
\sect{Boltzmann equation: particles between two plates}

%Baruch's D07.

Consider a classical gas of particles with mass $m$ between two plates separated by a
distance W. One plate at $y = 0$ is maintained at a temperature $T_1$, while the other
plate at $y = W$ is at a different temperature $T_2$. A zeroth order approximation to the
particle density is,
\[ f_0({\bf p},x,y,z)=\frac{n(y)}{[2\pi mk_BT(y)]^{3/2}}\eexp{-\frac{p^2}{2mk_B T(y)}}\]
\begin{itemize}
\item[(a)] The steady state solution has a uniform pressure; it does not have a uniform
chemical potential. Explain this statement and find the relation between $n(y)$
and $T(y)$.
\item[(b)] Show that $f_0$ does not solve Boltzmann's equation.

Consider a relaxation approximation, where the collision term of Boltzmann's
equation is replaced by a term that drives a solution $f_1$ towards $f_0$, i.e.
\[ [\frac{\partial}{\partial t}+\frac{p_y}{m}\frac{\partial}{\partial y}]f_0({\bf p},y)
=-\frac{f_1({\bf p},y)-f_0({\bf p},y)}{\tau}\]
and solve for $f_1$.
\item[(c)] The rate of heat transfer is $Q = n\langle p_yp^2\rangle_1/(2m^2)$;
$\langle ...\rangle_1$ is an average with respect
to $f_1$. Justify this form and evaluate $Q$ using the integrals
$\langle p_y^2p^4\rangle_0=35(mk_bT)^3$ and $\langle p_y^2p^2\rangle_0=5(mk_bT)^2$.
Identify the coefficient of thermal conductivity $\kappa$,
where $Q = -\kappa \frac{\partial T}{\partial y}$.
\item[(d)] Find the profile $T(y)$.
\item[(e)] Show that the current is $\langle J_y\rangle=0$. Explain why this result is to be expected.
\item[(f)] For particles with charge $e$ add an external field $E_y$ and extend Boltzmann's equation from (b). Evaluate, for uniform temperature, $J_y$ and the conductivity $\sigma$, where $J_y=\sigma E_y$. Check the Wiedemann-Franz law, $\kappa/\sigma T=$const.
\end{itemize}

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