\sect{Correlation functions from Langevin dynamics}

Consider the Langevin equation for a particle with mass ${M}$
and velocity ${{\bf v}\left(t\right)}$ in a medium with viscosity
${\gamma}$ and a random force ${\bf A}(t)$.

\begin{itemize}

\item [(a)]
Find the equilibrium value of $\langle {\bf v}(t){\bf A}(t)\rangle.

\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}

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