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6
latex/sections/abstract.tex
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@ -9,11 +9,11 @@
3000 Monte Carlo cycles, and used the following samples to find the probability 3000 Monte Carlo cycles, and used the following samples to find the probability
distribution at temperature $T_{1} = 1.0 J / k_{B}$, and $T_{2} = 2.4 J / k_{B}$. 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 $\langle \epsilon \rangle \approx -1.9969 J$, 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 with a variance $\text{Var} (\epsilon) = 0.0001$. And for $T_{2}$, close to the critical
temperature, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2370 J$, 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 with a variance $\text{Var} (\epsilon) = 0.0203$. In addition, we estimated
the expected energy and magnetization per spin, the heat capacity and magnetic the expected energy and magnetization per spin, the heat capacity and magnetic
susceptibility. We estimate the critical temperatures of finite lattice sizes, susceptibility. We have estimated the critical temperatures of finite lattice sizes,
and used these values to approximate the critical temperature of a lattice of and used these values to approximate the critical temperature of a lattice of
infinite size. Using linear regression, we estimated the critical temperature infinite size. Using linear regression, we estimated the critical temperature
to be $T_{c}(L = \infty) \approx 2.2695 J/k_{B}$. to be $T_{c}(L = \infty) \approx 2.2695 J/k_{B}$.

4
latex/sections/appendices.tex
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@ -92,7 +92,7 @@ and
&= \frac{4 (2e^{8 \beta J} + 4)}{4(\cosh(8 \beta J) + 3)} \\ &= \frac{4 (2e^{8 \beta J} + 4)}{4(\cosh(8 \beta J) + 3)} \\
&= \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} \ . &= \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} \ .
\end{align*} \end{align*}
The squared energy function The squared energy and magnetization functions are then
\begin{align*} \begin{align*}
\langle E^{2} \rangle &= (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J)^{2} \cdot \frac{1}{Z} e^{-8 \beta J} \\ \langle E^{2} \rangle &= (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J)^{2} \cdot \frac{1}{Z} e^{-8 \beta J} \\
& \quad + (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} \\ & \quad + (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} \\
@ -127,7 +127,7 @@ capacity per spin
\frac{C_{V}}{N} &= \frac{1}{N} \frac{1}{k_{B} T^{2}} (\mathbb{E}(E^{2}) - [\mathbb{E}(E)]^{2}) \\ \frac{C_{V}}{N} &= \frac{1}{N} \frac{1}{k_{B} T^{2}} (\mathbb{E}(E^{2}) - [\mathbb{E}(E)]^{2}) \\
&= \frac{1}{N k_{B} T^{2}} \mathbb{V}(E) \ . &= \frac{1}{N k_{B} T^{2}} \mathbb{V}(E) \ .
\end{align*} \end{align*}
Using Eq. \eqref{eq:susceptibility}, we find he susceptibility per spin Using Equation \eqref{eq:susceptibility}, we find the susceptibility per spin
\begin{align*} \begin{align*}
\frac{\chi}{N} &= \frac{1}{N} \frac{1}{k_{B} T} (\mathbb{E}(M^{2}) - [\mathbb{E}(M)]^{2}) \\ \frac{\chi}{N} &= \frac{1}{N} \frac{1}{k_{B} T} (\mathbb{E}(M^{2}) - [\mathbb{E}(M)]^{2}) \\
&= \frac{1}{N k_{B} T} \mathbb{V}(M) \ . &= \frac{1}{N k_{B} T} \mathbb{V}(M) \ .

8
latex/sections/conclusion.tex
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@ -11,7 +11,7 @@ of threads we observed a speed-up in runtime.
We initialized the lattices using both ordered and unordered spin configuration, We initialized the lattices using both ordered and unordered spin configuration,
and started sampling after the system reached an equilibrium. The number of Monte 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, Carlo cycles necessary to reach a system equilibrium, referred to as burn-in time,
was estimated to be 3000 cycles. We found that excluding the samples generated 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 during the burn-in time, improves the estimated expectation value of energy and
magnetization, in addition to the heat capacity and susceptibility, when samples magnetization, in addition to the heat capacity and susceptibility, when samples
are sparse. However, when we increase number of samples, excluding the burn-in are sparse. However, when we increase number of samples, excluding the burn-in
@ -19,11 +19,11 @@ samples does not affect the estimated values.
Continuing, we used the generated samples to compute energy per spin $\langle \epsilon \rangle$, 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 $\chi$. magnetization per spin $\langle |m| \rangle$, heat capacity $C_{V}$, and $\chi$.
