From 07b155a772eac4e4832e92e4253e80ca6679e1fe Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Tue, 14 May 2013 14:30:02 -0400 Subject: [PATCH] calibcant: Move Lorentzian discussion before the PSD derivation Prof. Cruz points out that the damped harmonic oscillator should be old hat. Lay out the confusion first to motivate the gory details that follow. --- src/calibcant/discussion.tex | 43 --------------------- src/calibcant/overview.tex | 75 +++++++++++++++++++++++++++++++++++- src/calibcant/procedure.tex | 2 +- src/calibcant/theory.tex | 18 ++++++--- 4 files changed, 86 insertions(+), 52 deletions(-) diff --git a/src/calibcant/discussion.tex b/src/calibcant/discussion.tex index f76ed46..2b6672c 100644 --- a/src/calibcant/discussion.tex +++ b/src/calibcant/discussion.tex @@ -1,49 +1,6 @@ \section{Discussion} \label{sec:calibcant:discussion} -\subsection{Fitting with a Lorentzian} -\label{sec:calibcant:lorentzian} - -It is popular to refer to the thermal power spectral density as a -``Lorentzian''\citep{howard88,hutter93,roters96,levy02,florin95} even -though \cref{eq:model-psd} differs from the classic -Lorentzian\citep{mathworld-lorentzian}. -\begin{equation} - L(x) = \frac{1}{\pi}\frac{\frac{1}{2}\Gamma} - {(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2} \;, - \label{eq:lorentzian} -\end{equation} -where $x_0$ sets the center and $\Gamma$ sets the width of the curve. -It is unclear whether the references are due to uncertainty about the -definition of the Lorentzian or to the fact that \cref{eq:model-psd} -is also peaked and therefore \cref{eq:lorentzian} a potential -substitute for \cref{eq:model-psd}. \citet{florin95} -likely \emph{are} using \cref{eq:lorentzian}, as the slope of the -fitted \PSD\ in their figure 2, has a slope at $f=0$. -Using \cref{eq:model-psd}, the derivative would have been zero, as we -can see by using the chain rule repeatedly, - -\begin{align} - \deriv{f}{\PSD_f} - &= \deriv{f}{}\p({\frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2}}) - = \frac{-G_{1f}}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2} - \deriv{f}{}\p({(f_0^2-f^2)^2 + \beta_f^2 f^2}) \\ - &= \frac{-G_{1f}}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2} - \p({2(f_0^2-f^2)\deriv{f}{}(f_0^2 - f^2) + 2\beta_f^2 f}) \\ - &= \frac{-G_{1f}}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2} - \p({-4f(f_0^2-f^2) + 2\beta_f^2 f}) \\ - &= \frac{2G_{1f}f}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2} - \p({2(f_0^2-f^2) - \beta_f^2}) \\ - \left.\deriv{f}{\PSD_f}\right|_{f=0} &= 0 \;. - \label{eq:model-psd-df} -\end{align} - -However, \citet{benedetti12} has a solid derivation of -\cref{eq:DHO-psd}, which he then refers to as the ``Lorentzian''. In -order to avoid any uncertainty, we leave \cref{eq:model-psd} unnamed. -I encourage future researchers to explicitly list the model they use, -ideally by citing their associated open source calibration package. - \subsection{Peak frequency} \label{sec:calibcant:peak-frequency} diff --git a/src/calibcant/overview.tex b/src/calibcant/overview.tex index a3856a9..4518233 100644 --- a/src/calibcant/overview.tex +++ b/src/calibcant/overview.tex @@ -42,7 +42,7 @@ $f_0$\index{$f_0$}, and $\beta_f$\index{$\beta_f$}, the expectation value for $V_p$ is given by \begin{equation} \avg{V_p(t)^2} = \frac{\pi G_{1f}}{2\beta_f f_0^2} \;. - \label{eq:Vp-from-freq-fit} + \label{eq:avg-Vp-Gone-f} \end{equation} % \nomenclature[PSDf]{$\PSD_f$}{Power spectral density in @@ -53,7 +53,7 @@ value for $V_p$ is given by \nomenclature{$f$}{Frequency (hertz)} \nomenclature{$f_0$}{Resonant frequency (hertz)} -Combining \cref{eq:equipart_k,eq:x-from-Vp,eq:Vp-from-freq-fit}, we +Combining \cref{eq:equipart_k,eq:x-from-Vp,eq:avg-Vp-Gone-f}, we have \begin{align} \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}} @@ -84,3 +84,74 @@ power spectral density. TODO (extend)%, %as do % see Gittes 1998 for more thermal noise details % see Berg-Sorensen05 for excellent overdamped treament. + +\section{Fitting with a Lorentzian} +\label{sec:calibcant:lorentzian} + +It is popular to refer to the thermal power spectral density as a +``Lorentzian''\citep{howard88,hutter93,roters96,levy02,florin95}, but +there is dissagreement on what this means. The classic Lorentzian +function is\citep{mathworld-lorentzian} +\begin{equation} + L(x) = \frac{1}{\pi}\frac{\frac{1}{2}\Gamma} + {(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2} \;, + \label{eq:lorentzian} +\end{equation} +where $x_0$ sets the center and $\Gamma$ sets the width of the curve. +However, the correct \PSD\ for a damped harmonic oscillator in bath of +white noise is given by \cref{eq:psd-Vp}\citep{burnham03,benedetti12}. + +These formula are fundamentally different. + +For example, the slope of \cref{eq:psd-Vp} is zero at $f=0$, as we can +see by using the chain rule repeatedly, +\begin{align} + \deriv{f}{\PSD_f} + &= \deriv{f}{}\p({\frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2}}) + = \frac{-G_{1f}}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2} + \deriv{f}{}\p({(f_0^2-f^2)^2 + \beta_f^2 f^2}) \\ + &= \frac{-G_{1f}}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2} + \p({2(f_0^2-f^2)\deriv{f}{}(f_0^2 - f^2) + 2\beta_f^2 f}) \\ + &= \frac{-G_{1f}}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2} + \p({-4f(f_0^2-f^2) + 2\beta_f^2 f}) \\ + &= \frac{2G_{1f}f}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2} + \p({2(f_0^2-f^2) - \beta_f^2}) + \label{eq:model-psd-df} \\ + \left.