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

\begin{document}
\section{Results}\label{sec:results}
\subsection{Burn-in time}\label{subsec:burnin_time}
We start with a lattice where $L = 20$, to study the burn-in time, that is the 
number of Monte Carlo cycles necessary for the system to reach an equilibrium. 
We consider two different temperatures $T_{1} = 1.0 J/k_{B}$ and $T_{2} = 2.4 J/k_{B}$, 
where $T_{2}$ is close to the critical temperature. We can use the correlation 
time $\tau \approx L^{d + z}$ to determine time, where $d$ is the dimensionality 
of the system and $z = 2.1665 \pm 0.0012$ \footnote{This value was determined by 
Nightingale and Blöte for the Metropolis algorithm.}
% Need to include a section of Onsager's analytical result.
We show the numerical estimates for temperature $T_{1}$ of $\langle \epsilon \rangle$ 
in Figure \ref{fig:burn_in_energy_1_0} and $\langle |m| \rangle$ in Figure 
\ref{fig:burn_in_magnetization_1_0}. For temperature $T_{2}$, the numercal estimate 
of $\langle \epsilon \rangle$ is shown in Figure \ref{fig:burn_in_energy_2_4} and 
$\langle |m| \rangle$ in Figure \ref{fig:burn_in_magnetization_2_4}. The lattice 
is initialized in both an ordered and an unordered state. We observe that for 
$T_{1}$ there is no change in either expectation value with increasing number of 
Monte Carlo cycles, when we start with an ordered state. As for the unordered 
lattice, we observe a change for the first 5000 MC cycles, where it stabilizes. 
The approximated expected energy is $-2$ and expected magnetization is $1.0$, 
which is to be expected for temperature 0f $1.0$. T is below the critical and the 
pdf using $T = 1.0$ result in
\begin{align*}
    p(s|T=1.0) &= \frac{1}{e^{-\beta \sum E(s)}} e^{-\beta E(s)} \\
    &= \frac{1}{e^{-(1/k_{B}) \sum E(s)}} e^{-(1/k_{B}) E(s)} \ .
\end{align*}
% Burn-in figures
\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/burn_in_time_magnetization_1_0.pdf}
    \caption{$\langle |m| \rangle$ as a function of time, for $T = 1.0 J / k_{B}$}
    \label{fig:burn_in_magnetization_1_0}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/burn_in_time_energy_1_0.pdf}
    \caption{$\langle \epsilon \rangle$ as a function of time, for $T = 1.0 J / k_{B}$}
    \label{fig:burn_in_energy_1_0}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/burn_in_time_magnetization_2_4.pdf}
    \caption{$\langle |m| \rangle$ as a function of time, for $T = 2.4 J / k_{B}$}
    \label{fig:burn_in_magnetization_2_4}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/burn_in_time_energy_2_4.pdf}
    \caption{$\langle \epsilon \rangle$ as a function of time, for $T = 2.4 J / k_{B}$}
    \label{fig:burn_in_energy_2_4}
\end{figure}

% Histogram figures
\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/pd_estimate_1_0.pdf}
    \caption{Histogram $T = 1.0 J / k_{B}$}
    \label{fig:histogram_1_0}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/pd_estimate_2_4.pdf}
    \caption{Histogram $T = 2.4 J / k_{B}$}
    \label{fig:histogram_2_4}
\end{figure}

% Phase transition figures
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/energy.pdf}
    \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
    \label{fig:phase_energy}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/magnetization.pdf}
    \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
    \label{fig:phase_magnetization}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/heat_capacity.pdf}
    \caption{$C_{V}$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
    \label{fig:phase_heat}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/susceptibility.pdf}
    \caption{$\chi$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
    \label{fig:phase_susceptibility}
\end{figure}

% Critical temp regression figure 
\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/linreg.pdf}
    \caption{Linear regression, where $\beta_{0}$ is the intercept $T_{c}(L = \infty)$ and $\beta_{1}$ is the slope.}
    \label{fig:linreg}
\end{figure}



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