From aa73d29876a0d96f9d715800238dade079f26e9d Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Sun, 3 Dec 2023 15:47:53 +0100 Subject: [PATCH 01/17] Need to rewrite burn-in time subsection, include speed-up, and fix appendices. --- latex/ising_model.tex | 15 +-- latex/sections/appendices.tex | 20 ++-- latex/sections/methods.tex | 6 +- latex/sections/results.tex | 179 ++++++++++++++++++---------------- 4 files changed, 115 insertions(+), 105 deletions(-) diff --git a/latex/ising_model.tex b/latex/ising_model.tex index f03d617..de5c0c5 100644 --- a/latex/ising_model.tex +++ b/latex/ising_model.tex @@ -90,15 +90,16 @@ % Method % Problem 1: OK % Problem 2: OK -% Problem 3: OK - missing theory of phase transition -% Problem 4: validate numerical results using the analytical results -% Problem 7: division of workload mpi and openmp -% Problem 9: rewrite theoretical background of phase transition and critical temperature +% Problem 3: OK +% Problem 4: OK +% Problem 9: OK % Results % Problem 5: burn-in time -% Problem 6: estimate probability distribution based on histogram +% Problem 6: OK +% Problem 7: OK % Problem 8: investigate phase transitions, and estimate critical temperature based on plot % Problem 9: use critical temperature to estimate the ctritical temperature of infinite lattice -% -% \ No newline at end of file + +% Appendices +% Problem 0: move in front of references, and change back to double column diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex index d04aecb..151e96b 100644 --- a/latex/sections/appendices.tex +++ b/latex/sections/appendices.tex @@ -180,34 +180,34 @@ $\Delta E = E_{\text{after}} - E_{\text{before}}$. We find the $3^{2}$ values as where the five distinct values are $\Delta E = \{-16J, -8J, 0, 8J, 16J\}$. -\section{Extra}\label{sec:extra_results} -We increased number of MC cycles to 10 million +\section{Additional results}\label{sec:additional_results} +Results of 1 million MC cycles. \begin{figure} \centering \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf} - \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.} - \label{fig:phase_energy_10M} + \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.} + \label{fig:phase_energy_1M} \end{figure} \begin{figure} \centering \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf} - \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.} - \label{fig:phase_magnetization_10M} + \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.} + \label{fig:phase_magnetization_1M} \end{figure} \begin{figure} \centering \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf} - \caption{$C_{V}$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.} - \label{fig:phase_heat_10M} + \caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.} + \label{fig:phase_heat_1M} \end{figure} \begin{figure} \centering \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf} - \caption{$\chi$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.} - \label{fig:phase_susceptibility_10M} + \caption{$\chi$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.} + \label{fig:phase_susceptibility_1M} \end{figure} \end{document} \ No newline at end of file diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 9352144..db5deb3 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -212,11 +212,13 @@ correlation length is proportional with the lattice size, resulting in the criti temperature scaling relation \begin{equation} T_{c}(L) - T_{c}(L = \infty) = aL^{-1} \ , + \label{eq:critical_infinite} \end{equation} where $a$ is a constant. For the Ising model in two dimensions, with an lattice of -infinite size, the critical temperature is +infinite size, the analytical solution of the critical temperature is \begin{equation} T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B} + \label{eq:critical_solution} \end{equation} This result was found analytically by Lars Onsager in 1944. We can estimate the critical temperature of an infinite lattice, using finite lattices critical temperature @@ -288,7 +290,7 @@ distribution of the system. \subsection{Implementation and testing}\label{subsec:implementation_test} We implemented a test suite, and compared the numerical estimates to the analytical -results from \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 runtimes during implementation. diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 531019b..83ebee2 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -2,22 +2,8 @@ \begin{document} \section{Results}\label{sec:results} - -% 2.1-2-4 divided into 40 steps, which gives us a step size of 0.0075. - -% 10 MPI processes -% - 10 threads per process -% = 100 threads total -% Not a lot of downtime for the threads - -However, when the temperature is close to the critical point, we observe an increase -in expected energy and a decrease in magnetization. Suggesting a higher energy and -a loss of magnetization close to the critical temperature. - -% We did not set the seed for the random number generator, which resulted in -% different numerical estimates each time we ran the model. However, all expectation -% values are calculated using the same data. The burn-in time varied each time. -% We see a burn-in time t = 5000-10000 MC cycles. However, this changed between runs. +\subsection{Burn-in time}\label{subsec:burnin_time} +$\boldsymbol{Draft}$ We decided with a burn-in time parallelization trade-off. That is, we set the burn-in time lower in favor of sampling. To take advantage of the parallelization and not to waste computational resources. The argument to discard samples generated @@ -26,25 +12,6 @@ parallelize using MPI. We generated samples for the temperature range $T \in [2.1, 2.4]$. Using Fox we generated both 1 million samples and 10 million samples. -It is worth mentioning that the time (number of MC cycles) necessary to get a -good numerical estimate, compared to the analytical result, foreshadowing the -burn-in time. - -Markov chain starting point can differ, resulting in different simulation. By -discarding the first samples, the ones generated before system equilibrium we can -get an estimate closer to the real solution. Since we want to estimate expectation -values at a given temperature, the samples should represent the system at that -temperature. - -Depending on number of samples used in numerical estimates, using the samples -generated during burn-in can in high bias and high variance if the ratio is skewed. -However, if most samples are generated after burn-in the effect is not as visible. -Can't remove randomness by starting around equilibrium, since samples are generated -using several ising models we need to sample using the same conditions that is -system state equilibrium. - -\subsection{Burn-in time}\label{subsec:burnin_time} -$\boldsymbol{Draft}$ We start with a lattice where $L = 20$, to study the burn-in time, that is the number of Monte Carlo cycles necessary for the system to reach an equilibrium. We consider two different temperatures $T_{1} = 1.0 J/k_{B}$ and $T_{2} = 2.4 J/k_{B}$, @@ -52,7 +19,7 @@ where $T_{2}$ is close to the critical temperature. We can use the correlation time $\tau \approx L^{d + z}$ to determine time, 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.} -% Need to include a section of Onsager's analytical result. +% We show the numerical estimates for temperature $T_{1}$ of $\langle \epsilon \rangle$ in Figure \ref{fig:burn_in_energy_1_0} and $\langle |m| \rangle$ in Figure \ref{fig:burn_in_magnetization_1_0}. For temperature $T_{2}$, the numercal estimate @@ -97,85 +64,125 @@ pdf using $T = 1.0$ result in \subsection{Probability distribution}\label{subsec:probability_distribution} -$\boldsymbol{Draft}$ % Histogram figures -We use the estimated burn-in time to set starting time for sampling, then generate -samples to plot in a histogram for $T_{1}$ in Figure \ref{fig:histogram_1_0} and -$T_{2}$ in Figure \ref{fig:histogram_2_4}. For $T_{1}$ we can see that most samples -have the expected value $-2$, we have a distribution with low variance. +We used the estimated burn-in time as starting time for sampling, and generated +samples. To visualize the distribution of energy per spin $\epsilon$, we used histograms +with a bin size (...). In Figure \ref{fig:histogram_1_0} we show the distribution +for $T_{1}$, where the majority of the energy per spin is $-2$. The resulting expectation +value of energy per spin is $\langle \epsilon \rangle = -1.9969$, with a low variance +of Var$(\epsilon) = 0.0001$. % \begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/pd_estimate_1_0.pdf} \caption{Histogram $T = 1.0 J / k_{B}$} \label{fig:histogram_1_0} -\end{figure} +\end{figure} % +In Figure \ref{fig:histogram_2_4}, for $T_{2}$, the samples of energy per spin is +centered around the expectation value $\langle \epsilon \rangle = -1.2370$. % \begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/pd_estimate_2_4.pdf} \caption{Histogram $T = 2.4 J / k_{B}$} \label{fig:histogram_2_4} -\end{figure} +\end{figure} % +However, we observe a higher variance of Var$(\epsilon) = 0.0203$. When the temperature +increase, the system moves from an ordered to an onordered state. The change in +system state, or phase transition, indicates the temperature is close to a critical +point. \subsection{Phase transition}\label{subsec:phase_transition} $\boldsymbol{Draft}$ % Phase transition figures +We continue investigating the behavior of the system around the critical temperature. +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 +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 $10$ +MPI processes. Each process was set to spawn $10$ thread, resulting in a total of +$100$ threads working in parallel. We include results for $1$ million MC cycles +in Appendix \ref{sec:additional_results} + +$\boldsymbol{Rewrite}$ We ran a profiler to make sure the program was fully optimized which found that the +workload was balanced, the threads was not left idle to long/not a lot of downtime. + +In Figure \ref{fig:phase_energy_10M}, for the larger lattices we observe a sharper +increase in $\langle \epsilon \rangle$ in the temperature range $T \in [2.25, 2.35]$. %] \begin{figure} \centering - \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/energy.pdf} - \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.} - \label{fig:phase_energy} -\end{figure} - + \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf} + \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.} + \label{fig:phase_energy_10M} +\end{figure} % +We observe a deacrese in $\langle |m| \rangle$ for the same temperature range in +Figure \ref{fig:phase_magnetization_10M}, suggesting the system moves from an ordered +magnetized state to a state of no net magnetization. The system energy increase, +however, there is a loss of magnetization close to the critical temperature. \begin{figure}[H] \centering - \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/magnetization.pdf} - \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.} - \label{fig:phase_magnetization} -\end{figure} - + \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf} + \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.} + \label{fig:phase_magnetization_10M} +\end{figure} % +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 +value of heat capacity the lattice size increase. \begin{figure}[H] \centering - \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/heat_capacity.pdf} - \caption{$C_{V}$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.} - \label{fig:phase_heat} -\end{figure} - + \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.} + \label{fig:phase_heat_10M} +\end{figure} % +The magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, showed +the sharp peak in the same temperature range as that of the heat capacity. Since +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. \begin{figure}[H] \centering - \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/susceptibility.pdf} - \caption{$\chi$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.} - \label{fig:phase_susceptibility} -\end{figure} -We include results for 10 million MC cycles in Appendix \ref{sec:extra_results} + \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf} + \caption{$\chi$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.} + \label{fig:phase_susceptibility_10M} +\end{figure} % +% Add something about the correlation length? \subsection{Critical temperature}\label{subsec:critical_temperature} -$\boldsymbol{Draft}$ -We use the critical temperatures found in previous section, in addition to the -scaling relation in Equation \eqref{eq:critical_intinite} -\begin{equation} - T_{c} - T_{c}(L = \infty) = \alpha L^{-1} - \label{eq:critical_intinite} -\end{equation} -to estimate the critical temperature for a lattize of infinte size. We also -compared the estimate with the analytical solution -\begin{equation} - T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B} -\end{equation} -using linear regression. In Figure \ref{fig:linreg} we find the critical -temperatures as function of the inverse lattice size. When the lattice size increase -toward infinity, $1/L$ goes toward zero, we find the intercept which gives us an -estimated value of the critical temperature for a lattice of infinite size. - -% Critical temp regression figure +Based on the heat capacity (Figure \ref{fig:phase_heat_10M}) and susceptibility +(Figure \ref{fig:phase_susceptibility_10M}), we estimated the critical temperatures +of lattices of size $L \in \{20, 40, 60, 80, 100\}$ found in Table \ref{tab:critical_temperatures}. +\begin{table}[H] + \centering + \begin{tabular}{cc} % @{\extracolsep{\fill}} + \hline + $L$ & $T_{c}(L)$ \\ + \hline + $20$ & $J$ \\ + $40$ & $1$ \\ + $60$ & $J / k_{B}$ \\ + $80$ & $k_{B}$ \\ + $100$ & $1 / J$ \\ + \hline + \end{tabular} + \caption{Estimated critical temperatures for lattices $L \times L$.