From aa6a2cb5d870f8c71e52b4f33351df4a3006d34e Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 7 Nov 2023 15:11:19 +0100 Subject: [PATCH 01/25] Add section headings to all files, and add analytical values for problem 1 --- latex/ising_model.tex | 4 +-- latex/references.bib | 5 --- latex/sections/abstract.tex | 3 ++ latex/sections/conclusion.tex | 2 ++ latex/sections/introduction.tex | 2 +- latex/sections/methods.tex | 55 +++++++++++++++++++++++++++++++++ latex/sections/results.tex | 2 ++ 7 files changed, 65 insertions(+), 8 deletions(-) diff --git a/latex/ising_model.tex b/latex/ising_model.tex index 2614b7d..3a0d6f2 100644 --- a/latex/ising_model.tex +++ b/latex/ising_model.tex @@ -30,7 +30,7 @@ \usepackage[noend]{algpseudocode} \usepackage{subfigure} \usepackage{tikz} -\usetikzlibrary{quantikz2} +% \usetikzlibrary{quantikz2} % defines the color of hyperref objects % Blending two colors: blue!80!black = 80% blue and 20% black \hypersetup{ % this is just my personal choice, feel free to change things @@ -49,7 +49,7 @@ \begin{document} -\title{Project 3} % self-explanatory +\title{Project 4} % self-explanatory \author{Cory Alexander Balaton \& Janita Ovidie Sandtrøen Willumsen \\ \faGithub \, \url{https://github.uio.no/FYS3150-G2-2023/Project-4}} % self-explanatory \date{\today} % self-explanatory \noaffiliation % ignore this, but keep it. diff --git a/latex/references.bib b/latex/references.bib index d1dfe6c..494f104 100644 --- a/latex/references.bib +++ b/latex/references.bib @@ -67,8 +67,3 @@ } # Miscellaneous things -@misc{penning:fig, - key = {something}, - url = {https://www.med.physik.uni-muenchen.de/research/nuclear-science/nuclear-masses/mlltrap/layout/traps/index.html}, - note = {Configuration figure a} -} diff --git a/latex/sections/abstract.tex b/latex/sections/abstract.tex index b05e0b6..b69617c 100644 --- a/latex/sections/abstract.tex +++ b/latex/sections/abstract.tex @@ -1,4 +1,7 @@ \documentclass[../ising_model.tex]{subfiles} \begin{document} +\begin{abstract} + We did some amazing stuff! +\end{abstract} \end{document} diff --git a/latex/sections/conclusion.tex b/latex/sections/conclusion.tex index b05e0b6..073c2e9 100644 --- a/latex/sections/conclusion.tex +++ b/latex/sections/conclusion.tex @@ -1,4 +1,6 @@ \documentclass[../ising_model.tex]{subfiles} \begin{document} +\section{Conclusion}\label{sec:conclusion} + \end{document} diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index 7322a58..62d86ae 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -1,7 +1,7 @@ \documentclass[../ising_model.tex]{subfiles} \begin{document} - +\section{Introduction}\label{sec:introduction} \cite{openmp:2018} \end{document} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 81ddfcc..3178903 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -1,5 +1,60 @@ \documentclass[../ising_model.tex]{subfiles} \begin{document} +\section{Methods}\label{sec:methods} +% Problem 1 +\begin{align*} + Z &= \sum_{all \ s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\ + &= \dots \\ + &= 4 \cosh (8 \beta J) + 12 \\ +\end{align*} +% +\begin{align*} + \langle \epsilon \rangle &= \frac{-2J \sinh(8 \beta J)}{ \cosh(8 \beta J) + 3} +\end{align*} +% +\begin{align*} + \langle \epsilon^{2} \rangle &= \frac{4 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} +\end{align*} +% +\begin{align*} + \langle |m| \rangle &= \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} +\end{align*} +\begin{align*} + \langle |m|^{2} \rangle &= \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} +\end{align*} +% +\begin{align*} + \langle E \rangle &= \frac{-8 J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} +\end{align*} +% +\begin{align*} + \langle E \rangle^{2} &= \frac{64 J^{2} \sinh(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} +\end{align*} +% +\begin{align*} + \langle E^{2} \rangle &= \frac{64 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} +\end{align*} +% +\begin{align*} + \langle M \rangle &= \frac{2 e^{8 \beta J}}{\cosh(8 \beta J) + 3} +\end{align*} +% +\begin{align*} + \langle M \rangle^{2} &= \frac{4e^{16 \beta J} + 16e^{8 \beta J} + 16}{(\cosh(8 \beta J) + 3)^{2}} +\end{align*} +% +\begin{align*} + \langle M^{2} \rangle &= \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3} +\end{align*} +% +\begin{align*} + C_{V} &= \frac{64 J^{2}}{N k_{\text{B}} T^{2}} \Big( \frac{3 \cosh(8 \beta J) + \cosh^{2}(8 \beta J) - \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \Big) +\end{align*} +% +\begin{align*} + \chi &= \frac{1}{N} \frac{1}{k_{\text{B}} T^{2}} (\langle M^{2} \rangle - \langle M \rangle^{2}) \\ + &= \frac{1}{N k_{\text{B}} T} \Big( \frac{12e^{8 \beta J} + 4 e^{-8 \beta J} + 12}{(\cosh(8 \beta J) + 3)^{2}} \Big) +\end{align*} \end{document} diff --git a/latex/sections/results.tex b/latex/sections/results.tex index b05e0b6..26924a0 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -1,4 +1,6 @@ \documentclass[../ising_model.tex]{subfiles} \begin{document} +\section{Results}\label{sec:results} + \end{document} From da97e7a6abeacf3498bf86541ae4458e58f88172 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Thu, 9 Nov 2023 08:15:32 +0100 Subject: [PATCH 02/25] Add appendix for calculations --- latex/appendices.tex | 13 +++++++++++++ 1 file changed, 13 insertions(+) create mode 100644 latex/appendices.tex diff --git a/latex/appendices.tex b/latex/appendices.tex new file mode 100644 index 0000000..e1f55de --- /dev/null +++ b/latex/appendices.tex @@ -0,0 +1,13 @@ +\documentclass[../ising_model.tex]{subfiles} + +\begin{document} +\appendix +\section{Ising model system states}\label{sec:system_states} + + +\section{Analytical expressions}\label{sec:analytical_expressions} + + + + +\end{document} \ No newline at end of file From 7404521d0bb66e6c411678a77fde26975d077c0d Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Thu, 9 Nov 2023 08:16:16 +0100 Subject: [PATCH 03/25] Update section layout --- latex/sections/methods.tex | 8 ++++++++ 1 file changed, 8 insertions(+) diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 3178903..64ed8b4 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -2,6 +2,7 @@ \begin{document} \section{Methods}\label{sec:methods} +\subsection{The Ising model}\label{sec:ising} % Problem 1 \begin{align*} Z &= \sum_{all \ s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\ @@ -56,5 +57,12 @@ \chi &= \frac{1}{N} \frac{1}{k_{\text{B}} T^{2}} (\langle M^{2} \rangle - \langle M \rangle^{2}) \\ &= \frac{1}{N k_{\text{B}} T} \Big( \frac{12e^{8 \beta J} + 4 e^{-8 \beta J} + 12}{(\cosh(8 \beta J) + 3)^{2}} \Big) \end{align*} + +\subsection{Markov Chain Monte Carlo methods} + + +\subsection{Implementation} + + \end{document} From fe6144d6214e8538e1613e990a878eb2d672694d Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Thu, 