
\sect{Baruch's  D21.}

Consider the Langevin equation for a particle with mass ${M}$
and veloscity ${{\bf v}\left(t\right)}$ in a medium with viscosity
${\gamma}$ and a random force ${\bf A}(t)$.
\begin{itemize}
\item [(a)]
Show that in equilibrium     $\langle {\bf v}(t){\bf A}(t)\rangle
= 3k_{B}T\gamma /M$.
\item [(b)]
Given $\langle {\bf v}(t){\bf v}(0)\rangle \sim e^{-\gamma |t|}$
and $\langle {\bf v}\rangle=0$, use ${\bf v}(t) = \dot{{\bf x}}(t)$ to evaluate ${\langle
{\bf x}^{2}(t)\rangle}$ [do not use Langevin's equation] .\\
\end{itemize}

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