In addition, we estimated the probability distribution for temperature $T_{1} = 1.0 J / k_{B}$, 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 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 variation $\text{Var} (\epsilon) = 0.0001$. 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 $\langle \epsilon \rangle \approx -1.2370 J$, And for $T_{2}$, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2370 J$,
with a variation $\text{Var} (\epsilon) = 0.0203$. with a variance $\text{Var} (\epsilon) = 0.0203$.
We estimated the expected energy and magnetization per spin, in addition to the We estimated the expected energy and magnetization per spin, in addition to the
heat capacity and susceptibility for lattices of size $L = {20, 40, 60, 80, 100}$. heat capacity and susceptibility for lattices of size $L = {20, 40, 60, 80, 100}$.

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latex/sections/introduction.tex
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@ -7,8 +7,8 @@ magnetic property of these materials is utilized in devices such as compasses
for navigation, in computer hard drives to store data \cite{yale:2023:hdd}, and for navigation, in computer hard drives to store data \cite{yale:2023:hdd}, and
MRI machines to produce detailed anatomical images. MRI machines to produce detailed anatomical images.
Magnetic materials can be classified based on their specific magnetic behavior, Magnetic materials care classified based on their specific magnetic behavior,
and one of the groups consist of ferromagnetic materials. These materials are and one of the groups consists of ferromagnetic materials. These materials are
easily magnetized, and when exposed to a strong magnetic field they can become easily magnetized, and when exposed to a strong magnetic field they can become
saturated. In addition, when these materials are heated to a critical point, they saturated. In addition, when these materials are heated to a critical point, they
lose their magnetic property \cite{britannica:2023:ferromagnetism}. lose their magnetic property \cite{britannica:2023:ferromagnetism}.
@ -23,7 +23,7 @@ exposed to temperatures near a critical point. In addition, we will numerically
estimate the critical temperature where the system experience phase transition. 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 the algorithms and tools used in the implementation.
Continuing with Section \ref{sec:results}, we will 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 will conclude our findings in Section \ref{sec:conclusion}.
\end{document} \end{document}

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latex/sections/methods.tex
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@ -3,8 +3,8 @@
\begin{document} \begin{document}
\section{Methods}\label{sec:methods} \section{Methods}\label{sec:methods}
\subsection{The Ising model}\label{subsec:ising_model} \subsection{The Ising model}\label{subsec:ising_model}
The Ising model consist of a lattice of spins, which can be though of as atoms The Ising model consists of a lattice of spins, which can be thought of as atoms
in a grid. In two dimensions, the length of the lattice is given by $L$, and the in a grid. In two dimensions, the length of a lattice is given by $L$, and the
number of spins within a lattice is given by $N = L^{2}$. When we consider the number of spins within a lattice is given by $N = L^{2}$. When we consider the
entire lattice, the system spin configuration is represented as a matrix $L \times L$ entire lattice, the system spin configuration is represented as a matrix $L \times L$
\begin{align*} \begin{align*}
@ -23,15 +23,15 @@ The spins interact with its nearest neighbors, and in a two-dimensional lattice
each spin has up to four nearest neighbors. In our experiments we will use periodic each spin has up to four nearest neighbors. In our experiments we will use periodic
boundary conditions, resulting in all spins having exactly four nearest neighbors. boundary conditions, resulting in all spins having exactly four nearest neighbors.
To find the analytical expressions necessary for validating our model implementation, To find the analytical expressions necessary for validating our model implementation,
we will assume a $2 \times 2$ lattice. we 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}
E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} - B \sum_{k}^{N} s_{k}\ , E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} - B \sum_{k}^{N} s_{k}\ ,
\label{eq:energy_hamiltonian} \label{eq:energy_hamiltonian}
\end{equation} \end{equation}
where $\langle k l \rangle$ denote a spin pair. $J$ is the coupling constant, and where $\langle k l \rangle$ denotes a spin pair. $J$ is the coupling constant, and
$B$ is the external magnetic field. For simplicity, we will consider the Ising model $B$ is the external magnetic field. For simplicity, we consider the Ising model
where $B = 0$, and find the total system energy given by where $B = 0$, and find the total system energy given by
\begin{equation} \begin{equation}
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} \ .
@ -89,13 +89,13 @@ in Appendix \ref{sec:energy_special}.