9 Nov 2023 08:16:57 +0100 Subject: [PATCH 04/25] Fix online ref style --- latex/references.bib | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/latex/references.bib b/latex/references.bib index 494f104..2513567 100644 --- a/latex/references.bib +++ b/latex/references.bib @@ -58,7 +58,7 @@ } # C++ libraries -@online{openmp:2018, +@misc{openmp:2018, author = {OpenMP}, title = {OpenMP Application Programming Interface}, year = {2018}, From 2512a0febf28c43c256f3592c39ecd8008c97748 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Thu, 9 Nov 2023 08:17:19 +0100 Subject: [PATCH 05/25] Include appendix --- latex/ising_model.tex | 9 +++------ 1 file changed, 3 insertions(+), 6 deletions(-) diff --git a/latex/ising_model.tex b/latex/ising_model.tex index 3a0d6f2..1c4155e 100644 --- a/latex/ising_model.tex +++ b/latex/ising_model.tex @@ -73,15 +73,12 @@ % Conclusion \subfile{sections/conclusion} - -%\appendix -%\subfile{appendix/appendix_a} -%\subfile{appendix/appendix_b} -%\subfile{appendix/appendix_c} +% Appendix +\subfile{appendices} \onecolumngrid -\bibliographystyle{unsrt} +\bibliographystyle{unsrtnat} \bibliography{references} From 3d94d31f43d499df2044b5cfc52323c17161b1c8 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Thu, 9 Nov 2023 10:04:06 +0100 Subject: [PATCH 06/25] Add illustration of 2D grid --- latex/appendices.tex | 34 ++++++++++++++++++++++++++++++++-- 1 file changed, 32 insertions(+), 2 deletions(-) diff --git a/latex/appendices.tex b/latex/appendices.tex index e1f55de..3525403 100644 --- a/latex/appendices.tex +++ b/latex/appendices.tex @@ -3,11 +3,41 @@ \begin{document} \appendix \section{Ising model system states}\label{sec:system_states} +To avoid counting duplicates, we used +\begin{figure}\label{fig:tikz_counting} + \centering + \begin{tikzpicture} + \draw (0, 0) grid (2, 2); + % \node[inner] (s1) at (0.5, 0.5) {s}; + \node (s1) at (0.5, 1.5) {$s_1$}; + \node (s2) at (1.5, 1.5) {$s_2$}; + \node (s3) at (0.5, 0.5) {$s_3$}; + \node (s4) at (1.5, 0.5) {$s_4$}; + \node[gray] (s12) at (2.5, 1.5) {$s_1$}; + \node[gray] (s34) at (2.5, 0.5) {$s_3$}; + \node[gray] (s13) at (0.5, -0.5) {$s_1$}; + \node[gray] (s24) at (1.5, -0.5) {$s_2$}; + + \draw[red, ->] (s1.east) -- (s2.west); + \draw[red, ->] (s2.east) -- (s12.west); + \draw[red, ->] (s1.south) -- (s3.north); + \draw[red, ->] (s2.south) -- (s4.north); + \draw[red, ->] (s3.east) -- (s4.west); + \draw[red, ->] (s4.east) -- (s34.west); + \draw[red, ->] (s3.south) -- (s13.north); + \draw[red, ->] (s4.south) -- (s24.north); + \end{tikzpicture} + \caption{Rules for multiplying spin pairs.} +\end{figure} \section{Analytical expressions}\label{sec:analytical_expressions} - - +The Boltzmann distribution is normalized using a partition function $Z$ given by +\begin{align*} + Z &= \sum_{\text{all possible } s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\ + &= \dots \\ + &= 4 \cosh (8 \beta J) + 12 \\ +\end{align*} \end{document} \ No newline at end of file From 20cb224511ca876d85378712827e54cd77cf9cc7 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Thu, 9 Nov 2023 10:06:13 +0100 Subject: [PATCH 07/25] Move appendix file --- latex/ising_model.tex | 4 ++-- latex/sections/appendices.tex | 43 +++++++++++++++++++++++++++++++++++ 2 files changed, 45 insertions(+), 2 deletions(-) create mode 100644 latex/sections/appendices.tex diff --git a/latex/ising_model.tex b/latex/ising_model.tex index 1c4155e..be6820d 100644 --- a/latex/ising_model.tex +++ b/latex/ising_model.tex @@ -30,7 +30,7 @@ \usepackage[noend]{algpseudocode} \usepackage{subfigure} \usepackage{tikz} -% \usetikzlibrary{quantikz2} +% \usetikzlibrary{quantikz} % defines the color of hyperref objects % Blending two colors: blue!80!black = 80% blue and 20% black \hypersetup{ % this is just my personal choice, feel free to change things @@ -74,7 +74,7 @@ \subfile{sections/conclusion} % Appendix -\subfile{appendices} +\subfile{sections/appendices} \onecolumngrid diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex new file mode 100644 index 0000000..3525403 --- /dev/null +++ b/latex/sections/appendices.tex @@ -0,0 +1,43 @@ +\documentclass[../ising_model.tex]{subfiles} + +\begin{document} +\appendix +\section{Ising model system states}\label{sec:system_states} +To avoid counting duplicates, we used +\begin{figure}\label{fig:tikz_counting} + \centering + \begin{tikzpicture} + \draw (0, 0) grid (2, 2); + % \node[inner] (s1) at (0.5, 0.5) {s}; + \node (s1) at (0.5, 1.5) {$s_1$}; + \node (s2) at (1.5, 1.5) {$s_2$}; + \node (s3) at (0.5, 0.5) {$s_3$}; + \node (s4) at (1.5, 0.5) {$s_4$}; + \node[gray] (s12) at (2.5, 1.5) {$s_1$}; + \node[gray] (s34) at (2.5, 0.5) {$s_3$}; + \node[gray] (s13) at (0.5, -0.5) {$s_1$}; + \node[gray] (s24) at (1.5, -0.5) {$s_2$}; + + \draw[red, ->] (s1.east) -- (s2.west); + \draw[red, ->] (s2.east) -- (s12.west); + \draw[red, ->] (s1.south) -- (s3.north); + \draw[red, ->] (s2.south) -- (s4.north); + \draw[red, ->] (s3.east) -- (s4.west); + \draw[red, ->] (s4.east) -- (s34.west); + \draw[red, ->] (s3.south) -- (s13.north); + \draw[red, ->] (s4.south) -- (s24.north); + \end{tikzpicture} + \caption{Rules for multiplying spin pairs.} +\end{figure} + + +\section{Analytical expressions}\label{sec:analytical_expressions} +The Boltzmann distribution is normalized using a partition function $Z$ given by +\begin{align*} + Z &= \sum_{\text{all possible } s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\ + &= \dots \\ + &= 4 \cosh (8 \beta J) + 12 \\ +\end{align*} + + +\end{document} \ No newline at end of file From 4de4af4cc7611dfa1aad4b0a52e84e1d94f0a008 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Thu, 9 Nov 2023 10:07:07 +0100 Subject: [PATCH 08/25] Add tool section --- latex/sections/methods.tex | 15 ++++++++++++--- 1 file changed, 12 insertions(+), 3 deletions(-) diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 64ed8b4..0b4bc61 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -2,7 +2,7 @@ \begin{document} \section{Methods}\label{sec:methods} -\subsection{The Ising model}\label{sec:ising} +\subsection{The Ising model}\label{sec:ising_model} % Problem 1 \begin{align*} Z &= \sum_{all \ s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\ @@ -57,11 +57,20 @@ \chi &= \frac{1}{N} \frac{1}{k_{\text{B}} T^{2}} (\langle M^{2} \rangle - \langle M \rangle^{2}) \\ &= \frac{1}{N k_{\text{B}} T} \Big( \frac{12e^{8 \beta J} + 4 e^{-8 \beta J} + 12}{(\cosh(8 \beta J) + 3)^{2}} \Big) \end{align*} +The derivation of analytical expressions can be found in appendix +\ref{sec:analytical_expressions} -\subsection{Markov Chain Monte Carlo methods} +\subsection{Markov Chain Monte Carlo methods}\label{sec:mcmc_methods} -\subsection{Implementation} +\subsection{Implementation}\label{sec:implementation} + + +\subsection{Tools}\label{sec:tools} +The