Finished first review, and added a few comments.
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Janita Willumsen
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\begin{document}
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\section{Methods}\label{sec:methods}
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\subsection{The Ising model}\label{subsec:ising_model}
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The Ising model consist of a lattice of spins, which can be though of as atoms
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in a grid. In two dimensions, the length of the lattice is given by $L$, and the
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The Ising model consists of a lattice of spins, which can be thought of as atoms
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in a grid. In two dimensions, the length of a lattice is given by $L$, and the
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number of spins within a lattice is given by $N = L^{2}$. When we consider the
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entire lattice, the system spin configuration is represented as a matrix $L \times L$
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\begin{align*}
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@ -23,15 +23,15 @@ The spins interact with its nearest neighbors, and in a two-dimensional lattice
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each spin has up to four nearest neighbors. In our experiments we will use periodic
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boundary conditions, resulting in all spins having exactly four nearest neighbors.
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To find the analytical expressions necessary for validating our model implementation,
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we will assume a $2 \times 2$ lattice.
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we assume a $2 \times 2$ lattice.
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The hamiltonian of the Ising model is given by
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\begin{equation}
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E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} - B \sum_{k}^{N} s_{k}\ ,
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\label{eq:energy_hamiltonian}
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\end{equation}
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where $\langle k l \rangle$ denote a spin pair. $J$ is the coupling constant, and
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$B$ is the external magnetic field. For simplicity, we will consider the Ising model
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where $\langle k l \rangle$ denotes a spin pair. $J$ is the coupling constant, and
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$B$ is the external magnetic field. For simplicity, we consider the Ising model
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where $B = 0$, and find the total system energy given by
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\begin{equation}
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E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} \ .
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@ -89,13 +89,13 @@ in Appendix \ref{sec:energy_special}.
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0 & -8J & -4 & 1 \\
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\hline
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\end{tabular}
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\caption{Values of total energy, total magnetization, and degeneracy for the
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\caption{Values of the total energy, total magnetization, and degeneracy for the
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possible states of a system for a $2 \times 2$ lattice, with periodic boundary
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conditions.}
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\label{tab:lattice_config}
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\end{table}
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We calculate the analytical values for a $2 \times 2$ lattice, found in Table \ref{tab:lattice_config}.
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However, we scale the total energy and total magnetization by number of spins,
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However, we scale the total energy and total magnetization by the number of spins,
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to compare these quantities for lattices where $L > 2$. Energy per spin is given by
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\begin{equation}
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\epsilon (\mathbf{s}) = \frac{E(\mathbf{s})}{N} \ ,
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@ -165,10 +165,10 @@ We also need to determine the heat capacity
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\label{eq:heat_capacity}
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\end{equation}
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and the magnetic susceptibility
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\begin{align*}
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\begin{equation}
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\chi = \frac{1}{k_{\text{B} T}} (\mathbb{E}(M^{2}) - [\mathbb{E}(M)]^{2}) \ .
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\label{eq:susceptibility}
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\end{align*}
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\end{equation}
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In Appendix \ref{sec:heat_susceptibility} we derive expressions for the heat
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capacity and the susceptibility, and find the heat capacity
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\begin{align*}
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@ -205,29 +205,29 @@ Boltzmann constant we derive the remaining units, which can be found in Table
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We consider the Ising model in two dimensions, with no external external magnetic
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field. At temperatures below the critical temperature $T_{c}$, the Ising model will
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magnetize spontaneously. When increasing the temperature of the external field,
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the Ising model transition from an ordered to an unordered phase. The spins become
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more correlated, and we can measure the discontinous behavior as an increase in
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the Ising model transitions from an ordered to an unordered phase. The spins become
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more correlated, and we can measure the discontinuous behavior as an increase in
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correlation length $\xi (T)$ \cite[p. 432]{hj:2015:comp_phys}. At $T_{c}$, the
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correlation length is proportional with the lattice size, resulting in the critical
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correlation length is proportional to the lattice size, resulting in the critical
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temperature scaling relation
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\begin{equation}
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T_{c}(L) - T_{c}(L = \infty) = aL^{-1} \ ,
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\label{eq:critical_infinite}
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\end{equation}
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where $a$ is a constant. For the Ising model in two dimensions, with an lattice of
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where $a$ is a constant. For the Ising model in two dimensions, with a lattice of
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infinite size, the analytical solution of the critical temperature is
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\begin{equation}
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T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B}
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\label{eq:critical_solution}
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\end{equation}
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This result was found analytically by Lars Onsager in 1944. We can estimate the
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critical temperature of an infinite lattice, using finite lattices critical temperature
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and linear regression.
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This result was found analytically by Lars Onsager in 1944 \cite{onsager:1944}. We can estimate the
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critical temperature of an infinite lattice, using the critical temperature of
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finite lattices of different sizes and linear regression.
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\subsection{The Markov chain Monte Carlo method}\label{subsec:mcmc_method}
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Markov chains consist of a sequence of samples, where the probability of the next
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sample depend on the probability of the current sample. Whereas the Monte Carlo
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sample depends on the probability of the current sample. Whereas the Monte Carlo
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method introduces randomness to the sampling, which allows us to approximate
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statistical quantities \cite{tds:2021:mcmc} without using the pdf directly. We
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generate samples of spin configuration, to approximate $\langle \epsilon \rangle$,
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@ -236,10 +236,10 @@ $\langle |m| \rangle$, $C_{V}$, and $\chi$, using the Markov chain Monte Carlo
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The Ising model is initialized in a random state, and the Markov chain is evolved
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until the model reaches an equilibrium state. However, generating new random states
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require ergodicity and detailed balance. A Markov chain is ergoditic when all system
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states can be reached at every current state, whereas detaild balance implies no
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net flux of probability. To satisfy these criterias we use the Metropolis-Hastings
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algorithm, found in Figure \ref{algo:metropolis}, to generate samples of microstates.
