feat: adds identifying downturn notes

Makefile

@@ -1,6 +1,6 @@
 ### Build tools
 #
-LATEXMK := latexmk -pdflatex="luahblatex %O %S" -pdf -dvi- -ps- -quiet -logfilewarninglist
+LATEXMK := latexmk -pdflatex="lualatex %O %S" -pdf -dvi- -ps- -quiet -logfilewarninglist
 WS := wolframscript -f
 
 ### Directory variables
@@ -11,7 +11,7 @@ CALC_DIR := calc
 
 ### Here we go
 #
-OUT_PDF:= $(PDF_DIR)/notes.pdf
+OUT_PDF:= $(PDF_DIR)/notes.pdf $(PDF_DIR)/identifying_downturn.pdf
 
 .PHONY: all
 all: $(OUT_PDF)
@@ -20,7 +20,7 @@ all: $(OUT_PDF)
 #
 
 ## setup main pdf deps as variable that subdirs can add to
-MAIN_PDF_DEPS := bibliography.bib
+COMMON_PDF_DEPS := bibliography.bib
 
 ## Defining common directory recipes
 $(PDF_DIR):
@@ -36,10 +36,15 @@ $(CALC_DIR):
 FIGURES := 
 ## Making main.pdf and other pdfs
 #
-$(PDF_DIR)/notes.pdf: notes.tex $(MAIN_PDF_DEPS) | $(PDF_DIR) $(FIGURES)
+$(PDF_DIR)/notes.pdf: notes.tex $(COMMON_PDF_DEPS) | $(PDF_DIR) $(FIGURES)
 	$(LATEXMK) $(<F)
 	cp $(@F) $@
 
+$(PDF_DIR)/identifying_downturn.pdf: identifying_downturn.tex $(COMMON_PDF_DEPS) | $(PDF_DIR) $(FIGURES)
+	$(LATEXMK) $(<F)
+	cp $(@F) $@
+
+
 ### Convenience scripts for tidying tex
 .PHONY: declutter
 declutter:

do.sh

@@ -6,7 +6,7 @@ set -Eeuo pipefail # -e "Automatic exit from bash shell script on error"  -u "Tr
 
 build() {
    echo "I am ${FUNCNAME[0]}ing"
-   make pdfs/notes.pdf
+   make all
 }
 