Ising model and MCMC methods are implemented in C++, and parallelized using +\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. \end{document} From 2b35ab592fb0197d515f9534095077b5ed4437a3 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Thu, 9 Nov 2023 10:07:26 +0100 Subject: [PATCH 09/25] Move file --- latex/appendices.tex | 43 ------------------------------------------- 1 file changed, 43 deletions(-) delete mode 100644 latex/appendices.tex diff --git a/latex/appendices.tex b/latex/appendices.tex deleted file mode 100644 index 3525403..0000000 --- a/latex/appendices.tex +++ /dev/null @@ -1,43 +0,0 @@ -\documentclass[../ising_model.tex]{subfiles} - -\begin{document} -\appendix -\section{Ising model system states}\label{sec:system_states} -To avoid counting duplicates, we used -\begin{figure}\label{fig:tikz_counting} - \centering - \begin{tikzpicture} - \draw (0, 0) grid (2, 2); - % \node[inner] (s1) at (0.5, 0.5) {s}; - \node (s1) at (0.5, 1.5) {$s_1$}; - \node (s2) at (1.5, 1.5) {$s_2$}; - \node (s3) at (0.5, 0.5) {$s_3$}; - \node (s4) at (1.5, 0.5) {$s_4$}; - \node[gray] (s12) at (2.5, 1.5) {$s_1$}; - \node[gray] (s34) at (2.5, 0.5) {$s_3$}; - \node[gray] (s13) at (0.5, -0.5) {$s_1$}; - \node[gray] (s24) at (1.5, -0.5) {$s_2$}; - - \draw[red, ->] (s1.east) -- (s2.west); - \draw[red, ->] (s2.east) -- (s12.west); - \draw[red, ->] (s1.south) -- (s3.north); - \draw[red, ->] (s2.south) -- (s4.north); - \draw[red, ->] (s3.east) -- (s4.west); - \draw[red, ->] (s4.east) -- (s34.west); - \draw[red, ->] (s3.south) -- (s13.north); - \draw[red, ->] (s4.south) -- (s24.north); - \end{tikzpicture} - \caption{Rules for multiplying spin pairs.} -\end{figure} - - -\section{Analytical expressions}\label{sec:analytical_expressions} -The Boltzmann distribution is normalized using a partition function $Z$ given by -\begin{align*} - Z &= \sum_{\text{all possible } s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\ - &= \dots \\ - &= 4 \cosh (8 \beta J) + 12 \\ -\end{align*} - - -\end{document} \ No newline at end of file From cfef3c7366de1e3384be7b2bc66438b785576039 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Thu, 9 Nov 2023 10:28:33 +0100 Subject: [PATCH 10/25] sections/methods.tex --- latex/sections/methods.tex | 91 ++++++++++++++++++++------------------ 1 file changed, 47 insertions(+), 44 deletions(-) diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 0b4bc61..9376258 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -3,62 +3,65 @@ \begin{document} \section{Methods}\label{sec:methods} \subsection{The Ising model}\label{sec:ising_model} +% Add definitions + +We will assume a $2 \times 2$ lattice to study the possible system states, and +to find analytical expressions necessary in the Markov Chain Monte Carlo method. +These results are used to test our code during implementation. % Problem 1 -\begin{align*} - Z &= \sum_{all \ s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\ - &= \dots \\ - &= 4 \cosh (8 \beta J) + 12 \\ -\end{align*} + +\begin{equation}\label{eq:partition_function} + Z = 4 \cosh (8 \beta J) + 12, +\end{equation} % -\begin{align*} - \langle \epsilon \rangle &= \frac{-2J \sinh(8 \beta J)}{ \cosh(8 \beta J) + 3} -\end{align*} +\begin{equation}\label{eq:energy_spin_first} + \langle \epsilon \rangle = \frac{-2J \sinh(8 \beta J)}{ \cosh(8 \beta J) + 3} +\end{equation} % -\begin{align*} - \langle \epsilon^{2} \rangle &= \frac{4 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} -\end{align*} +\begin{equation}\label{eq:energy_spin_second} + \langle \epsilon^{2} \rangle = \frac{4 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} +\end{equation} % -\begin{align*} - \langle |m| \rangle &= \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} -\end{align*} -\begin{align*} - \langle |m|^{2} \rangle &= \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} -\end{align*} +\begin{equation}\label{eq:magnetization_spin_first} + \langle |m| \rangle = \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} +\end{equation} % -\begin{align*} - \langle E \rangle &= \frac{-8 J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} -\end{align*} +\begin{equation}\label{eq:magnetization_spin_second} + \langle |m|^{2} \rangle = \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} +\end{equation} % -\begin{align*} - \langle E \rangle^{2} &= \frac{64 J^{2} \sinh(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} -\end{align*} +\begin{equation}\label{eq:energy_total_first} + \langle E \rangle = \frac{-8 J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} +\end{equation} % -\begin{align*} - \langle E^{2} \rangle &= \frac{64 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} -\end{align*} +\begin{equation}\label{eq:energy_total_first_squared} + \langle E \rangle^{2} = \frac{64 J^{2} \sinh(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} +\end{equation} % -\begin{align*} - \langle M \rangle &= \frac{2 e^{8 \beta J}}{\cosh(8 \beta J) + 3} -\end{align*} +\begin{equation}\label{eq:energy_total_second} + \langle E^{2} \rangle = \frac{64 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} +\end{equation} % -\begin{align*} - \langle M \rangle^{2} &= \frac{4e^{16 \beta J} + 16e^{8 \beta J} + 16}{(\cosh(8 \beta J) + 3)^{2}} -\end{align*} +\begin{equation}\label{eq:magnetization_total_first} + \langle |M| \rangle = \frac{2 e^{8 \beta J}}{\cosh(8 \beta J) + 3} +\end{equation} % -\begin{align*} - \langle M^{2} \rangle &= \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3} -\end{align*} +\begin{equation}\label{eq:magnetization_total_first_squared} + \langle |M| \rangle^{2} = \frac{4e^{16 \beta J} + 16e^{8 \beta J} + 16}{(\cosh(8 \beta J) + 3)^{2}} +\end{equation} % -\begin{align*} - C_{V} &= \frac{64 J^{2}}{N k_{\text{B}} T^{2}} \Big( \frac{3 \cosh(8 \beta J) + \cosh^{2}(8 \beta J) - \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \Big) -\end{align*} +\begin{equation}\label{eq:magnetization_total_second} + \langle M^{2} \rangle = \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3} +\end{equation} % -\begin{align*} - \chi &= \frac{1}{N} \frac{1}{k_{\text{B}} T^{2}} (\langle M^{2} \rangle - \langle M \rangle^{2}) \\ - &= \frac{1}{N k_{\text{B}} T} \Big( \frac{12e^{8 \beta J} + 4 e^{-8 \beta J} + 12}{(\cosh(8 \beta J) + 3)^{2}} \Big) -\end{align*} -The derivation of analytical expressions can be found in appendix -\ref{sec:analytical_expressions} +\begin{equation}\label{eq:specific_heat_capacity} + C_{V} = \frac{64 J^{2}}{N k_{\text{B}} T^{2}} \Big( \frac{3 \cosh(8 \beta J) + \cosh^{2}(8 \beta J) - \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \Big) +\end{equation} +% +\begin{equation}\label{eq:sesceptibility} + \chi = \frac{1}{N k_{\text{B}} T} \Big( \frac{12e^{8 \beta J} + 4 e^{-8 \beta J} + 12}{(\cosh(8 \beta J) + 3)^{2}} \Big) +\end{equation} +The derivation of analytical expressions can be found in appendix \ref{sec:analytical_expressions}. \subsection{Markov Chain Monte Carlo methods}\label{sec:mcmc_methods} From 7d402f4ded692201a0d1284ff058172927885404 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Wed, 15 Nov 2023 15:37:02 +0100 Subject: [PATCH 11/25] Edit analytical expressions and add calculation --- latex/ising_model.tex | 9 +- latex/references.bib | 17 +++ latex/sections/appendices.tex | 208 +++++++++++++++++++++++++++++++++- latex/sections/methods.tex | 84 ++++++++++++-- 4 files changed, 301 insertions(+), 17 deletions(-) diff --git a/latex/ising_model.tex b/latex/ising_model.tex index be6820d..82b7fbc 100644 --- a/latex/ising_model.tex +++ b/latex/ising_model.tex @@ -10,6 +10,9 @@ % Allows special characters (including æøå) \usepackage[utf8]{inputenc} \usepackage{fontawesome} +\usepackage{multirow} +\usepackage[table]{xcolor} +% \rowcolors{2}{gray!15}{white} % \usepackage{tabularx} % \usepackage[english]{babel} @@ -23,12 +26,12 @@ \usepackage{xcolor} % set colors \usepackage{hyperref} % automagic cross-referencing \usepackage{listings} % display code -\usepackage{subfigure} % imports a lot of cool and useful figure commands +% \usepackage{subfigure} % imports a lot of cool and useful figure commands +\usepackage{subcaption} % \usepackage{float} %\usepackage[section]{placeins} \usepackage{algorithm} \usepackage[noend]{algpseudocode} -\usepackage{subfigure} \usepackage{tikz} % \usetikzlibrary{quantikz} % defines the color of hyperref objects @@ -73,6 +76,8 @@ % Conclusion \subfile{sections/conclusion} +\clearpage +\newpage % Appendix \subfile{sections/appendices} diff --git a/latex/references.bib b/latex/references.bib index 2513567..391436b 100644 --- a/latex/references.bib +++ b/latex/references.bib @@ -67,3 +67,20 @@ } # Miscellaneous things +@misc{obermeyer:2020:ising, + author = {Johannes Obermeyer}, + title = {The Ising Model in One and Two Dimensions}, + year = {2020}, + url = {https://www.thphys.uni-heidelberg.de/~wolschin/statsem20_3s.pdf}, + urldate = {2023-11-13}, + pages = {16}, + note = {Seminar on Statistical Physics at the University of Heidelberg, summary} +} + +@book{springer:2012:modernstat, + author = {Jay L. Devore and Kenneth N. Berk}, + title = {Modern Mathematical Statistics with Application}, + publisher = {Cham: Springer International Publishing AG}, + year = {2021}, + edition = {3} +} \ No newline at end of file diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex index 3525403..4719731 100644 --- a/latex/sections/appendices.tex +++ b/latex/sections/appendices.tex @@ -3,6 +3,24 @@ \begin{document} \appendix \section{Ising model system states}\label{sec:system_states} +Units +\begin{table}[H] + \centering + \begin{tabular}[c]{cc} % @{\extracolsep{\fill}} + \hline + Value & Unit \\ + \hline + $[ E ]$ & $J$ \\ + $[ T ]$ & $J / k_{\text{B}}$ \\ + $[ M ]$ & $\dots$ \\ + $[ C_{\text{V}} ]$ & $k_{\text{B}}$ \\ + $[ \chi ]$ & $1 / J$ \\ + \hline + \end{tabular} + \caption{Units, given by the coupling constant $J$ and the Boltzmann constant $k_{\text{B}}$.} + \label{tab:units} +\end{table} + To avoid counting duplicates, we used \begin{figure}\label{fig:tikz_counting} \centering @@ -30,14 +48,196 @@ To avoid counting duplicates, we used \caption{Rules for multiplying spin pairs.} \end{figure} +\begin{figure}\label{fig:tikz_neighbor} + \centering + \begin{subfigure}{0.4\linewidth} + \begin{tikzpicture} + \draw (0, 0) grid (2, 2); + \node (s1) at (0.5, 1.5) {$\uparrow$}; + \node (s2) at (1.5, 1.5) {$\uparrow$}; + \node (s3) at (0.5, 0.5) {$\downarrow$}; + \node (s4) at (1.5, 0.5) {$\downarrow$}; + \end{tikzpicture} + \caption{} + \label{fig:sub_tikz_neighbor_a} + \end{subfigure} + \ + \begin{subfigure}{0.4\linewidth} + \begin{tikzpicture} + \draw (0, 0) grid (2, 2); + \node (s1) at (0.5, 1.5) {$\uparrow$}; + \node (s2) at (1.5, 1.5) {$\downarrow$}; + \node (s3) at (0.5, 0.5) {$\downarrow$}; + \node (s4) at (1.5, 0.5) {$\uparrow$}; + \end{tikzpicture} + \caption{} + \label{fig:sub_tikz_neighbor_b} + \end{subfigure} + \caption{Possible spin configurations for two spins up.} +\end{figure} + \section{Analytical expressions}\label{sec:analytical_expressions} -The Boltzmann distribution is normalized using a partition function $Z$ given by +The Boltzmann distribution is normalized using a partition function $Z$, given by +\begin{equation}\label{eq:partition} + Z = \sum_{\text{all possible } \mathbf{s}}^{N} +\end{equation} +We use the values estimated for the $2 \times 2$ case, found in \ref{tab:lattice_config}, +and find \begin{align*} - Z &= \sum_{\text{all possible } s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\ - &= \dots \\ - &= 4 \cosh (8 \beta J) + 12 \\ + 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)} \\ + &= 2e^{8 \beta J} + 2e^{-8 \beta J} + 12. +\end{align*} +We rewrite the expression using $\cosh(8 \beta J) = 1/2 \big( e^{8 \beta J} + e^{-8 \beta J})$, and get +\begin{align*} + z &= 4 \cosh (8 \beta J) + 12 +\end{align*} +The Boltzmann distribution is given by +\begin{equation}\label{eq:boltzmann} + p(\mathbf{s} \ | \ T) = \frac{1}{Z} e^{-\beta E(\mathbf{s})}, +\end{equation} +for a given temperature $T$. With our expression for the partition function, we +get the probability distribution +\begin{align*} + p(\mathbf{s} \ | \ T) &= \frac{1}{4 \cosh (8 \beta J) + 12} e^{-\beta E(\mathbf{s})} \end{align*} +For discrete random variables $X$, with a known probability distribution, the +expected value of $x$ is given by +\begin{align*} + \mathbb{E}(x) &= \sum_{x \in D} x \cdot p(x) & \text{\cite[p. 127]{springer:2012:modernstat}}. +\end{align*} +For a function of a stochastic random variable, the expected value of $x$ is +\begin{align*} + \mathbb{E}(h(X)) &= \sum_{x \in D} h(x) \cdot p(x) +\end{align*} +And in the case of a linear function we have +\begin{align*} + \mathbb{E}(aX + b) &= a \cdot \mathbb{E}(X) + 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 $E(\mathbf{s})$. In addition, we use +$\langle E \rangle$ as notation for expexted value of a given variable, since $\mathbf{s}$ is the stochastic +random variable. +The expression for total energy and total magnetization is given in eq. \eqref{eq:energy} and \eqref{eq:magnetization}. +The expected values for these is given by +\begin{equation*} + \langle E(\mathbf{s}) \rangle = \sum_{i=1}^{N} E(s_{i}) p(s_{i} \ | \ T) +\end{equation*} + +\begin{equation*} + \langle |M(\mathbf{s})| \rangle = \sum_{i=1}^{N} |M(s_{i})| p(s_{i} \ | \ T) +\end{equation*} + +Since we want to compare expected values for different lattice sizes, we have to +find the expected values per spin. We normalize the total