0 & -8J & -4 & 1 \\ 0 & -8J & -4 & 1 \\
\hline \hline
\end{tabular} \end{tabular}
\caption{Values of total energy, total magnetization, and degeneracy for the \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 possible states of a system for a $2 \times 2$ lattice, with periodic boundary
conditions.} conditions.}
\label{tab:lattice_config} \label{tab:lattice_config}
\end{table} \end{table}
We calculate the analytical values for a $2 \times 2$ lattice, found in Table \ref{tab:lattice_config}. We calculate the analytical values for a $2 \times 2$ lattice, found in Table \ref{tab:lattice_config}.
However, we scale the total energy and total magnetization by number of spins, However, we scale the total energy and total magnetization by the number of spins,
to compare these quantities for lattices where $L > 2$. Energy per spin is given by to compare these quantities for lattices where $L > 2$. Energy per spin is given by
\begin{equation} \begin{equation}
\epsilon (\mathbf{s}) = \frac{E(\mathbf{s})}{N} \ , \epsilon (\mathbf{s}) = \frac{E(\mathbf{s})}{N} \ ,
@ -165,10 +165,10 @@ We also need to determine the heat capacity
\label{eq:heat_capacity} \label{eq:heat_capacity}
\end{equation} \end{equation}
and the magnetic susceptibility and the magnetic susceptibility
\begin{align*} \begin{equation}
\chi = \frac{1}{k_{\text{B} T}} (\mathbb{E}(M^{2}) - [\mathbb{E}(M)]^{2}) \ . \chi = \frac{1}{k_{\text{B} T}} (\mathbb{E}(M^{2}) - [\mathbb{E}(M)]^{2}) \ .
\label{eq:susceptibility} \label{eq:susceptibility}
\end{align*} \end{equation}
In Appendix \ref{sec:heat_susceptibility} we derive expressions for the heat In Appendix \ref{sec:heat_susceptibility} we derive expressions for the heat
capacity and the susceptibility, and find the heat capacity capacity and the susceptibility, and find the heat capacity
\begin{align*} \begin{align*}
@ -205,29 +205,29 @@ Boltzmann constant we derive the remaining units, which can be found in Table
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 spontaneously. 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 transitions 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 discontinuous 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
correlation length is proportional with the lattice size, resulting in the critical correlation length is proportional to the lattice size, resulting in the critical
temperature scaling relation temperature scaling relation
\begin{equation} \begin{equation}
T_{c}(L) - T_{c}(L = \infty) = aL^{-1} \ , T_{c}(L) - T_{c}(L = \infty) = aL^{-1} \ ,
\label{eq:critical_infinite} \label{eq:critical_infinite}
\end{equation} \end{equation}
where $a$ is a constant. For the Ising model in two dimensions, with an lattice of where $a$ is a constant. For the Ising model in two dimensions, with a lattice of
infinite size, the analytical solution of the critical temperature is infinite size, the analytical solution of the critical temperature is
\begin{equation} \begin{equation}
T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B} T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B}
\label{eq:critical_solution} \label{eq:critical_solution}
\end{equation} \end{equation}
This result was found analytically by Lars Onsager in 1944. We can estimate the This result was found analytically by Lars Onsager in 1944 \cite{onsager:1944}. We can estimate the
critical temperature of an infinite lattice, using finite lattices critical temperature critical temperature of an infinite lattice, using the critical temperature of
and linear regression. finite lattices of different sizes and linear regression.
\subsection{The Markov chain Monte Carlo method}\label{subsec:mcmc_method} \subsection{The Markov chain Monte Carlo method}\label{subsec:mcmc_method}
Markov chains consist of a sequence of samples, where the probability of the next Markov chains consist of a sequence of samples, where the probability of the next
sample depend on the probability of the current sample. Whereas the Monte Carlo sample depends on the probability of the current sample. Whereas the Monte Carlo
method introduces randomness to the sampling, which allows us to approximate method introduces randomness to the sampling, which allows us to approximate
statistical quantities \cite{tds:2021:mcmc} without using the pdf directly. We statistical quantities \cite{tds:2021:mcmc} without using the pdf directly. We
generate samples of spin configuration, to approximate $\langle \epsilon \rangle$, generate samples of spin configuration, to approximate $\langle \epsilon \rangle$,
@ -236,10 +236,10 @@ $\langle |m| \rangle$, $C_{V}$, and $\chi$, using the Markov chain Monte Carlo
The Ising model is initialized in a random state, and the Markov chain is evolved The Ising model is initialized in a random state, and the Markov chain is evolved
until the model reaches an equilibrium state. However, generating new random states until the model reaches an equilibrium state. However, generating new random states
require ergodicity and detailed balance. A Markov chain is ergoditic when all system requires ergodicity and detailed balance. A Markov chain is ergoditic when all system
states can be reached at every current state, whereas detaild balance implies no states can be reached at every current state, whereas detailed balance implies no
net flux of probability. To satisfy these criterias we use the Metropolis-Hastings net flux of probability. To satisfy these criteria we use the Metropolis-Hastings
algorithm, found in Figure \ref{algo:metropolis}, to generate samples of microstates. algorithm, found in Algorithm \ref{algo:metropolis}, to generate samples of microstates.
