\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 ${0<t<t_c}$  
is assumed to be negligible. In item~(6) assume that the 
particle had been launched in the far past (${t=-\infty}$): 
accordingly there is no preferred location on the ring.

\begin{enumerate}

\item Find the exact correlation function $\langle v(t)v(0) \rangle$ in case (a).

\item Find the correlation function $\langle v(t)v(0) \rangle$ for $t\gg t_c$ in case (b).

\item Find the spreading $S(t)\equiv \langle \theta(t)^2 \rangle$  for $t \gg t_c$  in case (a).

\item Find the spreading $S(t)\equiv \langle \theta(t)^2 \rangle$  for $t \gg t_c$  in case (b).

\item Express $\langle x(t)^2 \rangle$ for a spreading scenario given $S(t)$.
 
\item Express the correlation function $\langle x(t)x(0) \rangle$ given $S(t)$.

\item Write the explicit long time expression for $\langle x(t)x(0) \rangle$ in case (b), 
and deduce what is the critical value $\eta_c$ above which a ``phase transition" is expected 
in the response characteristics of the system.   

\end{enumerate}

\Dn

Tips: For a Gaussian variable that has zero average $\langle \eexp{i\varphi}\rangle = \exp[-(1/2)\langle \varphi^2\rangle]$. \\
The Fourier transform of $|\omega|$ has zero area, with negative tails $-1/(\pi t^2)$. \\
If you fail to solve (6), assume that the answer is the same as in (5), and proceed to (7).