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Produced plot with correct layout

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Janita Willumsen
2023-12-04 15:47:40 +01:00
parent 90047decb4
commit eb589ee768
27 changed files with 29 additions and 14 deletions
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latex/sections/results.tex
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@@ -149,17 +149,18 @@ sharper when we increase lattice size, we are moving closer to the critical temp
Based on the heat capacity (Figure \ref{fig:phase_heat_10M}) and susceptibility
(Figure \ref{fig:phase_susceptibility_10M}), we estimated the critical temperatures
of lattices of size $L \in \{20, 40, 60, 80, 100\}$ found in Table \ref{tab:critical_temperatures}.
% Tc wide 10M = 2.37, 2.325, 2.3025, 2.295, 2.2875
\begin{table}[H]
\centering
\begin{tabular}{cc} % @{\extracolsep{\fill}}
\hline
$L$ & $T_{c}(L)$ \\
\hline
$20$ & $ J / k_{B}$ \\
$40$ & $ J / k_{B}$ \\
$60$ & $ J / k_{B}$ \\
$80$ & $ J / k_{B}$ \\
$100$ & $ J / k_{B}$ \\
$20$ & $2.37 J / k_{B}$ \\
$40$ & $2.325 J / k_{B}$ \\
$60$ & $2.3025 J / k_{B}$ \\
$80$ & $2.295 J / k_{B}$ \\
$100$ & $2.2875 J / k_{B}$ \\
\hline
\end{tabular}
\caption{Estimated critical temperatures for lattices $L \times L$, where $L$ denote the lattice size.}
@@ -172,16 +173,16 @@ we plot the critical temperatures $T_{c}(L)$ of the inverse lattice size $1/L$.
the lattice size increase toward infinity, $1/L$ approaches zero. %
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/linreg.pdf}
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/linreg.pdf}
\caption{Linear regression, where $\beta_{0}$ is the intercept approximating $T_{c}(L = \infty)$, and $\beta_{1}$ is the slope.}
\label{fig:linreg_10M}
\end{figure}
Using linear regression, we find the intercept which gives us an estimated value
of the critical temperature for a lattice of infinite size. We find the critical
temperature to be $T_{c \text{num}} \approx 2.2695 J/k_{B}$. We also compared the
We used linear regression to find the intercept $\beta_{0}$, which gives us an estimated value
of the critical temperature for a lattice of infinite size. The estimated critical temperature
is $T_{c \text{num}} \approx 2.2693 J/k_{B}$. We also compared the
estimate with the analytical solution, the relative error of our estimate is
\begin{equation*}
\text{Relative error} = \frac{T_{c \text{ numerical}} - T_{c \text{ analytical}}}{T_{c \text{ analytical}}} \approx 0.001 J/k_{B}
\text{Relative error} = \frac{T_{c \text{ numerical}} - T_{c \text{ analytical}}}{T_{c \text{ analytical}}} \approx 5.05405 \cdot 10^{-5} J/k_{B}
\end{equation*}
\end{document}