expressions for total +energy \eqref{eq:energy} and magnetizaation \eqref{eq:magnetization} by the +number of spins to get +\begin{equation}\label{eq:spin_energy} + \epsilon(\mathbf{s}) = \frac{E(\mathbf{s})}{N} +\end{equation} + +\begin{equation}\label{eq:spin_magnetization} + m(\mathbf{s}) = \frac{M(\mathbf{s})}{N} +\end{equation} + +Both energy per spin and magnetization per spin are functions of $\mathbf{s}$, +we will use the short hand notation $\langle \epsilon \rangle$ and $\langle |m| \rangle$. +In addition, the number of spins are given as a constant for each lattice. We can +rewrite and use this when we find the expectation values per spin, for energy per spin +\begin{align*} + \langle \epsilon \rangle &= \sum_{i=1}^{N} \epsilon(s_{i}) p(s_{i} \ | \ T) \\ + &= \sum_{i=1}^{N} \frac{E(\mathbf{s})}{N} p(s_{i} \ | \ T) \\ + &= \frac{1}{N} \sum_{i=1}^{N} E(\mathbf{s}) p(s_{i} \ | \ T) +\end{align*} +The same applies for 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 +\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} \\ + &= \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}, +\end{align*} +and +\begin{align*} + \langle |M| \rangle &= 4 \cdot \frac{1}{Z} \cdot e^{8 \beta J} + 4 \cdot 2 \cdot \frac{1}{Z} \cdot e^{0} \\ + & \quad + 4 \cdot | -2| \cdot \frac{1}{Z} \cdot e^{0} + | -4| \cdot e^{8 \beta J} \\ + &= \frac{8 e^{8 \beta J} + 16}{Z} \\ + &= \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 +\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} \\ + &= \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}, +\end{align*} +and +\begin{align*} + \langle M^{2} \rangle &= 4^{2} \cdot \frac{1}{Z} \cdot e^{8 \beta J} + 4 \cdot 2^{2} \cdot \frac{1}{Z} \cdot e^{0} \\ + & \quad + 4 \cdot (-2)^{2} \cdot \frac{1}{Z} \cdot e^{0} + (-4)^{2}\cdot e^{8 \beta J} \\ + &= \frac{32e^{8 \beta J} + 32}{Z} \\ + &= \frac{4 (8e^{8 \beta J} + 8)}{4(\cosh(8 \beta J) + 3)} \\ + &= \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3} +\end{align*} +The squared expectation value is given by +\begin{align*} + \langle E \rangle^{2} &= \bigg(-\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} \bigg)^{2} \\ + &= \frac{64J^{2} \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}}, +\end{align*} +and +\begin{align*} + \langle |M| \rangle^{2} &= \Big( \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} \Big)^{2} \\ + &= \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} +\end{align*} +Calculating the heat capacity and susceptibility, we need the variance of both total +energy and total magnetizaation. We obtain this using the definition +\begin{align*} + \mathbb{V}(X) &= \sum_{x \in D} [(x - \mathbb{E}(x))^2 \cdot p(x)] & \text{\cite[p. 132]{springer:2012:modernstat}}. \\ + &= \mathbb{E}(X^{2}) - [\mathbb{E}(X)]^{2} +\end{align*} +The variance of total energy is given by +\begin{align*} + \mathbb{V}(E) &= \mathbb{E}(E^{2}) - [\mathbb{E}(E)]^{2} \\ + &= \frac{64J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} - \frac{64J^{2} \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \\ + &= 64J^{2} \bigg( \frac{\cosh(8 \beta J)}{\cosh(8 \beta J) + 3} - \frac{\sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \bigg) \\ + &= 64J^{2} \bigg( \frac{(\cosh(8 \beta J)) \cdot (\cosh(8 \beta J) + 3)}{(\cosh(8 \beta J) + 3)^{2}} \\ + & \quad - \frac{\sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \bigg) \\ + &= 64J^{2} \bigg( \frac{\cosh^{2}(8 \beta J) + 3\cosh(8 \beta J) - \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \bigg) \\ + &= 64J^{2} \bigg( \frac{3\cosh(8 \beta J) + 1}{(\cosh(8 \beta J) + 3)^{2}} \bigg) +\end{align*} +\begin{align*} + \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(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}} +\end{align*} + +\begin{align*} + C_{\text{V}} &= \frac{1}{N} \frac{1}{k_{\text{B} T^{2}}} (\mathbb{E}(E^{2}) - [\mathbb{E}(E)]^{2}) \\ + &= \frac{1}{N k_{\text{B} T^{2}}} \mathbb{V}(E) \\ + &= \frac{64J^{2} }{N k_{\text{B}} T^{2}} \bigg( \frac{3\cosh(8 \beta J) + 1}{(\cosh(8 \beta J) + 3)^{2}} \bigg) +\end{align*} + +\begin{align*} + \chi &= \frac{1}{N} \frac{1}{k_{\text{B} T}} (\mathbb{E}(M^{2}) - [\mathbb{E}(M)]^{2}) \\ + &= \frac{1}{N k_{\text{B} T}} \mathbb{V}(M) \\ + &= \frac{4}{N k_{\text{B} T}} \bigg( \frac{3e^{8 \beta J} + e^{-8 \beta J} + 3}{(\cosh(8 \beta J) + 3)^{2}} \bigg) +\end{align*} +\begin{align*} + \langle \epsilon^{2} \rangle &= \frac{1}{N^{2}} \sum_{i=1}^{N} E(\mathbf{s})^{2} p(s_{i} \ | \ T) \\ + &= +\end{align*} +The same applies for magnetization per spin +\begin{align*} + \langle |m| \rangle = \frac{1}{N} \sum_{i=1}^{N} |M(s_{i})| p(s_{i} \ | \ T) +\end{align*} \end{document} \ No newline at end of file diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 9376258..690cefe 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -1,13 +1,75 @@ -\documentclass[../ising_model.tex]{subfiles} +\documentclass[../ising_model.tex]{subfiles} -\begin{document} +\begin{document} \section{Methods}\label{sec:methods} \subsection{The Ising model}\label{sec:ising_model} % Add definitions +The Ising model correspond to a lattice with a number of spin sites, where +each spin has the possible state up $\uparrow$ or down $\downarrow$. We will study the +two-dimensional Ising model for ferromagnets, however, the model is not resticted +to this dimentionality~\cite[p. 3]{obermeyer:2020:ising}. We will assume a square lattice, where the length $L$ of +the lattice give the number of spin sites $N = L^{2}$. When we +consider the entire 2D lattice we can study the system spin configuration, or the +microstate, given by $\mathbf{s} = (s_{1}, s_{2}, \dots, s_{N})$. The value of +each spin $s_{i}$, where $i \in [1, N]$, is given by the direction of the spin +$\uparrow = +1$ and $\downarrow = -1$. + +We find the total energy of the system using +\begin{equation}\label{eq:energy} + E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} +\end{equation} +where $\langle k l \rangle$ denotes the neighboring spins. + +The total magnetization of the system is given by +\begin{equation}\label{eq:magnetization} + M(\mathbf{s}) = \sum_{i}^{N} s_{i}, +\end{equation} +with degeneracy denoting how many states have the same values. We will consider +a 2D lattice with $L = 2$, with periodic boundary conditions, following the +counting pattern shown in figure~\ref{fig:tikz_counting}. The resulting values for +the $2 \times 2$ lattice can be found in table \ref{tab:lattice_config}. + +\begin{table}[H] + \centering + \begin{tabular}{cccc} % @{\extracolsep{\fill}} + \hline + Spins up & $E(\mathbf{s})$ & $M(\mathbf{s})$ & Degeneracy \\ + \hline + 4 & -8J & 4 & 1 \\ + 3 & 0 & 2 & 4 \\ + \multirow{2}*{2} & 0 & 0 & 4 \\ + & 8J & 0 & 2 \\ + 1 & 0 & -2 & 4 \\ + 0 & -8J & -4 & 1 \\ + \hline + \end{tabular} + \caption{Values} + \label{tab:lattice_config} +\end{table} + +Probability stuff, for a system with a given temperature $T$. We use the Boltzmann +distribution in the analytical expression, to compare with our results. This is +a probability distribution from the family of exponential +\begin{equation}\label{eq:boltzmann_distribution} + p(\mathbf{s} \ | \ T) = \frac{1}{Z} \exp^{-\beta E(\mathbf{s})}, +\end{equation} +where $Z$ is the partition function $\beta$ is the related to the inverse of the +temperature as +\begin{equation}\label{eq:beta} + \beta = \frac{1}{k_{\text{B}} T}. +\end{equation} +We derive an analytical expression for $Z$ in appendix \ref{sec:analytical_expressions} We will assume a $2 \times 2$ lattice to study the possible system states, and to find analytical expressions necessary in the Markov Chain Monte Carlo method. These results are used to test our code during implementation. + +We study the 2D lattice and find the total energy and the total magnetization of +of the system, in addition to the degeneracy. In addition, we want to work with +unitless spins. To obtain this, we introduce the base unit for energy $J$, and +with the Boltzmann constant we derive other units necessary. The resulting units +can be found in table \ref{tab:units}. + % Problem 1 \begin{equation}\label{eq:partition_function} @@ -31,42 +93,42 @@ These results are used to test our code during implementation. \end{equation} % \begin{equation}\label{eq:energy_total_first} - \langle E \rangle = \frac{-8 J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} + \langle E \rangle = -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} \end{equation} % \begin{equation}\label{eq:energy_total_first_squared} - \langle E \rangle^{2} = \frac{64 J^{2} \sinh(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} + \langle E \rangle^{2} = \frac{64J^{2} \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \end{equation} % \begin{equation}\label{eq:energy_total_second} - \langle E^{2} \rangle = \frac{64 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} + \langle E^{2} \rangle = \frac{64J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} \end{equation} % \begin{equation}\label{eq:magnetization_total_first} - \langle |M| \rangle = \frac{2 e^{8 \beta J}}{\cosh(8 \beta J) + 3} + \langle |M| \rangle = \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} \end{equation} % \begin{equation}\label{eq:magnetization_total_first_squared} - \langle |M| \rangle^{2} = \frac{4e^{16 \beta J} + 16e^{8 \beta J} + 16}{(\cosh(8 \beta J) + 3)^{2}} + \langle |M| \rangle^{2} = \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \end{equation} % \begin{equation}\label{eq:magnetization_total_second} - \langle M^{2} \rangle = \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3} + \langle M^{2} \rangle = frac{8(e^{8 \beta J} + 1)}{\cosh(8 \beta J) + 3} \end{equation} % \begin{equation}\label{eq:specific_heat_capacity} - C_{V} = \frac{64 J^{2}}{N k_{\text{B}} T^{2}} \Big( \frac{3 \cosh(8 \beta J) + \cosh^{2}(8 \beta J) - \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \Big) + C_{V} = \frac{64J^{2} }{N k_{\text{B}} T^{2}} \frac{(3\cosh(8 \beta J) + 1)}{(\cosh(8 \beta J) + 3)^{2}} \end{equation} % \begin{equation}\label{eq:sesceptibility} - \chi = \frac{1}{N k_{\text{B}} T} \Big( \frac{12e^{8 \beta J} + 4 e^{-8 \beta J} + 12}{(\cosh(8 \beta J) + 3)^{2}} \Big) + \chi = \frac{4}{N k_{\text{B} T}} \frac{(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}} \end{equation} The derivation of analytical expressions can be found in appendix \ref{sec:analytical_expressions}. \subsection{Markov Chain Monte Carlo methods}\label{sec:mcmc_methods} -\subsection{Implementation}\label{sec:implementation} +\subsection{Algorithm and implementation}\label{sec:algo_implementation} \subsection{Tools}\label{sec:tools} From 9c06592b90685c16a889bd49a703a52225a266e4 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Fri, 17 Nov 2023 12:17:25 +0100 Subject: [PATCH 12/25] latex/sections/methods.tex --- latex/sections/methods.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 690cefe..a9fc293 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -113,7 +113,7 @@ can be found in table \ref{tab:units}. \end{equation} % \begin{equation}\label{eq:magnetization_total_second} - \langle M^{2} \rangle = frac{8(e^{8 \beta J} + 1)}{\cosh(8 \beta J) + 3} + \langle M^{2} \rangle = \frac{8(e^{8 \beta J} + 1)}{\cosh(8 \beta J) + 3} \end{equation} % \begin{equation}\label{eq:specific_heat_capacity} @@ -121,7 +121,7 @@ can be found in table \ref{tab:units}. \end{equation} % \begin{equation}\label{eq:sesceptibility} - \chi = \frac{4}{N k_{\text{B} T}} \frac{(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}} + \chi = \frac{4}{N k_{\text{B}} T} \frac{(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}} \end{equation} The derivation of analytical expressions can be found in appendix \ref{sec:analytical_expressions}. From 7afb51357ea425079e0c179b9106d48a20d21f08 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Wed, 22 Nov 2023 11:46:59 +0100 Subject: [PATCH 13/25] Add notes from draft into latex files, needs sorting and rewriting! --- latex/references.bib | 138 ++++++++++++++++++-------------- latex/sections/introduction.tex | 20 ++++- latex/sections/methods.tex | 30 +++++++ latex/sections/results.tex | 22 +++++ 4 files changed, 148 insertions(+), 62 deletions(-) diff --git a/latex/references.bib b/latex/references.bib index 391436b..2c406fb 100644 --- a/latex/references.bib +++ b/latex/references.bib @@ -1,86 +1,102 @@ # Ising model related literature +@misc{britannica:2023:ferromagnetism, + author = {Britannica, The Editors of Encyclopaedia}, + title = {ferromagnetism}, + url = {https://www.britannica.com/science/ferromagnetism}, + urldate = {2023-11-21} +} -# Math related literature +@article{onsager:1944, + author = {Lars Onsager}, + title = {Crystal statistics. I. A two-dimensional model with an order-disorder transition}, + journal = {Physical review}, + year = {1944}, + volume = {65}, + number = {3--4}, + pages = {117--149} +} +@misc{obermeyer:2020:ising, + author = {Johannes Obermeyer}, + title = {The Ising Model in One and Two Dimensions}, + year = {2020}, + url = {https://www.thphys.uni-heidelberg.de/~wolschin/statsem20_3s.pdf}, + urldate = {2023-11-13}, + pages = {16}, + note = {Seminar on Statistical Physics at the University of Heidelberg, summary} +} + +# Math and statistics @inbook{lindstrom:2016:ch3, - author = {Tom Lindstrøm}, - title = {Kalkulus}, - publisher = {Universitetsforlaget}, - year = {2016}, - edition = {4}, - pages = {133--137} + author = {Tom Lindstrøm}, + title = {Kalkulus}, + publisher = {Universitetsforlaget}, + year = {2016}, + edition = {4}, + pages = {133--137} } @inbook{lindstrom:2016:ch10:5, - author = {Tom Lindstrøm}, - title = {Kalkulus}, - publisher = {Universitetsforlaget}, - year = {2016}, - edition = {4}, - pages = {589--606} + author = {Tom Lindstrøm}, + title = {Kalkulus}, + publisher = {Universitetsforlaget}, + year = {2016}, + edition = {4}, + pages = {589--606} } @inbook{gjevik:2014:appendix, - author = {Bjørn Gjevik and Morten Wang Fagerland}, - title = {Feltteori og vektoranalyse}, - publisher = {Farleia Forlag}, - year = {2018}, - pages = {162--163} + author = {Bjørn Gjevik and Morten Wang Fagerland}, + title = {Feltteori og vektoranalyse}, + publisher = {Farleia Forlag}, + year = {2018}, + pages = {162--163} +} + +@book{springer:2012:modernstat, + author = {Jay L. Devore and Kenneth N. Berk}, + title = {Modern Mathematical Statistics with Application}, + publisher = {Cham: Springer International Publishing AG}, + year = {2021}, + edition = {3} } # Python libraries @article{hunter:2007:matplotlib, - author = {Hunter, J. D.