) = \sum \exp(-\frac{pV+H_i}{k_BT})
\end{displaymath}
By differentiation of
\begin{displaymath}
Q = \sum H_i \exp(-\frac{H- pV}{k_BT})
\end{displaymath}
by T:
\begin{displaymath}
-^2 = k_{B}T^{2}C^{'}_{p} + pV\beta
\end{displaymath}
where $C^{'}_{p}$ is the heat capacity caused by the {\em potential} energy
and $\beta$ is the thermal expansion
\noindent
By differentiation of
\begin{displaymath}
Q \langle V \rangle = \sum V^{N+1} \sum \exp( -\frac{H_i - pV_i}{k_BT} )
\end{displaymath}
by V
\begin{displaymath}
\langle V^2 \rangle - \langle V \rangle^2 = \langle V \rangle \kappa k_B T
\end{displaymath}
where $\kappa$ is the isothermal compressibility
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Author
Per Stoltze
stoltze@fysik.dtu.dk