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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
parent ad18cabfdd
commit abb13ef2da
8 changed files with 254 additions and 96 deletions
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36
latex/sections/appendices.tex
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@@ -6,7 +6,7 @@
When two spins have the orientation up, the total magnetization is zero. However,
the total energy of the system have two possible values, due to the location of
the spin up as visualized in Figure \ref{fig:tikz_neighbor}.
\begin{figure}\label{fig:tikz_neighbor}
\begin{figure}
\centering
\begin{subfigure}{0.4\linewidth}
\begin{tikzpicture}
@@ -35,6 +35,7 @@ the spin up as visualized in Figure \ref{fig:tikz_neighbor}.
\label{fig:tikz_neighbor}
\end{figure}
\section{Partition function}\label{sec:partition_function}
Using the values estimated for the $2 \times 2$ case, found in \ref{tab:lattice_config},
we find the partition function
@@ -49,6 +50,7 @@ and get
z &= 4 \cosh (8 \beta J) + 12 \ .
\end{align*}
\section{Expectation values}\label{sec:expectation_values}
For a linear function of a stochastic random variable $X$, with a known probability
distribution, the expected value of $x$ is given by
@@ -160,6 +162,7 @@ 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
@@ -176,4 +179,35 @@ $\Delta E = E_{\text{after}} - E_{\text{before}}$. We find the $3^{2}$ values as
\end{align*}
where the five distinct values are $\Delta E = \{-16J, -8J, 0, 8J, 16J\}$.
\section{Extra}\label{sec:extra_results}
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/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}
\end{document}