\section*{Results}

\subsection*{Relaxation Time (Time to Equilibrium)}

The first question of modelling was to determine when the system had reached equilibrium. As discussed in Section \ref{system-convergence} \TKK, the method chosen to determine the number of sweeps after which this had occurred, $N_0$ (also known as the transient boundary) was autocorrelation. The data processed was the 1\textdegree thermodynamic variables, the mean dimensionless energy $\widetilde{E}$ and mean magnetisation $m$.

To start we can show that the system does indeed attain equilibrium by observing the behaviour at a singular value of $\beta_0 = 0.25$ as shown in Figure \ref{single-beta-convergence}. Here we can see the system attains a stable state at $N_s \approx 25$.

\begin{figure}[hbt]
  \label{single-beta-convergence}
  \includegraphics[width=\columnwidth]{figures/single-beta-convergence.png}
  \caption{The mean magnetisation $m$ and mean dimensionless energy $\widetilde{E}$ as they evolve over sweeps of the Monte-Carlo simulation. Both these quantities are normalised relative to their maximum modulus value, so that they can be shown on the same value and compared. This results in a scaling to the range of $[-1, 1]$ such that the sign of the value is preserved.}
\end{figure}

\begin{figure}[hbt]
  \label{beta-relaxation-time}
  \includegraphics[width=\columnwidth]{figures/transient-boundary-vs-beta.png}
  \caption{The number of sweeps required for equilibrium to be reached (aka the transient boundary) compared to the $\beta_0$, the dimensionless inverse temperature. Also shown is the a line showing the position of $\beta_c$. This data is comprised of a mean of value of $10$ runs with different seeds. Here we focus on the region surrounding the critical temperature $\beta_c$. A $\beta_0$ increment of $0.001$ was used, with $N_\mathrm{max}$, the maximum number of sweeps being $500$ far from the critical temperature, and $10,000$ close to it. TKK define all these terms}
\end{figure}

Next the question is show this varies across the range of $\beta_0$. This is shown in Figure \ref{beta-relaxation-time} where we can see the relaxation time when computed in terms of the $\widetilde E$ and $m$ both. Here we can see expected that far from the critical temperature (relatively speaking) $N_0$ decays quickly, spiking just before the critical temperature and decaying after it.

This data allows us to determine the portion of each run which is at equilibrium and hence to which we can apply the relevant statistical and statistical mechanical methods for analysis (most notably the Ergodic hypothesis TKK in thermodynamics section.)

\subsection*{Thermodynamic Observables}

Using this data to extract steady state information we are able to determine values of the other thermodynamic observables we desire including, the modulus magnetisation $\abs{m}$, the magnetic susceptibility $\chi$, and heat-capacity $c$.


\subsection*{Comparison of Models}

\subsection*{Correlation Length}

\subsection*{Critical Exponents}


\section*{Conclusion}
