\sect{Brownian particle on a ring}
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