From 36a10f224d6e0b2666b406587b46e0e3067410b9 Mon Sep 17 00:00:00 2001 From: Deepak Date: Thu, 18 Mar 2021 20:07:33 -0500 Subject: [PATCH] Adds noise calculation again where'd it go --- main.tex | 17 +++++++++++++++++ 1 file changed, 17 insertions(+) diff --git a/main.tex b/main.tex index bb070fe..243e7f4 100644 --- a/main.tex +++ b/main.tex @@ -242,6 +242,23 @@ We can look at the Nam conductivity now: As there is as yet no filtering based on the free energy, this is messy, so the comparison to \cref{fig:osRepro1b} is necessary to see which combinations of $T$ and $n$ are valid. However, this is still quite numerically unstable, even with the dimensional reduction procedure described above for $n$ and $\corr$. The information that higher $n$ leads to lower $\sigma$ is expected, and does suggest that the implementation is correct, although still very noisy. + +\section{Noise calculation} + +For our noise calculation, we assume a material parameterised by a Debye frequency $\debye$ and the interaction parameter $N(0) V$. +Because these aren't necessarily experimentally accessible, I tried to keep their values such that they lead to a physically reasonable $T_c$. +In order to keep $N(0) V$ small and $\debye$ bigger than the other parameters, I chose $N(0)V = 0.25$ and $\debye = \SI{1e13}{\per\s}$. +This leads to $T_{c0} = T_c(\mu = 0) = \SI{1.44e11}{\per\s}$, similar to the values used for the equilibrium Nam case. + +\begin{enumerate} + \item Use the Owen Scalapino coupled integral equations \cref{eq:gap,eq:n}, find $\mu$ and $\Delta$ for fixed $n$. + \item Find the expected gap from the approximation in OS, $T_c(n) \approx (1 - 4n) T_{c0}$. + If $T > T_c(n)$, then the calculation is skipped (a more complete handling would use either the Lindhard form or use a Nam expression that's been extended to $\Delta = 0$). + This is necessary because the coupled integral equations are very hard to solve + \item Using the modified Nam equations, calculate the dielectric function and create the approximated interpolation form, similar to the equilibrium case. + \item Calculate the noise as usual. +\end{enumerate} + \printbibliography \end{document}