}, - title = {Matplotlib: A 2D graphics environment}, - journal = {Computing in Science \& Engineering}, - volume = {9}, - number = {3}, - pages = {90--95}, - abstract = {Matplotlib is a 2D graphics package used for Python for - application development, interactive scripting, and publication-quality - image generation across user interfaces and operating systems.}, + author = {Hunter, J. D.}, + title = {Matplotlib: A 2D graphics environment}, + journal = {Computing in Science \& Engineering}, + volume = {9}, + number = {3}, + pages = {90--95}, + abstract = {Matplotlib is a 2D graphics package used for Python for + application development, interactive scripting, and publication-quality + image generation across user interfaces and operating systems.}, publisher = {IEEE COMPUTER SOC}, - doi = {10.1109/MCSE.2007.55}, - year = 2007 + doi = {10.1109/MCSE.2007.55}, + year = 2007 } @article{waskom:2021:seaborn, - author = {Michael L. Waskom}, - title = {seaborn: statistical data visualization}, - journal = {Journal of Open Source Software}, - volume = {6}, - number = {60}, - publisher = {The Open Journal}, - year = {2021}, - doi = {10.21105/joss.03021}, - url = {https://doi.org/10.21105/joss.03021}, - pages = {3021} + author = {Michael L. Waskom}, + title = {seaborn: statistical data visualization}, + journal = {Journal of Open Source Software}, + volume = {6}, + number = {60}, + publisher = {The Open Journal}, + year = {2021}, + doi = {10.21105/joss.03021}, + url = {https://doi.org/10.21105/joss.03021}, + pages = {3021} } # C++ libraries @misc{openmp:2018, - author = {OpenMP}, - title = {OpenMP Application Programming Interface}, - year = {2018}, - url = {https://www.openmp.org/wp-content/uploads/OpenMP-API-Specification-5.0.pdf}, - urldate = {2023-10-22} + author = {OpenMP}, + title = {OpenMP Application Programming Interface}, + year = {2018}, + url = {https://www.openmp.org/wp-content/uploads/OpenMP-API-Specification-5.0.pdf}, + urldate = {2023-10-22} } # Miscellaneous things -@misc{obermeyer:2020:ising, - author = {Johannes Obermeyer}, - title = {The Ising Model in One and Two Dimensions}, - year = {2020}, - url = {https://www.thphys.uni-heidelberg.de/~wolschin/statsem20_3s.pdf}, - urldate = {2023-11-13}, - pages = {16}, - note = {Seminar on Statistical Physics at the University of Heidelberg, summary} -} - -@book{springer:2012:modernstat, - author = {Jay L. Devore and Kenneth N. Berk}, - title = {Modern Mathematical Statistics with Application}, - publisher = {Cham: Springer International Publishing AG}, - year = {2021}, - edition = {3} -} \ No newline at end of file diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index 62d86ae..cea1be1 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -2,6 +2,24 @@ \begin{document} \section{Introduction}\label{sec:introduction} -\cite{openmp:2018} +We can study individual atoms +At a microscopic level we can study individual atoms, using microscopic theory. When a system consist of several atoms, we can study the atoms physical properties of matter using statistical physics. To determine quantities such as heat capacity and susceptibility, we have to use ... + +Some uncharged materials have the ability to acquire attractive powers, that is the atomic spins align as well as the magnetic moment. Neighboring spins interact, and the spins magnetic moment can influence each other. This physical phenomenon is called ferromagnetism \cite{britannica:2023:ferromagnetism}. + +The Ising model is a mathematical description of a physical system, of a ferromagnet. At a certain teperature, the critical point, the behavior of the ferromagnet changes. We want to study the phase transition of a ferromagnet, and find the critical temperature. + +However, as the lattice size increase, so does the number of calculations. Since computational resources are finite, we use parallelization to decrease runtime and required resources. In parallelizing our implementation, we can approximate the critical temperature of a material. + +In Section \ref{sec:methods}, we will state the theoretical background for this experiment, as well as the implementation and tools used. Continuing with the results in Section \ref{sec:results}, in addition to a discussion of our results. Lastly, we will conclude our findings in Section \ref{sec:conclusion}. + +\subsection{Draft} +Study physical processes, phase transition and their behavior near critical points. That is, study how a magnetic material changes properties when exposed to thermal energy. The Ising model was originally a mathematical description meant to simulate ferromagnets. Has found many uses like hemoglobin in absorbing oxygen. + +Ferromagnets exhibit long-range ordering phenomenon at atomic level, line up spins with each other by locking the magnetic moments. Thermal agitation randomizes. Critical temperature of iron is 1043 K. Ising model simply describes how a magnetic material responds to thermal energy and an external magnetic field. The direction of the spins influences the total potential energy. + +Def: Ensemble is a collection of microphysics systems from which we derive expectation values and thermodynamic properties related to experiment. + +Statistical physics bridge the gap between microscopic and macroscopic world, using microscopic theory we can derive thermodynamic quantities. \end{document} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index a9fc293..9db3ea5 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -137,6 +137,36 @@ The Ising model and MCMC methods are implemented in C++, and parallelized using \cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. +\subsection{Draft} +Def: Correlation length $\xi$ of phase transition, and is related to the spin-correlation function which defines the magnetic