182 lines
8.1 KiB
TeX
182 lines
8.1 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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% 2.1-2-4 divided into 40 steps, which gives us a step size of 0.0075.
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% 10 MPI processes
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% - 10 threads per process
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% = 100 threads total
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% Not a lot of downtime for the threads
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However, when the temperature is close to the critical point, we observe an increase
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in expected energy and a decrease in magnetization. Suggesting a higher energy and
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a loss of magnetization close to the critical temperature.
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% We did not set the seed for the random number generator, which resulted in
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% different numerical estimates each time we ran the model. However, all expectation
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% values are calculated using the same data. The burn-in time varied each time.
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% We see a burn-in time t = 5000-10000 MC cycles. However, this changed between runs.
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We decided with a burn-in time parallelization trade-off. That is, we set the
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burn-in time lower in favor of sampling. To take advantage of the parallelization
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and not to waste computational resources. The argument to discard samples generated
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during the burn-in time is ... Increasing number of samples outweigh the ...
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parallelize using MPI. We generated
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samples for the temperature range $T \in [2.1, 2.4]$. Using Fox we generated both
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1 million samples and 10 million samples.
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It is worth mentioning that the time (number of MC cycles) necessary to get a
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good numerical estimate, compared to the analytical result, foreshadowing the
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burn-in time.
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Markov chain starting point can differ, resulting in different simulation. By
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discarding the first samples, the ones generated before system equilibrium we can
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get an estimate closer to the real solution. Since we want to estimate expectation
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values at a given temperature, the samples should represent the system at that
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temperature.
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Depending on number of samples used in numerical estimates, using the samples
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generated during burn-in can in high bias and high variance if the ratio is skewed.
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However, if most samples are generated after burn-in the effect is not as visible.
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Can't remove randomness by starting around equilibrium, since samples are generated
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using several ising models we need to sample using the same conditions that is
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system state equilibrium.
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\subsection{Burn-in time}\label{subsec:burnin_time}
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$\boldsymbol{Draft}$
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We start with a lattice where $L = 20$, to study the burn-in time, that is the
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number of Monte Carlo cycles necessary for the system to reach an equilibrium.
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We consider two different temperatures $T_{1} = 1.0 J/k_{B}$ and $T_{2} = 2.4 J/k_{B}$,
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where $T_{2}$ is close to the critical temperature. We can use the correlation
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time $\tau \approx L^{d + z}$ to determine time, where $d$ is the dimensionality
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of the system and $z = 2.1665 \pm 0.0012$ \footnote{This value was determined by
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Nightingale and Blöte for the Metropolis algorithm.}
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% Need to include a section of Onsager's analytical result.
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We show the numerical estimates for temperature $T_{1}$ of $\langle \epsilon \rangle$
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in Figure \ref{fig:burn_in_energy_1_0} and $\langle |m| \rangle$ in Figure
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\ref{fig:burn_in_magnetization_1_0}. For temperature $T_{2}$, the numercal estimate
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of $\langle \epsilon \rangle$ is shown in Figure \ref{fig:burn_in_energy_2_4} and
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$\langle |m| \rangle$ in Figure \ref{fig:burn_in_magnetization_2_4}. The lattice
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is initialized in both an ordered and an unordered state. We observe that for
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$T_{1}$ there is no change in either expectation value with increasing number of
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Monte Carlo cycles, when we start with an ordered state. As for the unordered
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lattice, we observe a change for the first 5000 MC cycles, where it stabilizes.
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The approximated expected energy is $-2$ and expected magnetization is $1.0$,
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which is to be expected for temperature 0f $1.0$. T is below the critical and the
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pdf using $T = 1.0$ result in
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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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% 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{$\langle |m| \rangle$ as a function of time, 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{$\langle \epsilon \rangle$ as a function of time, 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{$\langle |m| \rangle$ as a function of time, 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{$\langle \epsilon \rangle$ as a function of time, 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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$\boldsymbol{Draft}$
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% Histogram figures
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We use the estimated burn-in time to set starting time for sampling, then generate
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samples to plot in a histogram for $T_{1}$ in Figure \ref{fig:histogram_1_0} and
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$T_{2}$ in Figure \ref{fig:histogram_2_4}. For $T_{1}$ we can see that most samples
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have the expected value $-2$, we have a distribution with low variance.
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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{Histogram $T = 1.0 J / k_{B}$}
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\label{fig:histogram_1_0}
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\end{figure}
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\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{Histogram $T = 2.4 J / k_{B}$}
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\label{fig:histogram_2_4}
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\end{figure}
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\subsection{Phase transition}\label{subsec:phase_transition}
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$\boldsymbol{Draft}$
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% Phase transition figures
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/energy.pdf}
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\caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
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\label{fig:phase_energy}
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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/phase_transition/fox/wide/1M/magnetization.pdf}
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\caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
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\label{fig:phase_magnetization}
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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/phase_transition/fox/wide/1M/heat_capacity.pdf}
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\caption{$C_{V}$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
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\label{fig:phase_heat}
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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/phase_transition/fox/wide/1M/susceptibility.pdf}
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\caption{$\chi$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
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\label{fig:phase_susceptibility}
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\end{figure}
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We include results for 10 million MC cycles in Appendix \ref{sec:extra_results}
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\subsection{Critical temperature}\label{subsec:critical_temperature}
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$\boldsymbol{Draft}$
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We use the critical temperatures found in previous section, in addition to the
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scaling relation in Equation \eqref{eq:critical_intinite}
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\begin{equation}
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T_{c} - T_{c}(L = \infty) = \alpha L^{-1}
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\label{eq:critical_intinite}
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\end{equation}
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to estimate the critical temperature for a lattize of infinte size. We also
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compared the estimate with the analytical solution
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\begin{equation}
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T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B}
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\end{equation}
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using linear regression. In Figure \ref{fig:linreg} we find the critical
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temperatures as function of the inverse lattice size. When the lattice size increase
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toward infinity, $1/L$ goes toward zero, we find the intercept which gives us an
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estimated value of the critical temperature for a lattice of infinite size.
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% Critical temp regression figure
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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 $T_{c}(L = \infty)$ and $\beta_{1}$ is the slope.}
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\label{fig:linreg}
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\end{figure}
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\end{document}
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