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\sect{Fluctuations of damped harmonic oscillator}

A particle of mass $\mass$ is described by its position $x$ and velocity $v$. 
It is bounded by a harmonic potential of frequency ${\Omega}$, 
and experiences a damping with a coefficient $\eta$. 
Additionally It is subject to an external force $f(t)$. The system is at temperature $T$.

\begin {itemize}

\item[(a)]
Write the generalized susceptibility that describes 
the response of ${x}$ to the driving by $f(t)$.

\item[(b)]
Using the FD relation deduce what are the power spectra of $x$ and of $v$.

\item[(c)]
Write an integral expression for the autocorrelation 
function $\langle v(t)v(0)\rangle$.  
Find explicit results in various limits, 
e.g. for damped particle ($\Omega\rightarrow0$). 

\item[(d)]
Find $\langle x^2 \rangle$ and  $\langle v^2 \rangle$ for $\eta\rightarrow0$, 
both in the quantum and in the classical case. Verify consistency with the canonical results.

\end {itemize}