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\section{Conclusion}\label{sec:conclusion}
We studied the ferromagnetic behavior of the Ising model, and observed a phase 
transition at temperatures close to the critical temperature. We used a Markov 
chain Monte Carlo sampling method to generate spin configurations, while utilizing 
parallel programming techniques. When increasing number of processes, and number 
of threads we observed a speed-up in runtime. 

We initialized the lattices using both ordered and unordered spin configuration, 
and started sampling after the system reached an equilibrium. The number of Monte 
Carlo cycles necessary to reach a system equilibrium, referred to as burn-in time, 
was estimated to be $5000$ cycles. We found that excluding the samples generated 
during the burn-in time, improves the estimated expectation value of energy and 
magnetization, in addition to the heat capacity and susceptibility, when samples 
are sparse. However, when we increase number of samples, excluding the burn-in 
samples does not affect the estimated values.

Continuing, we used the generated samples to compute energy per spin $\langle \epsilon \rangle$, 
magnetization per spin $\langle |m| \rangle$, heat capacity $C_{V}$, and $\chi$. 
In addition, we estimated the probability distribution for temperatures $T_{1} = 1.0 J / k_{B}$,
and $T_{2} = 2.4 J / k_{B}$. We found that for $T_{1}$ the expected mean energy 
per spin is $\langle \epsilon \rangle \approx -1.9969 J$, with a variance $\text{Var} (\epsilon) = 0.0001$. 
And for $T_{2}$, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2370 J$, 
with a variance $\text{Var} (\epsilon) = 0.0203$. 

We estimated the expected energy and magnetization per spin, in addition to the 
heat capacity and susceptibility for lattices of size $L = {20, 40, 60, 80, 100}$. 
We observed a phase transition in the temperature range $T \in [2.1, 2.4] J / k_{B}$.
Using the values from the finite lattices, we approximated the critical temperature 
of a lattice of infinite size. Using linear regression, we numerically estimated 
$T_{c}(L = \infty) \approx 2.2695 J/k_{B}$ which is close to the analytical solution 
$T_{C}(L = \infty) \approx 2.269 J/k_{B}$ Lars Onsager found in 1944. 
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