\sect{Baruch's  D06.}

A thermalized gas particle at temperature $T$ is suddenly confined to positions $q$ in a one dimensional trap. The corresponding
state is described by an initial density function $\rho(q, p, t = 0) = \delta(q)f(p)$
where $\delta(q)$ is Dirac's delta function and
\begin{equation}
f(p) = \frac{{\mbox e}^{-p^2/2mk_BT}}{\sqrt{2\pi mk_BT}}.
\end{equation}
\begin{itemize}
\item[(a)] Starting from Liouville's equation with the Hamiltonian ${\cal H}=p^2/2m$ derive $\rho(q, p, t)$. For a given time $t$ draw the points in the $(p,q)$ plane where $\rho(q, p, t)$ is finite and emphasize the segment where $f(p)$ is large, $p<\sqrt{mk_BT}\equiv p_0$.
\item[(b)] Derive the expressions for the averages $\langle q^2 \rangle$ and
$\langle p^2 \rangle$ at $t > 0$.
\item[(c)] Suppose that hard walls are placed at $q = \pm Q$. Repeat the plot of (a) and again emphasize the range $p<p_0$. What happens in this plot at long times $t>2Qm/p_0\equiv \tau_0$? What is the meaning of the time $\tau_0$?

\item[(d)] A "coarse grained" density ${\tilde \rho}$ is obtained by ignoring variations of $\rho$ below some small
resolution in the $(q, p)$ plane; e.g., by averaging $\rho$ over cells of the resolution area. Find
${\tilde \rho}(q, p)$ for the situation in part (c) at long time $t\gg\tau_0$, and show that it is stationary.
\end{itemize}

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