
\sect{Dissipation phase space volume and entropy}

%Baruch's  D02.

Consider the derivation of Liouville's theorem for the
ensemble density  $\rho(p, q, t)$ in phase space $(p, q)$
corresponding to the motion of a particle of mass $m$ with
friction $\gamma$
\[\frac{dq}{dt}= \frac{p}{m},   \,\,\,\,     \frac{dp}{dt}  = -\gamma  p\,.\]
\begin{itemize}
\item [(a)]
Show that Liouville's theorem is replaced by ${d\rho /dt =
\gamma\rho}$  .
\item [(b)]
Assume that the initial  ${\rho\left(p, q, t=0\right)}$ is uniform
in a volume  ${\omega_{0}}$ in phase space and zero outside of
this volume. Find  ${\rho\left(p, q, t\right)}$ if ${\omega_{ 0}}$
is a rectangle  ${-\bar{p}  <p<\bar{p} ,\,\,\,-\bar{q}
<q<\bar{q}}$. Find implicitly  ${\rho\left(p, q, t\right)}$ for a
general ${\omega_{ 0}}$.
\item [(c)]
what happens to the occupied volume  ${\omega_{0}}$ as time
evolves? (assume a general shape of  ${\omega_{0}}$). Explain at
what ${t}$ this description breaks down due to quantization.
\item [(d)]
Find the Boltzmann entropy as function of time for case (b).
Discuss the meaning of the result.\\
\end{itemize}

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