One Monte Carlo cycle consist of changing the initial configuration of the lattice, One Monte Carlo cycle consist of changing the initial configuration of the lattice,
by randomly flipping a spin. When a spin is flipped, the change in energy is evaluated as by randomly flipping a spin. When a spin is flipped, the change in energy is evaluated as
@ -258,41 +258,38 @@ $\Delta E = \{-16J, -8J, 0, 8J, 16J\}$, we derive these values in Appendix \ref{
We can avoid computing the Boltzmann factor, by using a look up table (LUT). We can avoid computing the Boltzmann factor, by using a look up table (LUT).
We use $\Delta E$ as an index to access the resulting value of the exponential function, We use $\Delta E$ as an index to access the resulting value of the exponential function,
in an array. in an array.
\begin{figure}[H] \begin{algorithm}[H]
\begin{algorithm}[H] \caption{Metropolis-Hastings Algorithm}
\caption{Metropolis-Hastings Algorithm} \label{algo:metropolis}
\label{algo:metropolis} \begin{algorithmic}
\begin{algorithmic} \Procedure{Metropolis}{$lattice, energy, magnetization$}
\Procedure{Metropolis}{$lattice, energy, magnetization$} \For{ Each spin in the lattice }
\For{ Each spin in the lattice } \State $ri \leftarrow \text{random index of the lattice}$
\State $ri \leftarrow \text{random index of the lattice}$ \State $rj \leftarrow \text{random index of the lattice}$
\State $rj \leftarrow \text{random index of the lattice}$ \State $dE \leftarrow 2 \cdot lattice_{ri,rj} \cdot (lattice_{ri,rj-1} + lattice_{ri,rj+1}
\State $dE \leftarrow 2 \cdot lattice_{ri,rj} \cdot (lattice_{ri,rj-1} + lattice_{ri,rj+1} + lattice_{ri-1,rj} + lattice_{ri+1,rj} )$
+ lattice_{ri-1,rj} + lattice_{ri+1,rj} )$
\State $rn \leftarrow \text{random number} \in [0,1)$ \State $rn \leftarrow \text{random number} \in [0,1)$
\If{$rn \leq e^{- \frac{dE}{T}}$} \If{$rn \leq e^{- \frac{dE}{T}}$}
\State $lattice_{ri,rj} = -1 \cdot lattice_{ri,rj}$ \State $lattice_{ri,rj} = -1 \cdot lattice_{ri,rj}$
\State $magnetization = magnetization + 2 \cdot lattice_{ri,rj}$ \State $magnetization = magnetization + 2 \cdot lattice_{ri,rj}$
\State $energy = energy + dE$ \State $energy = energy + dE$
\EndIf \EndIf
\EndFor \EndFor
\EndProcedure \EndProcedure
\end{algorithmic} \end{algorithmic}
\end{algorithm} \end{algorithm}
\end{figure} The Markov process reaches an equilibrium after a certain number of Monte Carlo cycles,
The Markov process reach an equilibrium after a certain number of Monte Carlo cycles, called burn-in time. After the burn-in time, the system state reflects the state of a real system.
where the system state reflects the state of a real system. After this burn-in time, and we can start sampling microstates. The probability distribution of the samples
given by number of Monte Carlo cycles, we can start sampling microstates. will tend toward the actual probability distribution of the system.
The probability distribution of the samples will tend toward the actual probability
distribution of the system.
\subsection{Implementation and testing}\label{subsec:implementation_test} \subsection{Implementation and testing}\label{subsec:implementation_test}
We implemented a test suite, and compared the numerical estimates to the analytical We implemented a test suite, and compared the numerical estimates to the analytical
results from Section \ref{subsec:statistical_mechanics}. In addition, we set a tolerance to results from Section \ref{subsec:statistical_mechanics}. In addition, we set a tolerance to
verify convergence, and a maximum number of Monte Carlo cycles to avoid longer verify convergence, and a maximum number of Monte Carlo cycles to avoid longer
runtimes during implementation. runtimes during execution.