\deriv{f}{\PSD_f}\right|_{f=0} &= 0 \;. + \label{eq:model-psd-df-zero} +\end{align} +On the other hand, the slope of \cref{eq:lorentzian} is only zero at +the peak (where $x=x_0$). +\begin{align} + \deriv{x}{L(x)} + &= \frac{1}{\pi}\frac{\frac{-1}{2}\Gamma} + {\p({(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2})^2} + \cdot \deriv{x}{}\p({(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2}) \\ + &= \frac{1}{\pi}\frac{\frac{-1}{2}\Gamma} + {\p({(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2})^2} + \cdot 2 (x-x_0) \\ + &= \frac{1}{\pi}\frac{-\Gamma (x-x_0)} + {\p({(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2})^2} + \label{eq:lorentzian-dx} +\end{align} + +It is unclear whether the ``Lorentzian'' references are due to +uncertainty about the definition of the Lorentzian or to the fact that +the two equations have similar behaviour near the +peak. \citet{florin95} likely \emph{are} using \cref{eq:lorentzian}, +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''. +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 +uncertainty, we leave \cref{eq:psd-Vp} unnamed. I encourage future +researchers to explicitly list the model they use, ideally by citing +their associated open source calibration package. diff --git a/src/calibcant/procedure.tex b/src/calibcant/procedure.tex index 2b3f61e..79fa069 100644 --- a/src/calibcant/procedure.tex +++ b/src/calibcant/procedure.tex @@ -252,7 +252,7 @@ $P_{0f}$\citep{burnham03}. Plots of \cref{eq:psd-Vp-offset} fits look better than \cref{eq:psd-Vp} fits (\cref{fig:calibcant:vibration}), but the significance on the variance calculated with -\cref{eq:Vp-from-freq-fit} depends on the amount of background noise +\cref{eq:avg-Vp-Gone-f} depends on the amount of background noise in the vibration data. With over an order of magnitude difference between the power of the damped harmonic oscillator peak and the background noise, the effect of $P_{0f}$ will be small. With noisier diff --git a/src/calibcant/theory.tex b/src/calibcant/theory.tex index 4939cd0..9c79d18 100644 --- a/src/calibcant/theory.tex +++ b/src/calibcant/theory.tex @@ -1,8 +1,12 @@ -\section{Theory} +\section{Power spectra of damped harmonic oscillators} \label{sec:calibcant:theory} -Our cantilever can be approximated as a damped harmonic -oscillator\index{damped harmonic oscillator} +As discussed in \cref{sec:calibcant:lorentzian}, the power spectral +density for a Hookean cantilever is surprisingly ambiguous. In this +section, I'll derive the frequency-space power spectra of the +deflection voltage (\cref{eq:avg-Vp-Gone-f,eq:model-psd}), modeling +the cantilever as a damped harmonic oscillator\index{damped harmonic + oscillator}. \begin{equation} m\ddt{x} + \gamma \dt{x} + \kappa x = F(t) \;, \label{eq:DHO} @@ -223,8 +227,8 @@ We compute the \PSD\ by plugging \cref{eq:DHO-xmag} into Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd-F} we have \begin{equation} - \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;, - \label{eq:model-psd} \;. + \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;. + \label{eq:model-psd} \end{equation} Integrating over positive $\omega$ to find the total power per unit @@ -390,7 +394,7 @@ The variance of the function $x(t)$ is then given by plugging into = \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot} = \iOInf{f}{\PSD_f(x,f)} \;. \end{align} -We can now extract \cref{eq:psd-Vp,eq:Vp-from-freq-fit} from +We can now extract \cref{eq:psd-Vp,eq:avg-Vp-Gone-f} from \cref{eq:psd-Vp-Gone,eq:avg-Vp-Gone}. \begin{align} \begin{split} @@ -399,11 +403,13 @@ We can now extract \cref{eq:psd-Vp,eq:Vp-from-freq-fit} from = \frac{2\pi G_1}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\ &= \frac{G_1/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}} = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} + %\label{eq:psd-Vp} \end{split} \\ \avg{V_p(t)^2} &= \frac{\pi \frac{G_1}{(2\pi)^3}} {2 \frac{\beta}{2\pi} \p({\frac{\omega_0}{2\pi}})^2} = \frac{\pi G_{1f}}{2 \beta_f f_0^2} \;. + %\label{eq:avg-Vp-Gone-f} \end{align} where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and $G_{1f}\equiv G_1/8\pi^3$. Finally, we can generate -- 2.26.2