25 lines
1.3 KiB
TeX
25 lines
1.3 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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$\boldsymbol{Draft}$
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We have studied ferromagnetism using the Ising model, and observed
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a phase transition when the temperature of the system is close to the critical
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temperature. We used the Markov chain Monte Carlo method to generate samples of
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spin configurations, while utilizing methods of parallelization.
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We use the 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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Finding the burn-in time to be approx. 3000 Monte Carlo cycles. For temperature
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$T = 1.0 J / k_{B}$ we found a propability distrobution with an expected mean
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value of $\mu \approx -1.9969$ and variation $\sigma^{2} = 0.0001$. Whereas the
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pdf close to the critical temperature is $\mu \approx -1.2370$ and variation
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$\sigma^{2} = 0.0203$. We estimated the expected energy and magnetization per spin,
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in addition to the heat capacity and susceptibility. Using the values from
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finite sized lattices, we approximated the critical temperature of an infinite
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sized lattice. Using linear regression, we numerically estimated $T_{c}$ $T_{C}(L = \infty) \approx 2.2695$
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which is close to the analytical solution $T_{C}(L = \infty) \approx 2.269 J/k_{B}$
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found by Lars Onsager.
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\end{document}
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