Start proofreading result section

latex/ising_model.tex

@@ -85,5 +85,20 @@
 % Appendix
 \subfile{sections/appendices}
 
-
 \end{document}
+
+% Method
+% Problem 1: OK
+% Problem 2: OK
+% Problem 3: OK - missing theory of phase transition
+% Problem 7:
+% Problem 9:
+
+% Results
+% Problem 4: 
+% Problem 5: 
+% Problem 6:  
+% Problem 8: 
+% Problem 9: 
+%
+%

latex/sections/appendices.tex

@@ -127,5 +127,20 @@ and susceptibility
     \frac{\chi}{N} &= \frac{4}{N k_{B} T} \bigg( \frac{3e^{8 \beta J} + e^{-8 \beta J} + 3}{(\cosh(8 \beta J) + 3)^{2}} \bigg) \ .
 \end{align*}
 
+\section{Change in total system energy}\label{sec:delta_energy}
+When we consider the change in energy after flipping a single spin, we evaluate 
+$\Delta E = E_{\text{after}} - E_{\text{before}}$. We find the $3^{2}$ values as
+\begin{align*}
+    \Delta E = -8J - (-8J) = 0 \\
+    \Delta E = -8J - 0 = -8J \\
+    \Delta E = -8J - 8J = -16J \\
+    \Delta E = 0 - (-8J) = 8J \\
+    \Delta E = 0 - 0 = 0 \\
+    \Delta E = 0 - 8J = -8J \\
+    \Delta E = 8J - (-8J) = 16J \\
+    \Delta E = 8J - 0 = 8J \\
+    \Delta E = 8J - 8J = 0,
+\end{align*}
+where the five distinct values are $\Delta E = \{-16J, -8J, 0, 8J, 16J\}$.
 
 \end{document}

latex/sections/methods.tex

@@ -177,6 +177,9 @@ Boltzmann constant we derive the remaining units, which can be found in Table
     \label{tab:units}
 \end{table}
 
+\subsection{Phase transition and critical temperature}\label{subsec:phase_critical}
+% P9 critical temperature
+
 \subsection{The Markov chain Monte Carlo method}\label{subsec:mcmc_method}
 Markov chains consist of a sequence of samples, where the probability of the next 
 sample depend on the probability of the current sample. Whereas the Monte Carlo 
@@ -202,8 +205,9 @@ At each step of flipping a spin, the change in energy is evaluated as
 \begin{align*}
     \Delta E &= E_{\text{after}} - E_{\text{before}} \ .
 \end{align*}
-We have to evaluate 
+Since the total system energy only takes three different values, the change in 
-
+energy can take $3^{2}$ values. However, there are only five distinct values 
+$\Delta E = \{-16J, -8J, 0, 8J, 16J\}$, which we find in Appendix \ref{sec:delta_energy}.
 \begin{figure}[H]
     \begin{algorithm}[H]
     \caption{Metropolis-Hastings Algorithm}
@@ -226,13 +230,22 @@ We have to evaluate
             \EndProcedure
         \end{algorithmic}
     \end{algorithm}
-    \caption{Algo}
 \end{figure}
-
+We can avoid computing the Boltzmann factor
-\subsection{Phase transition and critical temperature}\label{subsec:phase_critical}
+\begin{equation}
+    p(\mathbf{s} | T) = e^{-\beta \Delta E}
+    \label{eq:boltzmann_factor}
+\end{equation}
+at every spin flip, by using a look up table (LUT) with the possible values. We use 
+the change in energy $\Delta E$ as a key for the resulting value of the exponential 
+function in a hash map.
 
 \subsection{Implementation}\label{subsec:implementation}
-% P2
+% P3 boundary condition and if-tests
+To avoid the overhead of if-tests, and take advantage of the parallelization, we 
+define an index for every edge case. That is, for a spin at a given boundary index 
+we use a pre-set index (...), to avoid if-tests and reduce overhead and runtime.
+% P7 parallelization
 
 \subsection{Tools}\label{subsec:tools}
 The Ising model and MCMC methods are implemented in C++, and parallelized using 
@@ -244,17 +257,3 @@ addition, while optimizing our implementation we used the profiler
 
 \end{document}
 
-% Method
-% Problem 1: OK
-% Problem 2: 
-% Problem 3: 
-
-% Results
-% Problem 4: 
-% Problem 5: 
-% Problem 6: 
-% Problem 7: 
-% Problem 8: 
-% Problem 9: 
-%
-%

latex/sections/results.tex

@@ -2,5 +2,108 @@
 
 \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}
+
+
 
 \end{document}