 all() {

identifying_downturn.tex

@@ -0,0 +1,125 @@
+\documentclass{article}
+
+%other packages
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{physics}
+% 
+% \usepackage[
+% 	style=phys, articletitle=false, biblabel=brackets, chaptertitle=false, pageranges=false, url=true
+% ]{biblatex}
+
+\usepackage{graphicx}
+\usepackage{todonotes}
+\usepackage{siunitx}
+
+\usepackage{cleveref}
+
+\title{Identifying the downturn issue with clean case}
+
+% \addbibresource{./bibliography.bib}
+
+\graphicspath{{./figures/}}
+
+\newcommand{\vf}{v_{\mathrm{F}}}
+\newcommand{\qf}{q_{\mathrm{F}}}
+
+\begin{document}
+
+\maketitle
+
+\section{The problem}
+
+As discussed earlier there was an issue with a sharp drop in $T_1(z)$ as temperature approached $T_c$.
+The dirty case looks fine in \cref{fig:exdirtynodownturn}, but the clean system in \cref{fig:excleandownturn} shows the problem.
+Both of those are at $z$ around $20 \lambda_F$ ($\lambda_F$ is $\approx \SI{0.4}{\nm}$.)
+
+\begin{figure}[htp]
+	\centering
+	\includegraphics[width=\linewidth]{downturnExampleDirty-no-downturn}
+	\caption{$T_1(z)$, everything looks appropriately good for a dirty system.} \label{fig:exdirtynodownturn}
+\end{figure}
+
+\begin{figure}[htp]
+	\centering
+	\includegraphics[width=\linewidth]{downturnExampleClean}
+	\caption{$T_1(z)$, for a clean system we have the downturn.} \label{fig:excleandownturn}
+\end{figure}
+
+The obvious question is whether this matters, because it's $T$ so close to $T_c$ that cutting it off is probably not an issue.
+But the effect matters because in the intermediate values of $\tau$ both the turning up to the normal state and the problematic downturn are present, and that's not great.
+\Cref{fig:downturnintermediate} shows what that looks like.
+I \emph{think} that that's still probably explainable as having a rise up to the normal state for $T \rightarrow T_c$, masked by the downturn because it only rises close to $T_c$.
+But it's not really a convincing graph, so I proceed under the assumption that it's worth investigation.
+
+\begin{figure}[htp]
+	\centering
+	\includegraphics[width=\linewidth]{downturnIntermediate}
+	\caption{$T_1(z)$, for a clean system we have the downturn.} \label{fig:downturnintermediate}
+\end{figure}
+
+Interestingly, this problem is dependent on the specific $z$.
+$T_1$ plotted for different zs in \cref{fig:varioust1vsT}.
+The smaller $z$ doesn't have that effect.
+But that's the direction that's less helpful, because it's already a very small $z$.
+
+\begin{figure}[htp]
+	\centering
+	\includegraphics[width=\linewidth]{t1vsTexamples}
+	\caption{$T_1(T)$, for different $z$, some showing the downturn others not.} \label{fig:varioust1vsT}
+\end{figure}
+
+I plotted $T_1$ as a function of $z$ at very high temperatures ($T = 0.99999 T_c$), in the downturn, and again for $T = 0.99 T_c$, which is above the downturn.
+In \cref{fig:t1vszdirty}, the dirty case, everything looks sensible.
+However, in \cref{fig:t1vszclean} we can see that for $z > \lambda_F$ the $T_1(z)$ curve has a very different flat shape at the very high temperature, which ultimately leads to the downturn.
+
+\begin{figure}[htp]
+	\centering
+	\includegraphics[width=\linewidth]{t1VsZDirtyCase-no-problems}
+	\caption{$T_1(z)$, at two different temperatures.} \label{fig:t1vszdirty}
+\end{figure}
+
+\begin{figure}[htp]
+	\centering
+	\includegraphics[width=\linewidth]{t1VsZcleanCase-bad}
+	\caption{$T_1(z)$, at two different temperatures.} \label{fig:t1vszclean}
+\end{figure}
+
+I looked at where that might come from.
+We have that $T_1 \propto \frac{1}{\chi}$, and 
+\begin{equation}
+	\chi_{B,zz} \propto \int_0^\infty \dd{u} u^2 e^{-2 u z} \Im r_s(u),
+\end{equation}
+in a regime where $u \gg \frac{1}{\lambda}$ for vacuum wavelength $\lambda$.
+This is always the regime we care about.
+So $\chi(z) \approx \Im r_s(\frac{1}{z})$ or so.
+I plotted $\Im r_s(u)$ in \cref{fig:imrs}, and it doesn't really have any of the unusual features we might exepect.
+That figure is pretty ugly, but the takeaway is that for $u > 0.2 \lambda_F$, all that matters is the temperature.
+However, for $u < 0.2 \lambda_F$ the dirty and clean lines start to split, and for small enough $u$ the difference between the higher and lower temperature cases is less important than the clean-dirty difference.
+That makes sense, and doesn't at all explain why $T_1$ is problematic.
+
+\begin{figure}[htp]
+	\centering
+	\includegraphics[width=\linewidth]{Im rs-no-weirdness}
+	\caption{$\Im r_s$, showing no weirdness.} \label{fig:imrs}
+\end{figure}
+
+So that leaves the integral as a problem.
+I haven't been able to excise the issue yet though, either by increasing precision or rewriting the integral different ways to pull $z$ out.
+Using something like 
+\begin{equation}
+	\chi_{B,zz} \propto \frac{1}{z^3} \int_0^\infty \dd{u'} u'^2 e^{-2 u'} \Im r_s(\frac{u'}{z})
+\end{equation}
+should be numerically nicer, but I get about the same answers (with Mathematica's integral handler internally doing something similar to that transform anyway?).
+The problem is definitely still apparent in the $\chi$ calculation, which is plotted in \cref{fig:chi}.
+So I'm currently confused on this.
+
+\begin{figure}[htp]
+	\centering
+	\includegraphics[width=\linewidth]{chiVsZCleanCaseWithProblem}
+	\caption{$\chi(z)$, showing the same issue as $T_1$ (as expected).} \label{fig:chi}
+\end{figure}
+
+% \printbibliography
+
+\end{document}

notes.tex

@@ -15,7 +15,7 @@
 
 \usepackage{cleveref}
 
-\title{Noise on ewjn mag noise}
+\title{Notes on ewjn mag noise}
 
 \addbibresource{./bibliography.bib}