\sect{Sub diffusion of Brownian particle}

The motion of a brownian particle in 1D is given by the Hamiltonian:
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\[{H_{total}}(x,p;A(t)) = \frac{1}{{2m}}{\left( {p - A(t)} \right)^2} + {H_{bath}(x)}\]
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Assume that the equation of motion for the average velocity is:
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\[m\frac{{\partial  \langle v \rangle }}{{\partial t}} =  - \eta  \langle v \rangle + f(t)\]
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In items 5-6-7 assume a zero temperature bath, and define
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\[ S(t) = \left\langle {{{\left( {x(t) - x(0)} \right)}^2}} \right\rangle \]

\begin{enumerate}

\item Relate $f(t)$ to $A(t)$.

\item What is the generalised susceptability $\chi(\omega)$ that relates $v$ to $A$. 

\item Find the power spectrum  $\tilde{C}(\omega)$ of the velocity $v$. 

\item Find the explicit expression for the correlation function $C(\tau)$ in the limit of high and low temperatures.

\item Find the coefficient $C_0$ in $C(\tau) \sim  - C_0 / \tau ^2 $.

\item Express ${dS(t)}/{dt}$ using the correlation function $C(\tau)$.

\item Given $S(t_0)=S_0$, find what is $S(t)$ for $t>t_0$.

\end{enumerate}