
\sect{Adiabatic law for generalized dispersion}

%Baruch's  A08

Consider a gas of noninteracting particles with kinetic
energy of the form
$\varepsilon ({\bf p})= \alpha|{\bf p}|^{3(\gamma - 1)}$ where ${\alpha}$
is a constant; ${\bf p}$
is the momentum quantized in a box of size ${L^{3}}$ by
${p_{x}=hn_{x}/L, p_{y}=hn_{y}/L, p_{z}=hn_{z}/L}$ with ${n_{x},
n_{y}, n_{z}}$ integers. Examples are nonrelativistic particles with
${\gamma =5/3}$ and extreme relativistic particles
with ${\gamma  =4/3}$.
\begin{itemize}
\item [(a)]
Use the microcanonical ensemble to show that in an adiabatic process
(i.e. constant ${S, N}$)  ${PV^{\gamma}}$ =const.
\item [(b)]
Deduce from (a) that the energy is           ${E=Nk_{B}T/\left(\gamma-
1\right)}$
and the entropy is ${S = \frac{k_{B}N}{\gamma-  1}  \ln\left(PV^{\gamma} \right) + f \left(N\right) }$.
What is the most general form of the function f(N)?
\item [(c)]
Show that $C_p/C_v =  \gamma$.
\item [(d)]
Repeat (a) by using the canonical ensemble.\\
\end{itemize}

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