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\section{Conclusion}\label{sec:conclusion}
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We have studied ferromagnetism using the Ising model, and observed 
a phase transition when the temperature of the system is close to the critical 
temperature. We used the Markov chain Monte Carlo method to generate samples of 
spin configurations, while utilizing methods of parallelization. 


We use the samples to compute energy per spin $\langle \epsilon \rangle$, 
magnetization per spin $\langle |m| \rangle$, heat capacity $C_{V}$, and $\chi$. 
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. 
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