We used a pattern to access all neighboring spins, where all indices of the neighboring We used a pattern to access all neighboring spins, where all indices of the neighboring
spins are put in an $L \times 2$ matrix. The indices are accessed using pre-defined spins are put in an $L \times 2$ matrix. The indices are accessed using pre-defined
@ -315,11 +312,11 @@ Monte Carlo cycles when analyzing phase transitions.
\subsection{Tools}\label{subsec:tools} \subsection{Tools}\label{subsec:tools}
The Ising model and MCMC methods are implemented in C++, and parallelized using The Ising model and MCMC methods are implemented in C++, and parallelized using
both \verb|OpenMPI| \cite{gabriel:2004:open_mpi} and \verb|OpenMP| \cite{openmp:2018}. both \verb|OpenMPI| \cite{gabriel:2004:open_mpi} and \verb|OpenMP| \cite{openmp:2018}.
We used the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce We use the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce
all the plots, \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the all the plots, \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the
figures. We also used a linear regression method from the Python library \verb|SciPy|. figures. We also use a linear regression method and the normal distribution method
To optimize our implementation, we used a profiler tool \verb|Scalasca| \cite{scalasca} from the Python library \verb|SciPy|. To optimize our implementation, we used a
and \verb|Score-P| \cite{scorep}. profiling tool \verb|Score-P| \cite{scorep}.
\end{document} \end{document}

36
latex/sections/results.tex
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@ -13,14 +13,14 @@ per spin for $T_{1}$ in Figure \ref{fig:burn_in_energy_1_0}, and $T_{2}$ in Figu
per spin for $T_{1}$ in Figure \ref{fig:burn_in_magnetization_1_0}, and for $T_{2}$ per spin for $T_{1}$ in Figure \ref{fig:burn_in_magnetization_1_0}, and for $T_{2}$
in Figure \ref{fig:burn_in_energy_2_4}. in Figure \ref{fig:burn_in_energy_2_4}.
% Not sure about the relevance of the paragraph below % Not sure about the relevance of the paragraph below
We used random numbers to set the initial state and the index of spin to flip, and % We used random numbers to set the initial state and the index of spin to flip, and
observed different graphs ... However, when we set the seed of the random number % observed different graphs ... However, when we set the seed of the random number
engine, we could reproduce the results at every run. It is possible to determine % engine, we could reproduce the results at every run. It is possible to determine
the burn-in time analytically, using the correlation time given by $\tau \approx L^{d + z}$. % the burn-in time analytically, using the correlation time given by $\tau \approx L^{d + z}$.
Where $d$ is the dimensionality of the system and $z = 2.1665 \pm 0.0012$ % Where $d$ is the dimensionality of the system and $z = 2.1665 \pm 0.0012$
\footnote{This value was determined by Nightingale and Blöte for the Metropolis algorithm.}. % \footnote{This value was determined by Nightingale and Blöte for the Metropolis algorithm.}.
However, when we increased number of Monte Carlo cycles the sampled generated during % However, when we increased number of Monte Carlo cycles the sampled generated during
the burn-in time did not affect the mean value (...). % Should add result showing this % the burn-in time did not affect the mean value (...). % Should add result showing this
The lattice was initialized in an ordered and an unordered state, for both temperatures. We observed The lattice was initialized in an ordered and an unordered state, for both temperatures. We observed
no change in expectation value of energy or magnetization for $T_{1}$, when we no change in expectation value of energy or magnetization for $T_{1}$, when we
@ -86,7 +86,7 @@ centered around the expectation value $\langle \epsilon \rangle = -1.2370$. %
\label{fig:histogram_2_4} \label{fig:histogram_2_4}
\end{figure} % \end{figure} %
However, we observed a higher variance of Var$(\epsilon) = 0.0203$. When the temperature However, we observed a higher variance of Var$(\epsilon) = 0.0203$. When the temperature
increased, the system moved from an ordered to an onordered state. The change in increased, the system moved from an ordered to an unordered state. The change in
system state, or phase transition, indicates the temperature is close to a critical system state, or phase transition, indicates the temperature is close to a critical
point. point.
