
\sect{Linear response and Kubo}

%Baruch's  D27

Consider a classical system of charged particles with a
Hamiltonian ${H_{0}\left(p,q\right)}$. Turning on an external
field ${\bf E}(t)$ leads to the Hamiltonian ${H= H_{0}(p,q)- e
\Sigma _{i}{\bf q}_{i}\cdot{\bf E}(t)}$.
\begin{itemize}
\item [(a)]
Show that the solution of Liouville's  equation to first order in
${\bf E}(t)$ is
\[\rho(p,q,t) = e^{-\beta H_{0}\left(p,q\right)}
\left[1+ \beta e \Sigma_{i}\int _{-\infty}^{t}\dot{{\bf
q}}_{i}(t')\cdot{\bf E}(t')dt'\right]\,.\]
\item [(b)]
In terms of the current density ${{\bf j}({\bf r},t)=e\Sigma
_{i}\dot{{\bf q}}_{i}\delta^{3}({\bf r} -{\bf q}_{i})}$ show that
for ${{\bf E}={\bf E}(\omega )e^{i\omega t}}$ the linear response
is ${\langle j^{\mu}(t)\rangle=\sigma^{\mu\nu} (\omega )E^{\nu} (
\omega)e^{i\omega t}}$ where ${\mu,\,\nu}$ , are vector components
and
\[\ \sigma^{\mu\nu}\left(\omega \right) = \beta \int_{0}^{\infty}
d \tau e ^{-i\omega\tau}  d^{3}r \langle j^{\mu}(0,0)j^{\nu} ({\bf
r},-\tau )\rangle_0\] where ${\langle...\rangle_{0}}$ is an
average of the ${{\bf E}=0}$ system. This is the (classical)
Kubo's formula.

\item [c)]
Rewrite (b) for ${\bf j}({\bf r},t)$ in presence of a position
dependent ${\bf E}({\bf r},t)$. Integrating ${\bf j}({\bf r},t)$
over a cross section perpendicular to ${\bf E}({\bf r},t)$ yields
the current ${I\left(t\right)}$. Show that the resistance
${R\left(\omega \right)}$ satisfies
\[\ R ^{-1}(\omega ) =  \beta\int_{0}^{\infty}d\tau e^{-i\omega \tau}
\langle I(0)I(\tau)\rangle_0\]
For a real ${R(\omega )}$ (usually
valid below some frequency)
deduce Nyquist's theorem.\\
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
