\sect{Brownian particle on a ring (Exam 2012)} The motion of a classical Brownian particle on a 1D ring is described by the Langevin equation ${\mass \ddot{\theta}+\eta \dot{\theta} =f(t)}$, where~$f(t)$ is due to a noisy electromotive force that has a correlation function ${\langle f(t')f(t'')\rangle=C_f(t'-t'')}$. The power spectrum $\tilde{C}_f(\omega)$ is defined as the Fourier transform of the correlation function. We consider two cases: \begin{enumerate} \item[(a)] High temperature white noise ${\tilde{C}_f(\omega)=\nu}$. \item[(b)] Zero temperature noise ${\tilde{C}_f(\omega)=c|\omega|}$. \end{enumerate} We define the angular velocity of the particle as ${v=\dot{\theta}}$, and its Cartesian coordinate as ${x=\sin(\theta)}$. In the absence of noise the dynamics is characterized by the damping time $t_c=\mass/\eta$. \Dn In items (3)-(5) you should assume a spreading scenario: the particle is initially (${t=0}$) located at ${\theta \sim 0}$. The spreading during the transient period \${0