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Add draft for most of the method and result sections.

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Janita Willumsen
2023-11-30 15:24:05 +01:00
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latex/draft/draft.tex
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@@ -474,18 +474,18 @@
% \end{figure}
However, when the temperature is close to the critical point, we observe an increase in expected energy and a decrease in magnetization. Suggesting a higher energy and a loss of magnetization close to the critical temperature.
% However, when the temperature is close to the critical point, we observe an increase in expected energy and a decrease in magnetization. Suggesting a higher energy and a loss of magnetization close to the critical temperature.
% We did not set the seed for the random number generator, which resulted in different numerical estimates each time we ran the model. However, all expectation values are calculated using the same data. The burn-in time varied each time. We see a burn-in time t = 5000-10000 MC cycles. However, this changed between runs.
We decided with a burn-in time parallelization trade-off. That is, we set the burn-in time lower in favor of sampling. To take advantage of the parallelization and not to waste computational resources. The argument to discard samples generated during the burn-in time is ... Increasing number of samples outweigh the ...
% % We did not set the seed for the random number generator, which resulted in different numerical estimates each time we ran the model. However, all expectation values are calculated using the same data. The burn-in time varied each time. We see a burn-in time t = 5000-10000 MC cycles. However, this changed between runs.
% We decided with a burn-in time parallelization trade-off. That is, we set the burn-in time lower in favor of sampling. To take advantage of the parallelization and not to waste computational resources. The argument to discard samples generated during the burn-in time is ... Increasing number of samples outweigh the ...
It is worth mentioning that the time (number of MC cycles) necessary to get a good numerical estimate, compared to the analytical result, foreshadowing the burn-in time.
% It is worth mentioning that the time (number of MC cycles) necessary to get a good numerical estimate, compared to the analytical result, foreshadowing the burn-in time.
Markov chain starting point can differ, resulting in different simulation. By discarding the first samples, the ones generated before system equilibrium we can get an estimate closer to the real solution. Since we want to estimate expectation values at a given temperature, the samples should represent the system at that temperature.
% Markov chain starting point can differ, resulting in different simulation. By discarding the first samples, the ones generated before system equilibrium we can get an estimate closer to the real solution. Since we want to estimate expectation values at a given temperature, the samples should represent the system at that temperature.
Depending on number of samples used in numerical estimates, using the samples generated during burn-in can in high bias and high variance if the ratio is skewed. However, if most samples are generated after burn-in the effect is not as visible. Can't remove randomness by starting around equilibrium, since samples are generated using several ising models we need to sample using the same conditions that is system state equilibrium.
% Depending on number of samples used in numerical estimates, using the samples generated during burn-in can in high bias and high variance if the ratio is skewed. However, if most samples are generated after burn-in the effect is not as visible. Can't remove randomness by starting around equilibrium, since samples are generated using several ising models we need to sample using the same conditions that is system state equilibrium.
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.
% 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}
% \centering
@@ -562,17 +562,17 @@ When the speed-up was satisfactory, we investigated the phase transition for lat
% \label{fig:phase_susceptibility}
% \end{figure}
We include results using 10 million MC cycles in Appendix \ref{sec:extra}
% We include results using 10 million MC cycles in Appendix \ref{sec:extra}
We use the critical temperatures found studying the phase transition, in addition to the scaling relation
\begin{equation}
T_{c} - T_{c}(L = \infty) = \alpha L^{-1}
\end{equation}
to estimate the critical temperatur for a lattize of infinte size. We also compare 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 show the critical temperatures as function of the inverse lattice size. As size goes toward infinity we reach inv L = 0 and find the intercept which is equal to the estimated critical of L=infty.
% We use the critical temperatures found studying the phase transition, in addition to the scaling relation
% \begin{equation}
% T_{c} - T_{c}(L = \infty) = \alpha L^{-1}
% \end{equation}
% to estimate the critical temperatur for a lattize of infinte size. We also compare 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 show the critical temperatures as function of the inverse lattice size. As size goes toward infinity we reach inv L = 0 and find the intercept which is equal to the estimated critical of L=infty.