} + \label{tab:critical_temperatures} +\end{table} +We used the critical temperatures of finite lattices and the scaling relation in +Equation \eqref{eq:critical_infinite}, Section \ref{subsec:phase_critical}, to estimate +the critical temperature of a lattice of infinite size. In Figure \ref{fig:linreg_10M}, +we plot the critical temperatures $T_{c}(L)$ of the inverse lattice size $1/L$. When +the lattice size increase toward infinity, $1/L$ approaches zero. % \begin{figure}[H] \centering - \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/linreg.pdf} - \caption{Linear regression, where $\beta_{0}$ is the intercept $T_{c}(L = \infty)$ and $\beta_{1}$ is the slope.} - \label{fig:linreg} + \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/linreg.pdf} + \caption{Linear regression, where $\beta_{0}$ is the intercept approximating $T_{c}(L = \infty)$, and $\beta_{1}$ is the slope.} + \label{fig:linreg_10M} \end{figure} - - +Using linear regression, we find the intercept which gives us an estimated value +of the critical temperature for a lattice of infinite size. We find the critical +temperature to be $T_{c \text{num}} \approx 2.2695 J/k_{B}$. We also compared the +estimate with the analytical solution, the relative error of our estimate is +\begin{equation*} + \text{Relative error} = \frac{T_{c \text{ numerical}} - T_{c \text{ analytical}}}{T_{c \text{ analytical}}} \approx 0.001 J/k_{B} +\end{equation*} \end{document} From a13c8471734a83125f543898b105f5fc82ff9cc5 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Sun, 3 Dec 2023 21:53:38 +0100 Subject: [PATCH 02/17] Fixed a few things from review. --- latex/sections/introduction.tex | 2 +- latex/sections/methods.tex | 43 +++++++++++++++++---------------- 2 files changed, 23 insertions(+), 22 deletions(-) diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index dfcf5a1..0c65060 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -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 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}. \end{document} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index db5deb3..4de3bc7 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -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} \end{pmatrix} \ . \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 $s_{i} \in \{-1, +1\}$, -where $-1$ represent down and $+1$ spin up. The spins interact with its nearest neighbors, -and in a two-dimensional lattice each spin has up to four nearest neighbors. However, -the model is not restricted to this dimentionality \cite[p. 3]{obermeyer:2020:ising}. -In our experiment we will use periodic boundary conditions, meaning all spins -have exactly four nearest neighbors. To find the analytical expressions necessary -for validating our model implementation, we will assume a $2 \times 2$ lattice. +A given spin $i$ can take one of two possible discrete values $s_{i} \in \{-1, +1\}$. +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 +boundary conditions, resulting in all spins having exactly four nearest neighbors. +To find the analytical expressions necessary for validating our model implementation, +we will assume a $2 \times 2$ lattice. The hamiltonian of the Ising model is given by \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} \ . \label{eq:energy_total} \end{equation} -To avoid counting duplicated, we count the neighboring spins using the pattern -visualized in Figure \ref{fig:tikz_boundary}. +We count the neighboring spins using the pattern visualized in Figure \ref{fig:tikz_boundary}, +to avoid counting a given spin pair several times. \begin{figure} \centering \begin{tikzpicture} @@ -196,7 +196,7 @@ Boltzmann constant we derive the remaining units, which can be found in Table $[\chi]$ & $1 / J$ & \\ \hline \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} \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} 