
\sect{Boltzmann equation: distribution function}

%Baruch's  D01.

Consider an ideal gas in an external potential $\phi({\bf
r})$.
\begin{itemize}
\item [(a)]
Let     $\,\,\,H=\int d^{3}v\int d^{3}r f({\bf r},{\bf v},t)\ln
f({\bf r},{\bf v},t)$

where $f({\bf r},{\bf v},t)$ is arbitrary except for the
conditions on density ${n}$ and energy ${E}$
\[\ \int d^{3}r\int d^{3}v  f({\bf r},{\bf v},t) =
n  \,\,\,,  \,\, \int d^{3}r\int d^{3}v  \left[\half mv^{2}+
\phi({\bf r})\right] f({\bf r},{\bf v},t) = E\,.\]
Find $f({\bf r},{\bf v})$ (i.e. ${t}$ independent) which maximizes ${H}$.
(Note: do not assume binary collisions, i.e. the Boltzmann
equation).
\item [(b)]
Use Boltzmann's equation to show that the general form of the
equilibrium distribution of the ideal gas (i.e. no collision term)
is $f[\frac{1}{2}mv^2+\phi({\bf r})]$ where the local force is
$\nabla \phi$. Determine this solution by allowing for collisions
and requiring that the collision term vanishes. Find also the
average density ${n\left(r\right)}$.\\
\end{itemize}

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