% \begin{figure}
% \centering
@@ -582,7 +582,7 @@ using linear regression. In Figure \ref{fig:linreg} we show the critical tempera
% \end{figure}
\subsection{Draft}
%\subsection{Draft}
% Test that the numerical stuff gets close to the analytical
% - validate implementation a given number of times, find average number of cycles
@@ -658,33 +658,33 @@ using linear regression. In Figure \ref{fig:linreg} we show the critical tempera
% \caption{Rules for multiplying spin pairs.}
% \end{figure}
\begin{figure}\label{fig:tikz_neighbor}
\centering
\begin{subfigure}{0.4\linewidth}
\begin{tikzpicture}
\draw (0, 0) grid (2, 2);
\node (s1) at (0.5, 1.5) {$\uparrow$};
\node (s2) at (1.5, 1.5) {$\uparrow$};
\node (s3) at (0.5, 0.5) {$\downarrow$};
\node (s4) at (1.5, 0.5) {$\downarrow$};
\end{tikzpicture}
\caption{}
\label{fig:sub_tikz_neighbor_a}
\end{subfigure}
\
\begin{subfigure}{0.4\linewidth}
\begin{tikzpicture}
\draw (0, 0) grid (2, 2);
\node (s1) at (0.5, 1.5) {$\uparrow$};
\node (s2) at (1.5, 1.5) {$\downarrow$};
\node (s3) at (0.5, 0.5) {$\downarrow$};
\node (s4) at (1.5, 0.5) {$\uparrow$};
\end{tikzpicture}
\caption{}
\label{fig:sub_tikz_neighbor_b}
\end{subfigure}
\caption{Possible spin configurations for two spins up.}
\end{figure}
% \begin{figure}\label{fig:tikz_neighbor}
% \centering
% \begin{subfigure}{0.4\linewidth}
% \begin{tikzpicture}
% \draw (0, 0) grid (2, 2);
% \node (s1) at (0.5, 1.5) {$\uparrow$};
% \node (s2) at (1.5, 1.5) {$\uparrow$};
% \node (s3) at (0.5, 0.5) {$\downarrow$};
% \node (s4) at (1.5, 0.5) {$\downarrow$};
% \end{tikzpicture}
% \caption{}
% \label{fig:sub_tikz_neighbor_a}
% \end{subfigure}
% \
% \begin{subfigure}{0.4\linewidth}
% \begin{tikzpicture}
% \draw (0, 0) grid (2, 2);
% \node (s1) at (0.5, 1.5) {$\uparrow$};
% \node (s2) at (1.5, 1.5) {$\downarrow$};
% \node (s3) at (0.5, 0.5) {$\downarrow$};
% \node (s4) at (1.5, 0.5) {$\uparrow$};
% \end{tikzpicture}
% \caption{}
% \label{fig:sub_tikz_neighbor_b}
% \end{subfigure}
% \caption{Possible spin configurations for two spins up.}
% \end{figure}
%----------
% APPENDIX
@@ -856,35 +856,35 @@ using linear regression. In Figure \ref{fig:linreg} we show the critical tempera
% \langle |m| \rangle = \frac{1}{N} \sum_{i=1}^{N} |M(s_{i})| p(s_{i} \ | \ T)
% \end{align*}
\section{Extra}\label{sec:extra}
We increased number of MC cycles to 10 million
\begin{figure}
\centering
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf}
\caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
\label{fig:phase_energy_10M}
\end{figure}
% \section{Extra}\label{sec:extra}
% We increased number of MC cycles to 10 million
% \begin{figure}
% \centering
% \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf}
% \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
% \label{fig:phase_energy_10M}
% \end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf}
\caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
\label{fig:phase_magnetization_10M}
\end{figure}
% \begin{figure}
% \centering
% \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf}
% \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
% \label{fig:phase_magnetization_10M}
% \end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf}
\caption{$C_{V}$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
\label{fig:phase_heat_10M}
\end{figure}
% \begin{figure}
% \centering
% \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf}
% \caption{$C_{V}$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
% \label{fig:phase_heat_10M}
% \end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf}
\caption{$\chi$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
\label{fig:phase_susceptibility_10M}
\end{figure}
% \begin{figure}
% \centering
% \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf}
% \caption{$\chi$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
% \label{fig:phase_susceptibility_10M}
% \end{figure}
\end{document}