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 -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 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 @@ -282,7 +282,7 @@ in an array. \end{algorithm} \end{figure} 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. The probability distribution of the samples will tend toward the actual probability 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 takes advantage of the parallel optimization. -We parallelize our code using both a message passing interface (OpenMPI) and -multi-threading (OpenMP). First, we divide the temperatures into smaller ranges, -and each MPI process receives a set of temperatures. Every MPI process spawn a -set of threads, which initialize an Ising model and performs the Metropolis-Hastings -algorithm. We limit the number of times threads are spawned and joined, by using -single parallel regions, reducing parallel overhead. 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. +We parallelized our code using both message passing interface (OpenMPI) and +multi-threading (OpenMP). First, we divided the temperatures into smaller sub-ranges, +and each MPI process received a sub-range of temperatures. Every MPI initialize a +parallel region with a set of threads, which then initializes an Ising model and +performs the Metropolis-Hastings algorithm. We limited the number of times threads +are spawned and joined, by using single parallel regions, reducing parallel overhead. +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 for $1$ and $10$ million +Monte Carlo cycles when analyzing phase transitions. From 869a3c4aceaeaebb4eb719262416ce6e47e6406a Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Mon, 4 Dec 2023 10:08:43 +0100 Subject: [PATCH 03/17] Missing estimate of speed-up and critical temperatures in table. --- latex/ising_model.tex | 7 ++- latex/sections/appendices.tex | 56 +++++++++++--------- latex/sections/methods.tex | 13 ++--- latex/sections/results.tex | 99 +++++++++++++++++------------------ 4 files changed, 92 insertions(+), 83 deletions(-) diff --git a/latex/ising_model.tex b/latex/ising_model.tex index de5c0c5..96ba30a 100644 --- a/latex/ising_model.tex +++ b/latex/ising_model.tex @@ -77,13 +77,16 @@ \clearpage \newpage +% Appendix +\subfile{sections/appendices} + +\clearpage + \onecolumngrid \bibliographystyle{unsrtnat} \bibliography{references} -% Appendix -\subfile{sections/appendices} \end{document} diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex index 151e96b..f80f7bd 100644 --- a/latex/sections/appendices.tex +++ b/latex/sections/appendices.tex @@ -40,12 +40,15 @@ the spin up as visualized in Figure \ref{fig:tikz_neighbor}. Using the values estimated for the $2 \times 2$ case, found in \ref{tab:lattice_config}, we find the partition function \begin{align*} - Z &= 1 \cdot e^{-\beta (-8J)} + 4 \cdot e^{-\beta (0)} + 4 \cdot e^{-\beta (0)} + 2 \cdot e^{-\beta (8J)} \\ - & \quad + 4 \cdot e^{-\beta (0)} 1 \cdot e^{-\beta (-8J)} \\ + Z &= 1 \cdot e^{-\beta (-8J)} + 4 \cdot e^{-\beta (0)} + 4 \cdot e^{-\beta (0)} \\ + & \quad + 2 \cdot e^{-\beta (8J)} + 4 \cdot e^{-\beta (0)} 1 \cdot e^{-\beta (-8J)} \\ &= 2e^{8 \beta J} + 2e^{-8 \beta J} + 12. \end{align*} -We rewrite the expression using the identity $\cosh(8 \beta J) = 1/2 \big( e^{8 \beta J} + e^{-8 \beta J})$, -and get +We rewrite the expression using the identity +\begin{align*} + \cosh(8 \beta J) &= 1/2 \big( e^{8 \beta J} + e^{-8 \beta J}) +\end{align*} +and find \begin{align*} z &= 4 \cosh (8 \beta J) + 12 \ . \end{align*} @@ -55,12 +58,10 @@ and get For a linear function of a stochastic random variable $X$, with a known probability distribution, the expected value of $x$ is given by \begin{align*} - \mathbb{E}(aX + b) &= a \cdot \mathbb{E}(X) + b & \text{\cite[p. 131]{springer:2012:modernstat}} + \langle aX + b \rangle &= a \cdot \langle X \rangle + b & \text{\cite[p. 131]{springer:2012:modernstat}} \end{align*} In our case the discrete random variable is the spin configuration, and we want -to find the expected value of the function $E(\mathbf{s})$. Continuing, we will -use the notation $\langle E \rangle$ for the expectation value of a given variable, -in this case $E$. +to find the expected value of the function $E(\mathbf{s})$. Both energy per spin and magnetization per spin are functions of $\mathbf{s}$. In addition, the number of spins is given as a constant for each lattice. We can @@ -75,9 +76,10 @@ The same applies