\subsection{Highly damped case}

For highly damped systems, the inertial term becomes insignificant
 ($m \rightarrow 0$).
This model is commonly used for optically trapped beads. % \cite{}
Because it is simpler and solutions are more easily available,
 %cite{grossman05}{}{}{}
we'll use it outline the general approach before diving into the general case.

Fourier transforming eq. \ref{eq.DHO} with $m=0$ and applying \ref{eq.four_deriv} we have
% ODHO stands for very Over Damped Harmonic oscillator
\begin{align}
  (i \gamma \omega + k) x(\omega) &= F(\omega) \label{eq.ODHO_freq} \\
  |x(\omega)|^2 &= \frac{|F(\omega)|^2}{k^2 + \gamma^2 \omega^2}.
                                               \label{eq.ODHO_xmag}
\end{align}
We compute the \PSD\ by plugging eq. \ref{eq.ODHO_xmag} into \ref{eq.psd_def}
\begin{equation}
  \PSD(x, \omega)
        = \normLimT \frac{2\magSq{F(\omega)}}{k^2 + \gamma^2\omega^2}.
                                               \label{eq.ODHO_psd}
\end{equation}

Because thermal noise is white (not autocorrelated + Wiener-Khinchin Theorem),
we can denote the one sided thermal power spectral density per unit time by
\begin{equation}
  \PSD(F, \omega) = G_0
     = \normLimT 2 \magSq{F(\omega)} \label{eq.GOdef} % label O != zero
\end{equation}

Plugging eq. \ref {eq.GOdef} into \ref{eq.ODHO_psd} we have
\begin{equation}
  \PSD(x, \omega) = \frac{G_0}{k^2 + \gamma^2\omega^2}.
\end{equation}
This is the formula we would use to fit our measured \PSD, but let us go a
bit farther to find the expected \PSD\ and thermal noise
 given $m$, $\gamma$ and $k$.

Integrating over positive $\omega$ to find the total power per unit time yields
\begin{align}
  \iOInf{\omega}{\PSD(x, \omega)}
     &= \iOInf{\omega}{\frac{G_0}{k^2 + \gamma^2\omega^2}} \\
     &= \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{k^2 + z^2}} \\
     &= \frac{G_0 \pi}{2 \gamma k},
\end{align}
where the integral is solved in Section \ref{sec.integrals}.

Plugging into our corollary to Parseval's theorem (eq. \ref{eq.parseval_var}), 
\begin{equation}
  \avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma k} \label{eq.ODHO_var}
\end{equation}

Plugging eq. \ref{eq.ODHO_var} into eqn. \ref{eq.equipart} we have
\begin{align}
  k \frac{G_0 \pi}{2 \gamma k} &= k_BT \\
  G_0 &= \frac{2 \gamma k_BT}{\pi}.
\end{align}

So we expect $X(t)$ to have a power spectral density per unit time given by
\begin{equation}
  \PSD(x, \omega) = \frac{2}{\pi} 
                       \cdot
                    \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2}.
\end{equation}
