From: W. Trevor King <wking@tremily.us>
Date: Fri, 14 Jun 2013 03:24:05 +0000 (-0400)
Subject: calibcant/overview.tex: Flesh out related work
X-Git-Tag: v1.0~81
X-Git-Url: http://git.tremily.us/?a=commitdiff_plain;h=106d4026e41bdb7d22127734e463efe5284411f5;p=thesis.git

calibcant/overview.tex: Flesh out related work

Also, cite roters96 for using the proper form but calling it a
Lorentzian.
---

diff --git a/src/calibcant/overview.tex b/src/calibcant/overview.tex
index dcd0ec7..f70f33a 100644
--- a/src/calibcant/overview.tex
+++ b/src/calibcant/overview.tex
@@ -87,11 +87,23 @@ derivation of noise in damped simple harmonic oscillators that
 underlies all frequency-space methods for improving the basic
 $\kappa\avg{x^2} = k_BT$ method.
 
-\citet{roters96} describe a similar approach to deriving the
-power spectral density.  TODO (extend)%,
-%as do 
-% see Gittes 1998 for more thermal noise details
-% see Berg-Sorensen05 for excellent overdamped treament.
+\citet{roters96} derive the \PSD\ with a similar Fourier transform,
+but they use the fluctuation--dissipation theorem to extract the \PSD\
+from the susceptibility (see
+their \fref{equation}{4}).  \citet{benedetti12} has independently
+developed a Parseval's approach similar to mine (in
+his \fref{section}{8.2.1}), although he glosses over some of the
+integrals.  \citet{berg-sorensen04} has an extensive treatment of the
+extremely overdamped case and laser tweezer calibration, which they
+revisit a year later during a discussion of noise
+color\citep{berg-sorensen05}.  \citet{gittes98} derive some related
+results in the extremely overdamped case, such the fact that the
+signal to thermal noise ratio is independent of trap stiffness
+$\kappa$.  Despite this earlier work, I think it is worth explicitly
+deriving the \PSD\ of a damped harmonic oscillator here, as I have
+been unable to find a reference that I feel treats the problem with
+sufficient rigor.  An explicit derivation may also help clear up the
+confusion about the proper \PSD\ form discussed in the next section.
 
 \section{Fitting with a Lorentzian}
 \label{sec:calibcant:lorentzian}
@@ -151,12 +163,13 @@ as the slope of the fitted \PSD\ in their \fref{figure}{2}, has a
 slope at $f=0$.  If they were using \cref{eq:psd-Vp}, the derivative
 would have been zero (\cref{eq:model-psd-df-zero}).
 
-We have at least two models in use, one likely the
-``Lorentzian'' (\cref{eq:lorentzian}) and one that's not.  Perhaps
-researchers claiming to use the ``Lorentzian'' are consistently
-using \cref{eq:lorentzian}?  There is at least one
-counterexample: \citet{benedetti12} has a solid derivation of
-\cref{eq:DHO-psd}, which he then refers to as the ``Lorentzian''.
+We have at least two models in use, one likely the ``Lorentzian''
+(\cref{eq:lorentzian}) and one that's not.  Perhaps researchers
+claiming to use the ``Lorentzian'' are consistently
+using \cref{eq:lorentzian}?  There are at least two
+counterexamples---\citet{roters96,benedetti12}---with solid
+derivations of
+\cref{eq:DHO-psd} which they then refer to as the ``Lorentzian''.
 Which formula are the remaining ``Lorentzian'' fitters using?  What
 about groups that only reference their method as ``thermal
 calibration'' without specifying a \PSD\ model?  In order to avoid any