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latex/appendices.tex

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-\documentclass[../ising_model.tex]{subfiles}
-
-\begin{document}
-\appendix
-\section{Ising model system states}\label{sec:system_states}
-To avoid counting duplicates, we used 
-\begin{figure}\label{fig:tikz_counting}
-    \centering
-    \begin{tikzpicture}
-        \draw (0, 0) grid (2, 2);
-        % \node[inner] (s1) at (0.5, 0.5) {s};
-        \node (s1) at (0.5, 1.5) {$s_1$};
-        \node (s2) at (1.5, 1.5) {$s_2$};
-        \node (s3) at (0.5, 0.5) {$s_3$};
-        \node (s4) at (1.5, 0.5) {$s_4$};
-        \node[gray] (s12) at (2.5, 1.5) {$s_1$};
-        \node[gray] (s34) at (2.5, 0.5) {$s_3$};
-        \node[gray] (s13) at (0.5, -0.5) {$s_1$};
-        \node[gray] (s24) at (1.5, -0.5) {$s_2$};
-    
-        \draw[red, ->] (s1.east) -- (s2.west);
-        \draw[red, ->] (s2.east) -- (s12.west);
-        \draw[red, ->] (s1.south) -- (s3.north);
-        \draw[red, ->] (s2.south) -- (s4.north);
-        \draw[red, ->] (s3.east) -- (s4.west);
-        \draw[red, ->] (s4.east) -- (s34.west);
-        \draw[red, ->] (s3.south) -- (s13.north);
-        \draw[red, ->] (s4.south) -- (s24.north);
-      \end{tikzpicture}
-      \caption{Rules for multiplying spin pairs.}
-\end{figure}
-
-
-\section{Analytical expressions}\label{sec:analytical_expressions}
-The Boltzmann distribution is normalized using a partition function $Z$ given by
-\begin{align*}
-	Z &= \sum_{\text{all possible } s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\
-	&= \dots \\
-	&= 4 \cosh (8 \beta J) + 12 \\
-\end{align*}
-
-
-\end{document}