\sect{Baruch's  D26.}

Shot noise: The discreteness of the electron charge ${e}$ implies that the
current is not uniform in time and is a source of noise. Consider
a vacuum tube in which electrons are emitted from the negative
electrode and flow to the positive electrode; the probability of
emitting any one electron is independent of when other electrons
are emitted. Suppose that the current meter has a response time
$\tau$. If $T$ is the average time between the emission of two electrons, then the average current
is  $\langle I\rangle=e/T=\frac{e}{\tau}t$, where $t=\tau/T$ is the transmission probability,
$0\leq t\leq 1$.

\begin{itemize}
\item [(a)]
Show that the fluctuations in $I$ are $\langle \delta
I^{2}\rangle =\frac{e^2}{\tau^2}t(1-t)$. Why would you expect the fluctuations
to vanish at both $t=0$ and $t=1$?
[Hint: For each $\tau$ interval ${n_{i}=0}$ or ${n_{i}=1}$ and
$\langle n_i\rangle=\tau/T$; discretize time in units of $\tau$.]

\item [(b)]
Consider the meter response to be in the range ${0<|\omega |<2\pi
/\tau }$. Show that for $t\ll 1$ the fluctuations in a frequency interval
$d\omega$ are ${d\langle \delta I^{2}\rangle = e\langle I\rangle
d\omega /2\pi}$ . At what frequencies does this noise dominate
over the
Johnson noise in the circuit?

\item[(c)] Show that the 3rd order commulant is
$\langle (I-\langle I\rangle))^3\rangle=\frac{e^3}{\tau^3}t(1-t)(1-2t)$.
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\end{itemize}

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