194 lines
11 KiB
TeX
194 lines
11 KiB
TeX
\documentclass[../ising_model.tex]{subfiles}
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\begin{document}
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\section{Results}\label{sec:results}
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\subsection{Burn-in time}\label{subsec:burnin_time}
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We started with a lattice size $L = 20$ and considered the temperatures $T_{1} = 1.0 J/k_{B}$,
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and $T_{2} = 2.4 J/k_{B}$, where $T_{2}$ is close to the analytical critical
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temperature given by Equation \eqref{eq:critical_solution}.
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To determine the burn-in time, we used the numerical estimates of energy
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per spin for $T_{1}$ in Figure \ref{fig:burn_in_energy_1_0}, and $T_{2}$ in Figure
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\ref{fig:burn_in_energy_2_4}. We also considered the estimates of magnetization
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per spin for $T_{1}$ in Figure \ref{fig:burn_in_magnetization_1_0}, and for $T_{2}$
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in Figure \ref{fig:burn_in_energy_2_4}.
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% Not sure about the relevance of the paragraph below
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% We used random numbers to set the initial state and the index of spin to flip, and
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% observed different graphs ... However, when we set the seed of the random number
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% engine, we could reproduce the results at every run. It is possible to determine
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% the burn-in time analytically, using the correlation time given by $\tau \approx L^{d + z}$.
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% Where $d$ is the dimensionality of the system and $z = 2.1665 \pm 0.0012$
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% \footnote{This value was determined by Nightingale and Blöte for the Metropolis algorithm.}.
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% However, when we increased number of Monte Carlo cycles the sampled generated during
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% the burn-in time did not affect the mean value (...). % Should add result showing this
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The lattice was initialized in an ordered and an unordered state, for both temperatures. We observed
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no change in expectation value of energy or magnetization for $T_{1}$, when we
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initialized the lattice in an ordered state. As for the unordered initialized
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lattice, we first observed a change in expectation values, and a stabilization around
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$5000$ Monte Carlo cycles. The expected energy per spin is $\langle \epsilon \rangle = -2$
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and the expected magnetization per spin is $\langle |m| \rangle = 1.0$. % add something about what is expected for $T_{1}$ ?
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For $T_{2}$ we observed a change in expectation values for both the ordered and the
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unordered lattice.
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% \begin{align*}
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% p(s|T=1.0) &= \frac{1}{e^{-\beta \sum E(s)}} e^{-\beta E(s)} \\
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% &= \frac{1}{e^{-(1/k_{B}) \sum E(s)}} e^{-(1/k_{B}) E(s)} \ .
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% \end{align*}
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For $T_{2}$ we observe an increase in expected energy per spin $\langle \epsilon \rangle \approx -1.23$,
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and a decrease in expected magnetization per spin $\langle |m| \rangle \approx 0.46$.
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% Burn-in figures
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/burn_in_time_magnetization_1_0.pdf}
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\caption{Magnetization per spin $\langle |m| \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 1.0 J / k_{B}$}
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\label{fig:burn_in_magnetization_1_0}
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\end{figure}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/burn_in_time_energy_1_0.pdf}
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\caption{Energy per spin $\langle \epsilon \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 1.0 J / k_{B}$}
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\label{fig:burn_in_energy_1_0}
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\end{figure}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/burn_in_time_magnetization_2_4.pdf}
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\caption{Magnetization per spin $\langle |m| \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 2.4 J / k_{B}$}
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\label{fig:burn_in_magnetization_2_4}
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\end{figure}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/burn_in_time_energy_2_4.pdf}
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\caption{Energy per spin $\langle \epsilon \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 2.4 J / k_{B}$}
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\label{fig:burn_in_energy_2_4}
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\end{figure}
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\subsection{Probability distribution}\label{subsec:probability_distribution}
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% Histogram figures
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We used the estimated burn-in time of $5000$ Monte Carlo cycles as starting time, and generated
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samples. To visualize the distribution of energy per spin $\epsilon$, we used histograms
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with a bin size $0.02$. In Figure \ref{fig:histogram_1_0} we show the distribution
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for $T_{1}$. Where the resulting expectation
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value of energy per spin is $\langle \epsilon \rangle = -1.9969$, with a low variance
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of Var$(\epsilon) = 0.0001$. %
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/pd_estimate_1_0.pdf}
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\caption{Distribution of values of energy per spin, when temperature is $T = 1.0 J / k_{B}$}
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\label{fig:histogram_1_0}
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\end{figure} %
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In Figure \ref{fig:histogram_2_4}, for $T_{2}$, the samples of energy per spin is
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centered around the expectation value $\langle \epsilon \rangle = -1.2370$. %
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/pd_estimate_2_4.pdf}
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\caption{Distribution of values of energy per spin, when temperature is $T = 2.4 J / k_{B}$}
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\label{fig:histogram_2_4}
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\end{figure} %
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However, we observed a higher variance of Var$(\epsilon) = 0.0203$. When the temperature
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increased, the system moved from an ordered to an unordered state. The change in
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system state, or phase transition, indicates the temperature is close to a critical
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point.
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\subsection{Phase transition}\label{subsec:phase_transition}
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% Phase transition figures
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We continued investigating the behavior of the system around the critical temperature.
