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