to magnetization per spin \begin{align*} \langle |m| \rangle = \frac{1}{N} \sum_{i=1}^{N} |M(s_{i})| p(s_{i} \ | \ T) \ . \end{align*} -Continuing with the expectation values for a $2 \times 2$ lattice, excluding the terms which give zero we get +Continuing with the expectation values for a $2 \times 2$ lattice, excluding the terms which result in zero, we get \begin{align*} - \langle E \rangle &= (-8J) \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J) \cdot \frac{1}{Z} e^{-8 \beta J} + (-8J) \cdot \frac{1}{Z} e^{8 \beta J} \\ + \langle E \rangle &= (-8J) \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J) \cdot \frac{1}{Z} e^{-8 \beta J} \\ + & \quad + (-8J) \cdot \frac{1}{Z} e^{8 \beta J} \\ &= \frac{16J}{Z} \big(e^{-8 \beta J} - e^{8 \beta J}) \\ &= -\frac{32J \sinh(8 \beta J)}{4(\cosh(8 \beta J) + 3)} \\ &= -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} \ , @@ -90,9 +92,10 @@ and &= \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} \ . \end{align*} -The squared function +The squared energy function \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} + (-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} \\ &= \frac{128J^{2}}{Z} \big(e^{8 \beta J} + e^{-8 \beta J} \big) \\ &= \frac{128J^{2} \cosh(8 \beta J)}{4(\cosh(8 \beta J) + 3)} \\ &= \frac{64J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} \ , @@ -116,6 +119,7 @@ and &= \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \ . \end{align*} + \section{Heat capacity and magnetic susceptibility}\label{sec:heat_susceptibility} To find the heat capacity in Eq. \ref{eq:heat_capacity}, we normalize to heat capacity per spin @@ -149,8 +153,9 @@ and the variance of the total magnetization is given by \mathbb{V}(M) &= \mathbb{E}(M^{2}) - [\mathbb{E}(|M|)]^{2} \\ &= \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3} - \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \\ &= \frac{(8(e^{8 \beta J} + 1)) \cdot (\cosh(8 \beta J) + 3) - 4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \\ - &= \frac{4(e^{8 \beta J} + 1) \cdot (e^{8 \beta J} + e^{-8 \beta J}) + 24(e^{8 \beta J} + 1) - 4(e^{8 \beta J} + 1)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \\ - &= \frac{4e^{2(8 \beta J)} + 4e^{8 \beta J} 4e^{0} + 4e^{-8 \beta J} 24e^{8 \beta J} + 24 - 4e^{2(8 \beta J)} - 16e^{8 \beta J} - 16}{(\cosh(8 \beta J) + 3)^{2}} \\ + &= \frac{4(e^{8 \beta J} + 1) \cdot (e^{8 \beta J} + e^{-8 \beta J})}{(\cosh(8 \beta J) + 3)^{2}} \\ + & \quad + \frac{24(e^{8 \beta J} + 1) - 4(e^{8 \beta J} + 1)^{2}}{(\cosh(8 \beta J) + 3)^{2}}\\ + % &= \frac{4e^{2(8 \beta J)} + 4e^{8 \beta J} 4e^{0} + 4e^{-8 \beta J} 24e^{8 \beta J} + 24 - 4e^{2(8 \beta J)} - 16e^{8 \beta J} - 16}{(\cosh(8 \beta J) + 3)^{2}} \\ &= \frac{4(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}} \ . \end{align*} We find the heat capacity @@ -181,29 +186,30 @@ where the five distinct values are $\Delta E = \{-16J, -8J, 0, 8J, 16J\}$. \section{Additional results}\label{sec:additional_results} -Results of 1 million MC cycles. -\begin{figure} +We also did the phase transition experiment using 1 million MC cycles. In Figure \ref{fig:phase_energy_1M} +we show expected energy per spin, and in Figure \ref{fig:phase_magnetization_1M} +expected magnetization per spin. +\begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf} \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.} \label{fig:phase_energy_1M} -\end{figure} - -\begin{figure} +\end{figure} % +\begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf} \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.} \label{fig:phase_magnetization_1M} -\end{figure} - -\begin{figure} +\end{figure} % +In Figure \ref{fig:phase_heat_1M} we show heat capacity, and in Figure \ref{fig:phase_susceptibility_1M} +the magnetic susceptibility. +\begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf} \caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.