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First, we generated $10$ million samples of spin configurations, per temperature, for lattices of
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size $L \in \{20, 40, 60, 80, 100\}$, and temperatures $T \in [2.1, 2.4]$. We divided the
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temperature range into $40$ steps, with an equal step size of $0.0075$. The samples
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were generated in parallel, where the program allocated $4$ sequential temperatures to $10$
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MPI processes. Each process was set to spawn $10$ threads, resulting in a total of
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$100$ threads working in parallel. We include results for $1$ million MC cycles
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in Appendix \ref{sec:additional_results}
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To evaluate the performance of the parallelization, we used a profiler. The
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assessment output can be found in Appendix \ref{sec:additional_results} in Figure
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\ref{fig:scorep_assessment}. The assessment shows a lower score for the MPI load
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balance, compared to the OpenMP load balance. The master process gatheres all the
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data using blocking communication, resulting in the other processes waiting. This
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results in one process, the master, having to work more. The OpenMP load balance
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score is very good, suggesting that the threads are not left idle for long time periods.
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In Figure \ref{fig:phase_energy_10M}, for the larger lattices we observe a sharper
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increase in $\langle \epsilon \rangle$ in the temperature range $T \in [2.25, 2.35]$. %]
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf}
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\caption{Expected energy per spin $\langle \epsilon \rangle$ for temperatures $T \in [2.1, 2.4]$, and $10^7$ MC cycles.}
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\label{fig:phase_energy_10M}
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\end{figure} %
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We observe a decrease in $\langle |m| \rangle$ for the same temperature range in
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Figure \ref{fig:phase_magnetization_10M}, suggesting that the system moves from an ordered
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magnetized state to a state of no net magnetization. The system energy increases,
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however, there is a loss of magnetization close to the critical temperature.
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf}
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\caption{Expected magnetization per spin $\langle |m| \rangle$ for temperatures $T \in [2.1, 2.4]$, and $10^7$ MC cycles.}
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\label{fig:phase_magnetization_10M}
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\end{figure} %
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In Figure \ref{fig:phase_heat_10M}, we observe an increase in heat capacity in the
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temperature range $T \in [2.25, 2.35]$. In addition, we observe a sharper peak
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value of heat capacity when the lattice size increase.
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf}
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\caption{Heat capacity $C_{V}$ for temperatures $T \in [2.1, 2.4]$, and $10^7$ MC cycles.}
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\label{fig:phase_heat_10M}
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\end{figure} %
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The magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, shows
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the sharp peak in the same temperature range as that of the heat capacity. Since the
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shape of the curve for both heat capacity and the magnetic susceptibility become
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sharper when we increase lattice size, we are moving closer to the critical temperature.
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf}
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\caption{Magnetic susceptibility $\chi$ for temperatures $T \in [2.1, 2.4]$, and $10^7$ MC cycles.}
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\label{fig:phase_susceptibility_10M}
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\end{figure} %
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% Add something about the correlation length?
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\subsection{Critical temperature}\label{subsec:critical_temperature}
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Based on the heat capacity (Figure \ref{fig:phase_heat_10M}) and susceptibility
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(Figure \ref{fig:phase_susceptibility_10M}), we estimated the critical temperatures
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of lattices of size $L \in \{20, 40, 60, 80, 100\}$ found in Table \ref{tab:critical_temperatures}.
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% Tc wide 10M = 2.37, 2.325, 2.3025, 2.295, 2.2875
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\begin{table}[H]
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\centering
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\begin{tabular}{cc} % @{\extracolsep{\fill}}
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\hline
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$L$ & $T_{c}(L)$ \\
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\hline
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$20$ & $2.37 J / k_{B}$ \\
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$40$ & $2.325 J / k_{B}$ \\
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$60$ & $2.3025 J / k_{B}$ \\
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$80$ & $2.295 J / k_{B}$ \\
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$100$ & $2.2875 J / k_{B}$ \\
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\hline
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\end{tabular}
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\caption{Estimated critical temperatures for lattices $L \times L$, where $L$ denote the lattice size.}
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\label{tab:critical_temperatures}
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\end{table}
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We used the critical temperatures of finite lattices and the scaling relation in
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Equation \eqref{eq:critical_infinite}, Section \ref{subsec:phase_critical}, to estimate
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the critical temperature of a lattice of infinite size. In Figure \ref{fig:linreg_10M},
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we plot the critical temperatures $T_{c}(L)$ of the inverse lattice size $1/L$. When
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the lattice size increase toward infinity, $1/L$ approaches zero. %
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/linreg.pdf}
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\caption{Linear regression, where $\beta_{0}$ is the intercept approximating $T_{c}(L = \infty)$, and $\beta_{1}$ is the slope.}
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\label{fig:linreg_10M}
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\end{figure}
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We used linear regression to find the intercept $\beta_{0}$, which gives us an estimated value
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of the critical temperature for a lattice of infinite size. The estimated critical temperature
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is $T_{c}^{*}(L = \infty) \approx 2.2693 J/k_{B}$. We also compared the
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estimate with the analytical solution, the relative error of our estimate is
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\begin{equation*}
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\text{Relative error} = \frac{T_{c}^{*} - T_{c}}{T_{c}} \approx 5.05405 \cdot 10^{-5} J/k_{B}
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\end{equation*}
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\end{document}
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