From 4de4af4cc7611dfa1aad4b0a52e84e1d94f0a008 Mon Sep 17 00:00:00 2001
From: Janita Willumsen <j.willu@me.com>
Date: Thu, 9 Nov 2023 10:07:07 +0100
Subject: [PATCH] Add tool section

---
 latex/sections/methods.tex | 15 ++++++++++++---
 1 file changed, 12 insertions(+), 3 deletions(-)

diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex
index 64ed8b4..0b4bc61 100644
--- a/latex/sections/methods.tex
+++ b/latex/sections/methods.tex
@@ -2,7 +2,7 @@
 
 \begin{document}
 \section{Methods}\label{sec:methods}
-\subsection{The Ising model}\label{sec:ising}
+\subsection{The Ising model}\label{sec:ising_model}
 % Problem 1
 \begin{align*}
 	Z &= \sum_{all \ s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\
@@ -57,11 +57,20 @@
 	\chi &= \frac{1}{N} \frac{1}{k_{\text{B}} T^{2}} (\langle M^{2} \rangle - \langle M \rangle^{2}) \\
 	&= \frac{1}{N k_{\text{B}} T} \Big( \frac{12e^{8 \beta J} + 4 e^{-8 \beta J} + 12}{(\cosh(8 \beta J) + 3)^{2}}  \Big)
 \end{align*}
+The derivation of analytical expressions can be found in appendix 
+\ref{sec:analytical_expressions}
 
-\subsection{Markov Chain Monte Carlo methods}
+\subsection{Markov Chain Monte Carlo methods}\label{sec:mcmc_methods}
 
 
-\subsection{Implementation}
+\subsection{Implementation}\label{sec:implementation}
+
+
+\subsection{Tools}\label{sec:tools}
+The Ising model and MCMC methods are implemented in C++, and parallelized using 
+\verb|OpenMP| \cite{openmp:2018}. We used the Python library \verb|matplotlib| 
+\cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| 
+\cite{waskom:2021:seaborn} to set the theme in the figures. 
 
 
 \end{document}