53 lines
1.9 KiB
TeX
53 lines
1.9 KiB
TeX
\documentclass{article}
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%other packages
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{physics}
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% \usepackage[
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% style=phys, articletitle=false, biblabel=brackets, chaptertitle=false, pageranges=false, url=true
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% ]{biblatex}
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\usepackage{graphicx}
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\usepackage{todonotes}
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\usepackage{siunitx}
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\usepackage{cleveref}
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\title{Cliff Notes}
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% \addbibresource{./bibliography.bib}
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\graphicspath{{./figures/}}
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\newcommand{\vf}{v_{\mathrm{F}}}
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\newcommand{\qf}{q_{\mathrm{F}}}
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\begin{document}
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\maketitle
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We wanted to see what happened for sufficiently low temperatures to the noise from varying mean free paths.
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We should expect that the effect of impurities is to broaden the cliff, but for sufficiently cold or warm systems, the effect of impurities should be minimal.
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However, in \cref{fig:nc}, we see that for low $T$, the gap between curves does not decrease.
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It's unclear why that is.
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The lower limit in the figure is the smallest $T$ such that the code works, so it's possible that that convergence could occur for even lower $T$.
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However, the curves do not even begin to approach each other.
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Also, this range of $T$ values covers a couple orders of magnitude in $\Delta$, so it seems that some effect on $\tau$ should be visible, at least in the values closest to $\tau T_c \approx 1$.
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Varying aspects of the numerical calculation like the cutoff for interpolation don't change the relationship between the curves.
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I think the most likely explanation is that somewhere a numerical error proportional to $\tau$ gets incorporated somewhere, which is not unreasonable given how $\tau$ gets added to the frequency (so there's likely a sum of two terms of different order of magnitude).
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More investigation would likely be good.
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\begin{figure}[htp]
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\centering
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\includegraphics[width=\linewidth]{low_t_no_converge}
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\caption{$\chi(T)$ for low temperatures, for different $\tau$.} \label{fig:nc}
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\end{figure}
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% \printbibliography
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\end{document}
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