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

\begin{document}
\section{Results}\label{sec:results}
\subsection{Burn-in time}\label{subsec:burnin_time}
$\boldsymbol{Draft}$
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}


\subsection{Probability distribution}\label{subsec:probability_distribution}
$\boldsymbol{Draft}$
% Histogram figures
We use the estimated burn-in time to set starting time for sampling, then generate 
samples to plot in a histogram for $T_{1}$ in Figure \ref{fig:histogram_1_0} and 
$T_{2}$ in Figure \ref{fig:histogram_2_4}. For $T_{1}$ we can see that most samples 
have the expected value $-2$, we have a distribution with low variance.
\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}


\subsection{Phase transition}\label{subsec:phase_transition}
$\boldsymbol{Draft}$
% 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}
We include results for 10 million MC cycles in Appendix \ref{sec:extra_results}


\subsection{Critical temperature}\label{subsec:critical_temperature}
$\boldsymbol{Draft}$
We use the critical temperatures found in previous section, in addition to the 
scaling relation in Equation \eqref{eq:critical_intinite}
\begin{equation}
    T_{c} - T_{c}(L = \infty) = \alpha L^{-1}
    \label{eq:critical_intinite}
\end{equation}
to estimate the critical temperature for a lattize of infinte size. We also 
compared the estimate with the analytical solution 
\begin{equation}
    T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B}
\end{equation}
using linear regression. In Figure \ref{fig:linreg} we find the critical 
temperatures as function of the inverse lattice size. When the lattice size increase 
toward infinity, $1/L$ goes toward zero, we find the intercept which gives us an 
estimated value of the critical temperature for a lattice of infinite size.

% 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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