36 lines
2.3 KiB
TeX
36 lines
2.3 KiB
TeX
\documentclass[../ising_model.tex]{subfiles}
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\begin{document}
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\section{Conclusion}\label{sec:conclusion}
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We studied the ferromagnetic behavior of the Ising model, and observed a phase
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transition at temperatures close to the critical temperature. We used a Markov
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chain Monte Carlo sampling method to generate spin configurations, while utilizing
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parallel programming techniques. When increasing number of processes, and number
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of threads we observed a speed-up in runtime.
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We initialized the lattices using both ordered and unordered spin configuration,
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and started sampling after the system reached an equilibrium. The number of Monte
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Carlo cycles necessary to reach a system equilibrium, referred to as burn-in time,
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was estimated to be $5000$ cycles. We found that excluding the samples generated
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during the burn-in time, improves the estimated expectation value of energy and
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magnetization, in addition to the heat capacity and susceptibility, when samples
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are sparse. However, when we increase number of samples, excluding the burn-in
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samples does not affect the estimated values.
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Continuing, we used the generated samples to compute energy per spin $\langle \epsilon \rangle$,
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magnetization per spin $\langle |m| \rangle$, heat capacity $C_{V}$, and $\chi$.
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In addition, we estimated the probability distribution for temperatures $T_{1} = 1.0 J / k_{B}$,
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and $T_{2} = 2.4 J / k_{B}$. We found that for $T_{1}$ the expected mean energy
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per spin is $\langle \epsilon \rangle \approx -1.9969 J$, with a variance $\text{Var} (\epsilon) = 0.0001$.
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And for $T_{2}$, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2370 J$,
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with a variance $\text{Var} (\epsilon) = 0.0203$.
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We estimated the expected energy and magnetization per spin, in addition to the
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heat capacity and susceptibility for lattices of size $L = {20, 40, 60, 80, 100}$.
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We observed a phase transition in the temperature range $T \in [2.1, 2.4] J / k_{B}$.
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Using the values from the finite lattices, we approximated the critical temperature
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of a lattice of infinite size. Using linear regression, we numerically estimated
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$T_{c}^{*}(L = \infty) \approx 2.2693 J/k_{B}$ which is close to the analytical solution
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$T_{c}(L = \infty) \approx 2.269 J/k_{B}$ Lars Onsager found in 1944.
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\end{document}
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