From a6f9cd75382e399bd8eadc65705f072e502f8ae7 Mon Sep 17 00:00:00 2001
From: Deepak <dmallubhotla+github@gmail.com>
Date: Thu, 18 Mar 2021 19:52:54 -0500
Subject: [PATCH] adds noise calc sections

---
 .gitignore |  1 +
 main.tex   | 17 +++++++++++++++++
 2 files changed, 18 insertions(+)

diff --git a/.gitignore b/.gitignore
index 04800ed..bf2a9c2 100644
--- a/.gitignore
+++ b/.gitignore
@@ -9,6 +9,7 @@ dist
 *.log
 *.pdf
 *.jpg
+*.png
 *.bbl
 *.bcf
 *.blg
diff --git a/main.tex b/main.tex
index b37577e..04f51c3 100644
--- a/main.tex
+++ b/main.tex
@@ -209,6 +209,23 @@ with
 	S(\kappa, E) &= \frac{1}{\kappa} \left(E - i \left(\Im[\sqrt{(\xi + \xi')^2 - 1} + \sqrt{\xi'^2 - 1}] + 2 \nu \right) \right) \\
 	g  &= \frac{\xi' \left( \xi + \xi'\right) + 1}{\sqrt{\xi'^2 - 1}\sqrt{(\xi + \xi')^2 - 1}}
 \end{align}
+
+\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}