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\sect{Random walk with correlations}

The total displacement of a particle is a sum over steps
${X\left(t\right)}$, where ${t}$ is discrete.
If we define the velocity as ${v\left(t\right) = \frac{X\left(t\right)}{\tau\ 0}}$, where ${\tau 0}$ is
the time between steps, then the random walk is described by the
equation ${\left(\frac{dx}{dt}\right) = v\left(t\right)}$.

\begin {itemize}
\item[(a)]
Given the velocity-velocity correlation function ${c\left(t2-t1\right) =
\langle v\left(t1\right)v\left(t2\right)\rangle}$, write down an
expression for the spreading ${S\left(t\right)=\sqrt{\left[\langle
\left(x\left(t\right)-x\left(0\right)\right)^{2}\rangle\right]}}$.

\item[(b)]
Find an expression the diffusion coefficient, assuming that
${c\left(\tau\right)}$ is short range.

\item[(d)]
More generally, show that ${\frac{dS\left(t\right)}{dt}}$ is equal to the
${\left[-t,t\right]}$ integral of ${c\left(\tau\right)}$.

\item[(e)]
Assume that ${c\left(\tau\right)}$ has zero integral and power law tails ${c\left(\tau\right) =
\frac{-c_{0}}{\tau^{\alpha}}}$. Determine the sub-diffusive behavior
of ${S\left(t\right)}$ depending on the value of ${\alpha}$.

\end {itemize}



