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Janita Willumsen 2023-11-09 10:07:07 +01:00
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commit 4de4af4cc7
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latex/sections/methods.tex
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@ -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}