Add draft for most of the method and result sections.

latex/sections/results.tex

@@ -3,6 +3,7 @@
 \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 $T_{2} = 2.4 J/k_{B}$, 
@@ -53,7 +54,14 @@ pdf using $T = 1.0$ result in
     \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}
@@ -67,6 +75,9 @@ pdf using $T = 1.0$ result in
     \label{fig:histogram_2_4}
 \end{figure}
 
+
+\subsection{Phase transition}\label{subsec:phase_transition}
+$\boldsymbol{Draft}$
 % Phase transition figures
 \begin{figure}
     \centering
@@ -95,6 +106,26 @@ pdf using $T = 1.0$ result in
     \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]