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requires ergodicity and detailed balance. A Markov chain is ergoditic when all system
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states can be reached at every current state, whereas detailed balance implies no
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net flux of probability. To satisfy these criteria we use the Metropolis-Hastings
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algorithm, found in Algorithm \ref{algo:metropolis}, to generate samples of microstates.
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One Monte Carlo cycle consist of changing the initial configuration of the lattice,
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by randomly flipping a spin. When a spin is flipped, the change in energy is evaluated as
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@ -258,41 +258,38 @@ $\Delta E = \{-16J, -8J, 0, 8J, 16J\}$, we derive these values in Appendix \ref{
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We can avoid computing the Boltzmann factor, by using a look up table (LUT).
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We use $\Delta E$ as an index to access the resulting value of the exponential function,
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in an array.
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\begin{figure}[H]
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\begin{algorithm}[H]
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\caption{Metropolis-Hastings Algorithm}
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\label{algo:metropolis}
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\begin{algorithmic}
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\Procedure{Metropolis}{$lattice, energy, magnetization$}
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\For{ Each spin in the lattice }
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\State $ri \leftarrow \text{random index of the lattice}$
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\State $rj \leftarrow \text{random index of the lattice}$
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\State $dE \leftarrow 2 \cdot lattice_{ri,rj} \cdot (lattice_{ri,rj-1} + lattice_{ri,rj+1}
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+ lattice_{ri-1,rj} + lattice_{ri+1,rj} )$
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\begin{algorithm}[H]
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\caption{Metropolis-Hastings Algorithm}
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\label{algo:metropolis}
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\begin{algorithmic}
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\Procedure{Metropolis}{$lattice, energy, magnetization$}
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\For{ Each spin in the lattice }
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\State $ri \leftarrow \text{random index of the lattice}$
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\State $rj \leftarrow \text{random index of the lattice}$
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\State $dE \leftarrow 2 \cdot lattice_{ri,rj} \cdot (lattice_{ri,rj-1} + lattice_{ri,rj+1}
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+ lattice_{ri-1,rj} + lattice_{ri+1,rj} )$
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\State $rn \leftarrow \text{random number} \in [0,1)$
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\If{$rn \leq e^{- \frac{dE}{T}}$}
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\State $lattice_{ri,rj} = -1 \cdot lattice_{ri,rj}$
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\State $magnetization = magnetization + 2 \cdot lattice_{ri,rj}$
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\State $energy = energy + dE$
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\EndIf
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\EndFor
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\EndProcedure
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\end{algorithmic}
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\end{algorithm}
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\end{figure}
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The Markov process reach an equilibrium after a certain number of Monte Carlo cycles,
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where the system state reflects the state of a real system. After this burn-in time,
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given by number of Monte Carlo cycles, we can start sampling microstates.
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The probability distribution of the samples will tend toward the actual probability
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distribution of the system.
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\State $rn \leftarrow \text{random number} \in [0,1)$
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\If{$rn \leq e^{- \frac{dE}{T}}$}
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\State $lattice_{ri,rj} = -1 \cdot lattice_{ri,rj}$
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\State $magnetization = magnetization + 2 \cdot lattice_{ri,rj}$
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\State $energy = energy + dE$
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\EndIf
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\EndFor
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\EndProcedure
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\end{algorithmic}
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\end{algorithm}
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The Markov process reaches an equilibrium after a certain number of Monte Carlo cycles,
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called burn-in time. After the burn-in time, the system state reflects the state of a real system.
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and we can start sampling microstates. The probability distribution of the samples
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will tend toward the actual probability distribution of the system.
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\subsection{Implementation and testing}\label{subsec:implementation_test}
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We implemented a test suite, and compared the numerical estimates to the analytical
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results from Section \ref{subsec:statistical_mechanics}. In addition, we set a tolerance to
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verify convergence, and a maximum number of Monte Carlo cycles to avoid longer
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runtimes during implementation.
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runtimes during execution.
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We used a pattern to access all neighboring spins, where all indices of the neighboring
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spins are put in an $L \times 2$ matrix. The indices are accessed using pre-defined
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@ -315,11 +312,11 @@ Monte Carlo cycles when analyzing phase transitions.
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\subsection{Tools}\label{subsec:tools}
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The Ising model and MCMC methods are implemented in C++, and parallelized using
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both \verb|OpenMPI| \cite{gabriel:2004:open_mpi} and \verb|OpenMP| \cite{openmp:2018}.
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We used the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce
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We use the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce
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all the plots, \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the
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figures. We also used a linear regression method from the Python library \verb|SciPy|.
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To optimize our implementation, we used a profiler tool \verb|Scalasca| \cite{scalasca}
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and \verb|Score-P| \cite{scorep}.
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figures. We also use a linear regression method and the normal distribution method
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from the Python library \verb|SciPy|. To optimize our implementation, we used a
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profiling tool \verb|Score-P| \cite{scorep}.
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\end{document}
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