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\sect{Diffusion of Brownian particle from Langevin}


Brownian motion is formally obtained as the ${\Omega-->0}$ limit of
the previous problem.

\begin {itemize}

\item[(a)]

Calculate the velocity-velocity correlation function of the Brownian
particle in the limit of high temperature.

\item[(b)]
Show that it is an exponential function, and identify the
correlation time.

\item[(c)]
Write the relation between the dispersion
${\sqrt{\left[\langle\left(x\left(t\right)-x\left(0\right)\right)\rangle
2\right]}}$ and the velocity correlation function.

\item[(d)]
Deduce that the particle diffuses in space and write the
expression for the diffusion coefficient.

\item[(e)]
Show that in the limit of zero temperature the velocity-velocity
correlation function has a zero integral and power law tails (recall
Exe.701).

\item[(f)]
In the latter case deduce that instead of diffusive spreading
one should observe slow logarithmic growth of the variance.

\end {itemize}