@ -97,8 +97,8 @@ We continued investigating the behavior of the system around the critical temper
First, we generated $10$ million samples of spin configurations for lattices of First, we generated $10$ million samples of spin configurations for lattices of
size $L \in \{20, 40, 60, 80, 100\}$, and temperatures $T \in [2.1, 2.4]$. We divided the size $L \in \{20, 40, 60, 80, 100\}$, and temperatures $T \in [2.1, 2.4]$. We divided the
temperature range into $40$ steps, with an equal step size of $0.0075$. The samples temperature range into $40$ steps, with an equal step size of $0.0075$. The samples
were generated in parallel, where we allocated $4$ sequential temperatures to $10$ were generated in parallel, where the program allocated $4$ sequential temperatures to $10$
MPI processes. Each process was set to spawn $10$ thread, resulting in a total of MPI processes. Each process was set to spawn $10$ threads, resulting in a total of
$100$ threads working in parallel. We include results for $1$ million MC cycles $100$ threads working in parallel. We include results for $1$ million MC cycles
in Appendix \ref{sec:additional_results} in Appendix \ref{sec:additional_results}
@ -113,9 +113,9 @@ increase in $\langle \epsilon \rangle$ in the temperature range $T \in [2.25, 2.
\caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.} \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
\label{fig:phase_energy_10M} \label{fig:phase_energy_10M}
\end{figure} % \end{figure} %
We observe a deacrese in $\langle |m| \rangle$ for the same temperature range in We observe a decrease in $\langle |m| \rangle$ for the same temperature range in
Figure \ref{fig:phase_magnetization_10M}, suggesting the system moves from an ordered Figure \ref{fig:phase_magnetization_10M}, suggesting that the system moves from an ordered
magnetized state to a state of no net magnetization. The system energy increase, magnetized state to a state of no net magnetization. The system energy increases,
however, there is a loss of magnetization close to the critical temperature. however, there is a loss of magnetization close to the critical temperature.
\begin{figure}[H] \begin{figure}[H]
\centering \centering
@ -124,16 +124,16 @@ however, there is a loss of magnetization close to the critical temperature.
\label{fig:phase_magnetization_10M} \label{fig:phase_magnetization_10M}
\end{figure} % \end{figure} %
In Figure \ref{fig:phase_heat_10M}, we observe an increase in heat capacity in the In Figure \ref{fig:phase_heat_10M}, we observe an increase in heat capacity in the
temperature range $T \in [2.25, 2.35]$. In addition, we observed a sharper peak temperature range $T \in [2.25, 2.35]$. In addition, we observe a sharper peak
value of heat capacity the lattice size increase. value of heat capacity when the lattice size increase.
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf} \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf}
\caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.} \caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
\label{fig:phase_heat_10M} \label{fig:phase_heat_10M}
\end{figure} % \end{figure} %
The magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, show The magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, shows
the sharp peak in the same temperature range as that of the heat capacity. Since the sharp peak in the same temperature range as that of the heat capacity. Since the
shape of the curve for both heat capacity and the magnetic susceptibility become shape of the curve for both heat capacity and the magnetic susceptibility become
sharper when we increase lattice size, we are moving closer to the critical temperature. sharper when we increase lattice size, we are moving closer to the critical temperature.
\begin{figure}[H] \begin{figure}[H]

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review.md
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@ -14,79 +14,102 @@ Nothing
## ./latex/sections/abstract.tex ## ./latex/sections/abstract.tex
[ ] 12: variation => variance [x] 12: variation => variance
[ ] 14: variation => variance [x] 14: variation => variance
[ ] 16: "We estimate" => "We have estimated" as everything else is in past tense, and the rest of the sentence is in past tense as well. [x] 16: "We estimate" => "We have estimated" as everything else is in past tense, and the rest of the sentence is in past tense as well.
## ./latex/sections/introduction.tex ## ./latex/sections/introduction.tex
[ ] 11: consist => consists. "One of the groups" is singular. [x] 10: "Magnetic materials can be classified" => "Magnetic materials are classified". I changed this, since they are actually classified
[x] 11: consist => consists. "One of the groups" is singular.
[ ] 12: Is the saturation optional, or will they become saturated when exposed to a magnetic field? [ ] 12: Is the saturation optional, or will they become saturated when exposed to a magnetic field?
[ ] 26: "as well as algorithms ..." => "as well as the algorithms ..." Valid point, they do become saturated so maybe rephrase
[ ] 28: "we conclude ..." => "we will conclude ..." [x] 26: "as well as algorithms ..." => "as well as the algorithms ..."
[x] 28: "we conclude ..." => "we will conclude ..."
## ./latex/sections/methods.tex ## ./latex/sections/methods.tex
[ ] 6: consist => consists. "The Ising model" is singular. [x] 6: consist => consists. "The Ising model" is singular.
