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

\begin{document}
\begin{abstract}
    We have studied the ferromagnetic behavior of the Ising model at a critical 
    temperature, when undergoing phase transition. To generate spin configurations, 
    we used the Metropolis-Hastings algorithm, which applies a Markov chain Monte 
    Carlo sampling method. We determined the time of equilibrium to be approximately 
    3000 Monte Carlo cycles, and used the following samples to find the probability 
    distribution at temperature $T_{1} = 1.0 J / k_{B}$, and $T_{2} = 2.4 J / k_{B}$.
    For $T_{1}$ the mean energy per spin is $\langle \epsilon \rangle \approx -1.9969 J$, 
    with a variation $\text{Var} (\epsilon) = 0.0001$. And for $T_{2}$, close to the critical 
    temperature, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2370 J$, 
    with a variation $\text{Var} (\epsilon) = 0.0203$. In addition, we estimated 
    the expected energy and magnetization per spin, the heat capacity and magnetic 
    susceptibility. We estimate the critical temperatures of finite lattice sizes, 
    and used these values to approximate the critical temperature of a lattice of 
    infinite size. Using linear regression, we estimated the critical temperature 
    to be $T_{c}(L = \infty) \approx 2.2695 J/k_{B}$.
\end{abstract}
\end{document}
