From 106d4026e41bdb7d22127734e463efe5284411f5 Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Thu, 13 Jun 2013 23:24:05 -0400 Subject: [PATCH] calibcant/overview.tex: Flesh out related work Also, cite roters96 for using the proper form but calling it a Lorentzian. --- src/calibcant/overview.tex | 35 ++++++++++++++++++++++++----------- 1 file changed, 24 insertions(+), 11 deletions(-) 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 -- 2.26.2