22 lines
1.3 KiB
TeX
22 lines
1.3 KiB
TeX
\documentclass[../ising_model.tex]{subfiles}
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\begin{document}
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\begin{abstract}
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We have studied the ferromagnetic behavior of the Ising model at a critical
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temperature, when undergoing phase transition. To generate spin configurations,
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we used the Metropolis-Hastings algorithm, which applies a Markov chain Monte
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Carlo sampling method. We determined the time of equilibrium to be approximately
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$5000$ Monte Carlo cycles, and used the following samples to find the probability
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distribution at temperature $T_{1} = 1.0 J / k_{B}$, and $T_{2} = 2.4 J / k_{B}$.
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For $T_{1}$ the mean energy per spin is $\langle \epsilon \rangle \approx -1.9969 J$,
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with a variance $\text{Var} (\epsilon) = 0.0001$. And for $T_{2}$, close to the critical
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temperature, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2370 J$,
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with a variance $\text{Var} (\epsilon) = 0.0203$. In addition, we estimated
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the expected energy and magnetization per spin, the heat capacity and magnetic
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susceptibility. We have estimated the critical temperatures of finite lattice sizes,
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and used these values to approximate the critical temperature of a lattice of
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infinite size. Using linear regression, we estimated the critical temperature
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to be $T_{c}^{*}(L = \infty) \approx 2.2693 J/k_{B}$.
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\end{abstract}
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\end{document}
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