20 lines
1.1 KiB
TeX
20 lines
1.1 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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% Draft based on abstract
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We have used the Ising model to study the behavior in ferromagnets, when undergoing
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a phase transition near a critical temperature. We generated samples using the
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Markov chain Monte Carlo method, while utilizing methods of parallelization.
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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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