} \label{fig:phase_heat_1M} -\end{figure} - -\begin{figure} +\end{figure} % +\begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf} \caption{$\chi$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 4de3bc7..c008006 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -314,12 +314,13 @@ Monte Carlo cycles when analyzing phase transitions. \subsection{Tools}\label{subsec:tools} 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}. We used the Python library -\verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and -\verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. In -addition, we used \verb|Scalasca| \cite{scalasca} and \verb|Score-P| \cite{scorep} -to profile and optimize our implementation. -% Add profiler +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 +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|. +To optimize our implementation, we used a profiler tool \verb|Scalasca| \cite{scalasca} +and \verb|Score-P| \cite{scorep}. + \end{document} diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 83ebee2..43aa2b4 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -3,72 +3,72 @@ \begin{document} \section{Results}\label{sec:results} \subsection{Burn-in time}\label{subsec:burnin_time} -$\boldsymbol{Draft}$ -We decided with a burn-in time parallelization trade-off. That is, we set the -burn-in time lower in favor of sampling. To take advantage of the parallelization -and not to waste computational resources. The argument to discard samples generated -during the burn-in time is ... Increasing number of samples outweigh the ... -parallelize using MPI. We generated -samples for the temperature range $T \in [2.1, 2.4]$. Using Fox we generated both -1 million samples and 10 million samples. +We started with a lattice size $L = 20$ and considered the temperatures $T_{1} = 1.0 J/k_{B}$, +and $T_{2} = 2.4 J/k_{B}$, where $T_{2}$ is close to the analytical critical +temperature given by Equation \eqref{eq:critical_solution}. -We start with a lattice where $L = 20$, to study the burn-in time, that is the -number of Monte Carlo cycles necessary for the system to reach an equilibrium. -We consider two different temperatures $T_{1} = 1.0 J/k_{B}$ and $T_{2} = 2.4 J/k_{B}$, -where $T_{2}$ is close to the critical temperature. We can use the correlation -time $\tau \approx L^{d + z}$ to determine time, 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.} -% -We show the numerical estimates for temperature $T_{1}$ of $\langle \epsilon \rangle$ -in Figure \ref{fig:burn_in_energy_1_0} and $\langle |m| \rangle$ in Figure -\ref{fig:burn_in_magnetization_1_0}. For temperature $T_{2}$, the numercal estimate -of $\langle \epsilon \rangle$ is shown in Figure \ref{fig:burn_in_energy_2_4} and -$\langle |m| \rangle$ in Figure \ref{fig:burn_in_magnetization_2_4}. The lattice -is initialized in both an ordered and an unordered state. We observe that for -$T_{1}$ there is no change in either expectation value with increasing number of -Monte Carlo cycles, when we start with an ordered state. As for the unordered -lattice, we observe a change for the first 5000 MC cycles, where it stabilizes. -The approximated expected energy is $-2$ and expected magnetization is $1.0$, -which is to be expected for temperature 0f $1.0$. T is below the critical and the -pdf using $T = 1.0$ result in -\begin{align*} - p(s|T=1.0) &= \frac{1}{e^{-\beta \sum E(s)}} e^{-\beta E(s)} \\ - &= \frac{1}{e^{-(1/k_{B}) \sum E(s)}} e^{-(1/k_{B}) E(s)} \ . -\end{align*} +To determine the burn-in time, we used the numerical estimates of energy +per spin for $T_{1}$ in Figure \ref{fig:burn_in_energy_1_0}, and $T_{2}$ in Figure +\ref{fig:burn_in_energy_2_4}. We also considered the estimates of magnetization +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}. +% 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 +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 +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 $z = 2.1665 \pm 0.0012$ +\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 +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 +no change in expectation value of energy or magnetization for $T_{1}$, when we +initialized the lattice in an ordered state. As for the unordered initialized +lattice, we first observed a change in expectation values, and a stabilization around +$5000$ Monte Carlo cycles. The expected energy per spin is $\langle \epsilon \rangle = -2$ +and the expected magnetization per spin is $\langle |m| \rangle = 1.0$. % add something about what is expected for $T_{1}$ ? +For $T_{2}$ we observed a change in expectation values for both the ordered and the +unordered lattice. +% \begin{align*} +% p(s|T=1.0) &= \frac{1}{e^{-\beta \sum E(s)}} e^{-\beta E(s)} \\ +% &= \frac{1}{e^{-(1/k_{B}) \sum E(s)}} e^{-(1/k_{B}) E(s)} \ . +% \end{align*} +For $T_{2}$ we observe an increase in expected energy per spin $\langle \epsilon \rangle \approx -1.23$, +and a decrease in expected magnetization per spin $\langle |m| \rangle \approx 0.46$. % Burn-in figures \begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/burn_in_time_magnetization_1_0.pdf} - \caption{$\langle |m| \rangle$ as a function of time, for $T = 1.0 J / k_{B}$} + \caption{Magnetization per spin $\langle |m| \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 1.0 J / k_{B}$} \label{fig:burn_in_magnetization_1_0} \end{figure} \begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/burn_in_time_energy_1_0.pdf} - \caption{$\langle \epsilon \rangle$ as a function of time, for $T = 1.0 J / k_{B}$} + \caption{Energy per spin $\langle \epsilon \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 1.0 J / k_{B}$} \label{fig:burn_in_energy_1_0} \end{figure} \begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/burn_in_time_magnetization_2_4.pdf} - \caption{$\langle |m| \rangle$ as a function of time, for $T = 2.4 J / k_{B}$} + \caption{Magnetization per spin $\langle |m| \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 2.4 J / k_{B}$} \label{fig:burn_in_magnetization_2_4} \end{figure} \begin{figure}[H] \centering \includegraphics[width=\linewidth]{../images/burn_in_time_energy_2_4.pdf} - \caption{$\langle \epsilon \rangle$ as a function of time, for $T = 2.4 J / k_{B}$} + \caption{Energy per spin $\langle \epsilon \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 2.4 J / k_{B}$} \label{fig:burn_in_energy_2_4} \end{figure} \subsection{Probability distribution}\label{subsec:probability_distribution} % Histogram figures -We used the estimated burn-in time as starting time for sampling, and generated +We used the estimated burn-in time of $5000$ Monte Carlo cycles as starting time, and generated samples. To visualize the distribution of energy per spin $\epsilon$, we used histograms -with a bin size (...). In Figure \ref{fig:histogram_1_0} we show the distribution -for $T_{1}$, where the majority of the energy per spin is $-2$. The resulting expectation +with a bin size $0.02$. In Figure \ref{fig:histogram_1_0} we show the distribution +for $T_{1}$. Where the resulting expectation value of energy per spin is $\langle \epsilon \rangle = -1.9969$, with a low variance of Var$(\epsilon) = 0.0001$. % \begin{figure}[H] @@ -85,16 +85,15 @@ centered around the expectation value $\langle \epsilon \rangle = -1.2370$. % \caption{Histogram $T = 2.4 J / k_{B}$} \label{fig:histogram_2_4} \end{figure} % -However, we observe a higher variance of Var$(\epsilon) = 0.0203$. When the temperature -increase, the system moves from an ordered to an onordered state. The change in +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 system state, or phase transition, indicates the temperature is close to a critical point. \subsection{Phase transition}\label{subsec:phase_transition} -$\boldsymbol{Draft}$ % Phase transition figures -We continue investigating the behavior of the system around the critical temperature. +We continued investigating the behavior of the system around the critical temperature. 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 temperature range into $40$ steps, with an equal step size of $0.0075$. The samples @@ -133,7 +132,7 @@ value of heat capacity the lattice size increase. \caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.} \label{fig:phase_heat_10M} \end{figure} % -The magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, showed +The magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, show the sharp peak in the same temperature range as that of the heat capacity. Since 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. @@ -156,14 +155,14 @@ of lattices of size $L \in \{20, 40, 60, 80, 100\}$ found in Table \ref{tab:crit \hline $L$ & $T_{c}(L)$ \\ \hline - $20$ & $J$ \\ - $40$ & $1$ \\ - $60$ & $J / k_{B}$ \\ - $80$ & $k_{B}$ \\ - $100$ & $1 / J$ \\ + $20$ & $ J / k_{B}$ \\ + $40$ & $ J / k_{B}$ \\ + $60$ & $ J / k_{B}$ \\ + $80$ & $ J / k_{B}$ \\ + $100$ & $ J / k_{B}$ \\ \hline \end{tabular} - \caption{Estimated critical temperatures for lattices $L \times L$.} + \caption{Estimated critical temperatures for lattices $L \times L$, where $L$ denote the lattice size.} \label{tab:critical_temperatures} \end{table} We used the critical temperatures of finite lattices and the scaling relation in From 90047decb4310dfec10b59e6846317f356677261 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Mon, 4 Dec 2023 15:09:37 +0100 Subject: 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