\documentclass[../ising_model.tex]{subfiles}

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