\documentclass[../ising_model.tex]{subfiles}

\begin{document}
\section{Methods}\label{sec:methods}
\subsection{The Ising model}\label{sec:ising}
% Problem 1
\begin{align*}
	Z &= \sum_{all \ s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\
	&= \dots \\
	&= 4 \cosh (8 \beta J) + 12 \\
\end{align*}
%
\begin{align*}
	\langle \epsilon \rangle &= \frac{-2J \sinh(8 \beta J)}{ \cosh(8 \beta J) + 3}
\end{align*}
%
\begin{align*}
	\langle \epsilon^{2} \rangle &= \frac{4 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3}
\end{align*}
%
\begin{align*}
	\langle |m| \rangle &= \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)}
\end{align*}
\begin{align*}
	\langle |m|^{2} \rangle &= \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)}
\end{align*}
%
\begin{align*}
	\langle E \rangle &= \frac{-8 J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} 
\end{align*}
%
\begin{align*}
	\langle E \rangle^{2} &= \frac{64 J^{2} \sinh(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}}
\end{align*}
%
\begin{align*}
	\langle E^{2} \rangle &= \frac{64 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3}
\end{align*}
%
\begin{align*}
	\langle M \rangle &= \frac{2 e^{8 \beta J}}{\cosh(8 \beta J) + 3}
\end{align*}
%
\begin{align*}
	\langle M \rangle^{2} &= \frac{4e^{16 \beta J} +  16e^{8 \beta J} + 16}{(\cosh(8 \beta J) + 3)^{2}}
\end{align*}
%
\begin{align*}
	\langle M^{2} \rangle &= \frac{8e^{8 \beta J} +  8}{\cosh(8 \beta J) + 3}
\end{align*}
%
\begin{align*}
	C_{V} &= \frac{64 J^{2}}{N k_{\text{B}} T^{2}} \Big( \frac{3 \cosh(8 \beta J) + \cosh^{2}(8 \beta J) - \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \Big)
\end{align*}
%
\begin{align*}
	\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*}

\subsection{Markov Chain Monte Carlo methods}


\subsection{Implementation}


\end{document}