
\sect{The Kubo formula}

%Baruch's  D30.

Particles with charge $e$  and velocities ${\bf v}_i$ couple to an
external vector potential by $V_{int}=-\frac{e}{c}\sum_i{\bf
v}_i\cdot {\bf A}$ and the electric field is ${\bf
E}=-\frac{1}{c}\frac{\partial {\bf A}}{\partial t}$. The current
density (per unit volume) is ${\bf j}=e\sum_i{\bf v}_i$.
\begin{itemize}
\item[(a)] Identify the response  function for an $a$ component
field with a given frequency, $E_a(\omega)$,  in terms of the
conductivity $\sigma(\omega)$ where  ${\bf j}_a=\sigma (\omega)
{\bf E}_a$ (assume an isotropic system so that $\sigma(\omega)$ is
a scaler). Deduce the energy dissipation rate in terms of
$\sigma(\omega)$ and $E_a(\omega)$. Compare with Ohm's law. What
is the symmetry of Re$\sigma(\omega)$ when $\omega$ changes sign?
\item[(b)] Use the fluctuation dissipation theorem  to show the (classical)
Kubo formula:
\[\re\sigma (\omega)= \frac{1}{k_BT}\int_0^{\infty}\langle
j_a(0)\cdot j_a(t)\rangle\cos(\omega t) dt \]

\item[(c)]Write the Diffusion constant  ${D}$ in terms of the
velocity-velocity correlation function, assuming that this
correlation has a finite range in time.

Use Kubo's formula from (b) in the DC limit of zero frequency to
derive the Einstein-Nernst formula
for the mobility ${\mu =\frac{\sigma}{ne}= eD/k_{B}T}$, where $n$
is the particle density. (assume here uncorrelated particles).
\item[(d)] The quantum current noise is defined as
\[S(\omega)=\int_0^{\infty}dt \langle j_a(t)j_a(0)+j_a(0)j_a(t)\rangle \cos(\omega t).\]
Use the quantum FDT to relate this noise to the conductivity. When is the classical result (b) valid? What is the noise at $T=0$?
\end{itemize}

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