[ ] 6: though => thought [x] 6: though => thought
[ ] 7: "the length of the lattice ..." => "the length of a lattice ...". We're not talking about a specific lattice. [x] 7: "the length of the lattice ..." => "the length of a lattice ...". We're not talking about a specific lattice.
[ ] 26: remove "will", as we are assuming a 2x2 now and not later in the paper. [x] 26: remove "will", as we are assuming a 2x2 now and not later in the paper.
[ ] 33: denote => denotes. $\langle k l \rangle$ is singular. [x] 33: denote => denotes. $\langle k l \rangle$ is singular.
[ ] 34: Change to present tense, as we are considering it in this section and not later in the paper. [ ] 34: Change to present tense, as we are considering it in this section and not later in the paper.
[ ] 92: "Values of total energy ..." => "Values of the total energy ..." I think I've fixed it, however, tense is apparently not my jam these days XP
[ ] 98: "by number of spins ..." => "by the number of spins ..." [x] 92: "Values of total energy ..." => "Values of the total energy ..."
[ ] 162: Why are the equations for heat capacity and susceptibility in different environments? [x] 98: "by number of spins ..." => "by the number of spins ..."
[ ] 208: transition => transitions. "the Ising model" is singular. [x] 162: Why are the equations for heat capacity and susceptibility in different environments?
[ ] 209: discontinous => discontinuous Well, that is an excellent question! That I do not have an answer to, but I changed it so that both are in equation environment.
[ ] 211: with => to [x] 208: transition => transitions. "the Ising model" is singular.
[ ] 217: "with an lattice" => "with a lattice" The singular thing is really something I should have a look into!
[ ] 223: Should perhaps create a reference to his paper? [x] 209: discontinous => discontinuous
[ ] 224: "using finite lattices ..." => "using the finite lattices'". We are referring to specific lattices and "lattices" should be possessive. I would rewrite the second part to be "using the critical temperature of finite lattices of different sizes and linear regression." [x] 211: with => to
[ ] 230: depend => depends. "the next sample" is singular. [x] 217: "with an lattice" => "with a lattice"
[ ] 239: require => requires. "generating new random states" is singular. [x] 223: Should perhaps create a reference to his paper?
[ ] 240: detaild => detailed I agree, I had already added a reference but had forgotten to cite it in the report. I did not include a page number, don't think it is necessary since its what the paper finding?
[ ] 241: criterias => criteria. the plural of criterion is criteria. [x] 224: "using finite lattices ..." => "using the finite lattices'". We are referring to specific lattices and "lattices" should be possessive. I would rewrite the second part to be "using the critical temperature of finite lattices of different sizes and linear regression."
[ ] 242: Figure => Algorithm, since we can remove the figure environment enclosing the algorithm environment. I agree, it does make more sense
[x] 230: depend => depends. "the next sample" is singular.
[x] 239: require => requires. "generating new random states" is singular.
[x] 240: detaild => detailed
[x] 241: criterias => criteria. the plural of criterion is criteria.
[x] 242: Figure => Algorithm, since we can remove the figure environment enclosing the algorithm environment.
[ ] More details are specified in the next comment below. [ ] More details are specified in the next comment below.
[ ] 244: A Monte Carlo cycle consists of attempts of flipping a spin, and not a single attempt to flip a spin. [ ] 244: A Monte Carlo cycle consists of attempts of flipping a spin, and not a single attempt to flip a spin.
[ ] 261: Instead of having the algorithm inside a figure, you can use the H specifier in the algorithm environment (which you already do). You can read more about it [here](https://tex.stackexchange.com/questions/231191/algorithm-in-revtex4-1). I have already tested that it works, and it makes it so we can actually refer to it as an algorithm and not a figure. Not sure how to rephrase the sentence to make it correct, any specific suggestions?
[ ] 284: reach => reaches. "The Markov process" is singular. [x] 261: Instead of having the algorithm inside a figure, you can use the H specifier in the algorithm environment (which you already do). You can read more about it [here](https://tex.stackexchange.com/questions/231191/algorithm-in-revtex4-1). I have already tested that it works, and it makes it so we can actually refer to it as an algorithm and not a figure.
[ ] 295: implementation => execution. Implementation is the writing of the code, while the runtime is during execution of the program. I would also say to avoid potentially having the program run endlessly. This might have fallen through the cracks for me since I just copy-pasted from metropolis.tex
[x] 284: reach => reaches. "The Markov process" is singular.
[ ] 283: I changed the paragraph to include a better introduction of the concept burn-in time, do you think the "flow" is better now?
[x] 295: implementation => execution. Implementation is the writing of the code, while the runtime is during execution of the program. I would also say to avoid potentially having the program run endlessly.
