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Fixed a few things from review.

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Janita Willumsen 2023-12-03 21:53:38 +01:00
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latex/sections/introduction.tex
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@ -24,6 +24,6 @@ estimate the critical temperature where the system experience phase transition.
In Section \ref{sec:methods}, we will present the theoretical background for In Section \ref{sec:methods}, we will present the theoretical background for
this experiment, as well as algorithms and tools used in the implementation. this experiment, as well as algorithms and tools used in the implementation.
Continuing with Section \ref{sec:results}, where we present our results and Continuing with Section \ref{sec:results}, we will present our results and
discuss our findings. Lastly, we conclude our findings in Section \ref{sec:conclusion}. discuss our findings. Lastly, we conclude our findings in Section \ref{sec:conclusion}.
\end{document} \end{document}

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latex/sections/methods.tex
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@ -15,15 +15,15 @@ entire lattice, the system spin configuration is represented as a matrix $L \tim
s_{L,1} & s_{L,2} & \dots & s_{L,L} s_{L,1} & s_{L,2} & \dots & s_{L,L}
\end{pmatrix} \ . \end{pmatrix} \ .
\end{align*} % $\mathbf{s} = [s_{1}, s_{2}, \dots, s_{N}]$. \end{align*} % $\mathbf{s} = [s_{1}, s_{2}, \dots, s_{N}]$.
The total number of possible spin configurations, also called microstates, is $2^{N}$. The total number of possible spin configurations, also called microstates, is
$|\mathbf{s}| = 2^{N}$.
A given spin $i$ can take one of two possible discrete values $s_{i} \in \{-1, +1\}$, A given spin $i$ can take one of two possible discrete values $s_{i} \in \{-1, +1\}$.
where $-1$ represent down and $+1$ spin up. The spins interact with its nearest neighbors, The spins interact with its nearest neighbors, and in a two-dimensional lattice
and in a two-dimensional lattice each spin has up to four nearest neighbors. However, each spin has up to four nearest neighbors. In our experiments we will use periodic
the model is not restricted to this dimentionality \cite[p. 3]{obermeyer:2020:ising}. boundary conditions, resulting in all spins having exactly four nearest neighbors.
In our experiment we will use periodic boundary conditions, meaning all spins To find the analytical expressions necessary for validating our model implementation,
have exactly four nearest neighbors. To find the analytical expressions necessary we will assume a $2 \times 2$ lattice.
for validating our model implementation, we will assume a $2 \times 2$ lattice.
The hamiltonian of the Ising model is given by The hamiltonian of the Ising model is given by
\begin{equation} \begin{equation}
@ -37,8 +37,8 @@ where $B = 0$, and find the total system energy given by
E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} \ . E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} \ .
\label{eq:energy_total} \label{eq:energy_total}
\end{equation} \end{equation}
To avoid counting duplicated, we count the neighboring spins using the pattern We count the neighboring spins using the pattern visualized in Figure \ref{fig:tikz_boundary},
visualized in Figure \ref{fig:tikz_boundary}. to avoid counting a given spin pair several times.
\begin{figure} \begin{figure}
\centering \centering
\begin{tikzpicture} \begin{tikzpicture}
@ -196,7 +196,7 @@ Boltzmann constant we derive the remaining units, which can be found in Table
$[\chi]$ & $1 / J$ & \\ $[\chi]$ & $1 / J$ & \\
\hline \hline
\end{tabular} \end{tabular}
\caption{Values of total energy, total magnetization, and degeneracy for the possible states of a system for a $2 \times 2$ lattice, with periodic boundary conditions.} \caption{Values of the total energy, total magnetization, and degeneracy for the possible states of a system for a $2 \times 2$ lattice, with periodic boundary conditions.}
\label{tab:units} \label{tab:units}
\end{table} \end{table}
@ -204,7 +204,7 @@ Boltzmann constant we derive the remaining units, which can be found in Table
\subsection{Phase transition and critical temperature}\label{subsec:phase_critical} \subsection{Phase transition and critical temperature}\label{subsec:phase_critical}
We consider the Ising model in two dimensions, with no external external magnetic We consider the Ising model in two dimensions, with no external external magnetic
field. At temperatures below the critical temperature $T_{c}$, the Ising model will field. At temperatures below the critical temperature $T_{c}$, the Ising model will
magnetize spontaneous. When increasing the temperature of the external field, magnetize spontaneously. When increasing the temperature of the external field,
the Ising model transition from an ordered to an unordered phase. The spins become the Ising model transition from an ordered to an unordered phase. The spins become
more correlated, and we can measure the discontinous behavior as an increase in more correlated, and we can measure the discontinous behavior as an increase in
correlation length $\xi (T)$ \cite[p. 432]{hj:2015:comp_phys}. At $T_{c}$, the correlation length $\xi (T)$ \cite[p. 432]{hj:2015:comp_phys}. At $T_{c}$, the
@ -282,7 +282,7 @@ in an array.
\end{algorithm} \end{algorithm}
\end{figure} \end{figure}
The Markov process reach an equilibrium after a certain number of Monte Carlo cycles, The Markov process reach an equilibrium after a certain number of Monte Carlo cycles,
where the system state is reflecting the state of a real system. After this burn-in time, where the system state reflects the state of a real system. After this burn-in time,
given by number of Monte Carlo cycles, we can start sampling microstates. given by number of Monte Carlo cycles, we can start sampling microstates.
The probability distribution of the samples will tend toward the actual probability The probability distribution of the samples will tend toward the actual probability
distribution of the system. distribution of the system.
@ -300,14 +300,15 @@ constants, where the first column contain the indices for neighbors to the left
and up, and the second column right and down. This method avoids the use of if-tests, and up, and the second column right and down. This method avoids the use of if-tests,
and takes advantage of the parallel optimization. and takes advantage of the parallel optimization.
We parallelize our code using both a message passing interface (OpenMPI) and We parallelized our code using both message passing interface (OpenMPI) and
multi-threading (OpenMP). First, we divide the temperatures into smaller ranges, multi-threading (OpenMP). First, we divided the temperatures into smaller sub-ranges,
and each MPI process receives a set of temperatures. Every MPI process spawn a and each MPI process received a sub-range of temperatures. Every MPI initialize a
set of threads, which initialize an Ising model and performs the Metropolis-Hastings parallel region with a set of threads, which then initializes an Ising model and
algorithm. We limit the number of times threads are spawned and joined, by using performs the Metropolis-Hastings algorithm. We limited the number of times threads
single parallel regions, reducing parallel overhead. We used Fox \footnote{Technical are spawned and joined, by using single parallel regions, reducing parallel overhead.
specifications for Fox can be found at \url{https://www.uio.no/english/services/it/research/platforms/edu-research/help/fox/system-overview.md}}, We used Fox \footnote{Technical specifications for Fox can be found at \url{https://www.uio.no/english/services/it/research/platforms/edu-research/help/fox/system-overview.md}},
a high-performance computing cluster, to run our program. a high-performance computing cluster, to run our program for $1$ and $10$ million
Monte Carlo cycles when analyzing phase transitions.