[ ] 292: Do you have a good way of including "I would also say to avoid potentially having the program run endlessly" in the sentence?
[ ] 297: Change of tense. We have used present tense for most of this section, so we should continue with that. [ ] 297: Change of tense. We have used present tense for most of this section, so we should continue with that.
Added this into a new point, since the line numbers have shifted now
[ ] 294: So for the paragraph starting with "We used a pattern..." should be rewritten to present tense?
[ ] 300: Lack of if-tests does not hinder parallelization, but it can hinder compiler optimizations. [ ] 300: Lack of if-tests does not hinder parallelization, but it can hinder compiler optimizations.
[ ] 318: used => use. Keep to the present tense for consistency. Suggested change "... avoids the use of if-tests, so we can take advantage of the compiler optimization."
[ ] 320: used => use. [x] 318: used => use. Keep to the present tense for consistency.
[ ] 320: "the Python ..." => "the Python ...". Double space. [x] 320: used => use.
[ ] 321: used => use. [x] 320: "the Python ..." => "the Python ...". Double space.
[ ] 321: profiler => profiling. [x] Included the normal distribution method from SciPy as well
[ ] 321: Remove Scalasca as we only use Score-p now. (Easier to deal with). [x] 321: used => use.
[x] 321: profiler => profiling.
[x] 321: Remove Scalasca as we only use Score-p now. (Easier to deal with).
## ./latex/sections/results.tex ## ./latex/sections/results.tex
[ ] 15: Should have a line break to indicate new paragraph. [x] 15: Should have a line break to indicate new paragraph.
[ ] 15: It might be relevant, but it's a lot more work. we have all the tools necessary to produce the plots, so it depends on how much work we can do. [ ] 15: It might be relevant, but it's a lot more work. we have all the tools necessary to produce the plots, so it depends on how much work we can do.
[ ] 89: onordered => unordered I suggest we leave it out for now, I'll comment it out in case we want to use it
[x] 89: onordered => unordered
[ ] 97: More accurately, we generated 10M samples per temperature point. [ ] 97: More accurately, we generated 10M samples per temperature point.
[ ] 100: Should be that the program allocated, so that the reader understands that it does the division automatically. True, so "...$10$ million samples of spin configurations for lattices..." => "...$10$ million samples of spin configurations, per temperature, for lattices..."
[ ] 101: thread => threads [x] 100: Should be that the program allocated, so that the reader understands that it does the division automatically.
I agree!
[x] 101: thread => threads
[ ] 105: We used a profiler. And it's not just for ensuring load balance among threads, it's also for each process, and we use it to see that data transfer is minimal. [ ] 105: We used a profiler. And it's not just for ensuring load balance among threads, it's also for each process, and we use it to see that data transfer is minimal.
So instead of "We ran a profiler" it should be "We used a profiler"? Could you suggest a way of formulating the sentence so that the info is correct?
[ ] 106: was => were [ ] 106: was => were
[ ] 116: deacrese => decrease My previous comment might make this unnecessary?
[ ] 117: "suggesting the system ..." => "suggesting that the system ...". [x] 116: deacrese => decrease
[ ] 118: increase => increases. "The system energy" is singular. [x] 117: "suggesting the system ..." => "suggesting that the system ...".
[ ] 127: observed => observe. [x] 118: increase => increases. "The system energy" is singular.
[ ] 128: "capacity the" => "capacity when the" [x] 127: observed => observe.
[ ] 135: show => shows. [x] 128: "capacity the" => "capacity when the"
[ ] 136: "Since shape ..." => "Since the shape" [x] 135: show => shows.
[x] 136: "Since shape ..." => "Since the shape"
## ./latex/sections/conclusion.tex ## ./latex/sections/conclusion.tex
[ ] 14: Didn't we arrive at 5000? [x] 14: Didn't we arrive at 5000?
[ ] 22: temperature => temperatures Haha, ofc we did!
[ ] 24: variation => variance [x] 22: temperature => temperatures
[ ] 26: variation => variance [x] 24: variation => variance
[x] 26: variation => variance
## ./latex/sections/appendices.tex ## ./latex/sections/appendices.tex
[ ] 95: "The squared energy" => "The squared energy and magnetization functions are then" [x] 95: "The squared energy" => "The squared energy and magnetization functions are then"
[ ] 130: he => the I agree, that sound a lot better! I just ended up wtiting something somewhat descriptive in lack of words.
[x] 130: he => the
Also, I changed the Eq. => to Equation