susceptibility. The spin-correlation function is the covariance of the magnetization, expressing the degree of correlation between spins. $\xi$ is normally finite at critical points. Proportional to the lattice size at the critical point, can be used to relate behavior for infinitly large lattice $T_{C}(L) - T_{C}(L = \infty) = \propto \alpha T^{-1}$ + +Implementation: The metropolis algo only consider ratio, not necessary to calculate partition function. + +Metropolis algo: +\begin{algorithm}[H] + \caption{Metropolis algorithm p. 435 in lecture notes} + \label{algo:metropolis_algo} + \begin{algorithmic} + \Procedure{Metropolis algorithm}{$args$} + \State $E_{\text{before}} \leftarrow \text{Set initial random state}$ + \State $ \text{Flip spin}$ + \State $E_{\text{after}} \text{Calculate energy}$ + \State $\Delta E \leftarrow E_{\text{after}} - E_{\text{before}}$ + \If{$\Delta E \leq 0$} Accept new config if + \State $\text{Update expectation values}$ + \ElsIf{$\Delta E > 0$} + \State $\text{Calculate}$ + \EndIf + \EndProcedure + \end{algorithmic} +\end{algorithm} + +Probability: Using the expectation value require a stochastic random variable X (stoch being that it is tied to an event), a function from the "utfallsrommet" to the real numbers. + +Understanding a systems properties close to a critical point. For the 2D ising model, where L is large, we can observe discontinuity in $C_{V}$ and $\chi$. They diverge at the critical point in the thermosynamic limit, meaning the variance in energy and magnetization diverge. For lattice of finite size the variance scale as $1 / \sqrt{M}$ where $M = 2^{N}$ is the number of microstates. +\begin{align*} + e^{-\beta \Delta E} &= \frac{p_{after}}{p_{before}} = \frac{e^{-\beta E_{after}}}{e^{-\beta E_{before}}} +\end{align*} \end{document} diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 26924a0..4d1836a 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -3,4 +3,26 @@ \begin{document} \section{Results}\label{sec:results} +\subsection{Draft} +Test that the numerical stuff gets close to the analytical +- validate implementation a given number of times, find average number of cycles + +Initial state ordered vs random +- start ordered both at up=1 and random + +burn-in time +- burn-in 5000 vs 10000 mc cycles, only mention how a longer burn-in time can in some cases result in better cv and x + +probability using histogram of samples +- Comparing the histogram of $T_1 = 1.0$ and $T_2 = 2.4$. For $T_1 = 1.0$ we see a lower acceptance of flips, resulting in a low variance. The histogram looks more like an exponential function + +parallelization +- temp mpi +- each mpi process spawn thread that make own ising model +- each ising model goes through burn in time (no benefit) +- sampling done in parallel +- mcmc sampling openmp + +phase transition +critical temperature \end{document} From cd1820b1a44f26132eacf19e6acb03e2615b49a6 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Sat, 25 Nov 2023 13:48:13 +0100 Subject: [PATCH 14/25] Add separate plot for each temperature --- src/scripts/burn_in_time.py | 59 ++++++++++++++++++++++++++----------- 1 file changed, 42 insertions(+), 17 deletions(-) diff --git a/src/scripts/burn_in_time.py b/src/scripts/burn_in_time.py index 517ee01..e4010ee 100644 --- a/src/scripts/burn_in_time.py +++ b/src/scripts/burn_in_time.py @@ -1,22 +1,43 @@ import matplotlib.pyplot as plt +import seaborn as sns + +sns.set_theme() +params = { + "font.family": "Serif", + "font.serif": "Roman", + "text.usetex": True, + "axes.titlesize": "large", + "axes.labelsize": "large", + "xtick.labelsize": "large", + "ytick.labelsize": "large", + "legend.fontsize": "medium" +} +plt.rcParams.update(params) def plot_from_file(): files = [ - "output/burn_in_time/unordered_1_0.txt", - "output/burn_in_time/ordered_1_0.txt", - "output/burn_in_time/unordered_2_4.txt", - "output/burn_in_time/ordered_2_4.txt", + # "output/burn_in_time/unordered_1_0_1421110368.txt", + # "output/burn_in_time/ordered_1_0_611577739.txt", + "output/burn_in_time/unordered_2_4_1212892317.txt", + "output/burn_in_time/ordered_2_4_2408603856.txt", ] labels = [ - "1.0, unordered", - "1.0, ordered", - "2.4, unordered", - "2.4, ordered" + "Unordered", + "Ordered", + # "2.4, unordered", + # "2.4, ordered" + ] + colors = [ + "seagreen", + "darkred", + # "darkgoldenrod", + # "steelblue" ] - figure1, ax1 = plt.subplots() - figure2, ax2 = plt.subplots() - for infile, label in zip(files, labels): + fig1, ax1 = plt.subplots() + fig2, ax2 = plt.subplots() + + for infile, label, color in zip(files, labels, colors): with open(infile) as f: lines = f.readlines() @@ -29,13 +50,17 @@ def plot_from_file(): energy.append(float(items[1])) magnetization.append(float(items[5])) - ax1.plot(t, energy, label=fr"$\langle \epsilon \rangle$ {label}") - ax2.plot(t, magnetization, label=fr"$\langle | m | \rangle$ {label}") + ax1.plot(t, energy, label=f"{label}", color=color) + ax2.plot(t, magnetization, label=f"{label}", color=color) - figure1.legend() - figure1.savefig("../latex/images/burn_in_time_energy.pdf") - figure2.legend() - figure2.savefig("../latex/images/burn_in_time_magnetization.pdf") + ax1.set_xlabel("t") + ax1.set_ylabel(r"$\langle \epsilon \rangle$") + ax1.legend(title="Initial state") + fig1.savefig("../latex/images/burn_in_time_energy_2_4.pdf", bbox_inches="tight") + ax2.set_ylabel(r"$\langle |m| \rangle$") + ax2.set_xlabel("t") + ax2.legend(title="Initial state") + fig2.savefig("../latex/images/burn_in_time_magnetization_2_4.pdf", bbox_inches="tight") 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4} % self-explanatory +\title{The Ising model} % self-explanatory \author{Cory Alexander Balaton \& Janita Ovidie Sandtrøen Willumsen \\ \faGithub \, \url{https://github.uio.no/FYS3150-G2-2023/Project-4}} % self-explanatory \date{\today} % self-explanatory \noaffiliation % ignore this, but keep it. diff --git a/latex/references.bib b/latex/references.bib index 2c406fb..63f9a7b 100644 --- a/latex/references.bib +++ b/latex/references.bib @@ -26,6 +26,13 @@ note = {Seminar on Statistical Physics at the University of Heidelberg, summary} } +@article{bialek:2008:ising, + author = {Gašper Tkačik and Elad Schneidman and Michal J. Berry II and William Bialek}, + title = {Ising models for network of real neurons}, + journal = {arXiv}, + year = {2008} +} + # Math and statistics @inbook{lindstrom:2016:ch3, author = {Tom Lindstrøm}, @@ -100,3 +107,17 @@ } # Miscellaneous things +@misc{tds:2021:mcmc, + author = {Shivam Agari}, + title = {Monte Carlo Markov Chain (MCMC), Explained}, + url = {https://towardsdatascience.com/monte-carlo-markov-chain-mcmc-explained-94e3a6c8de11}, + urldate = {2023-11-24} +} + +@misc{yale:2023:hdd, + author = {Ismail-Beigi Research Group}, + title = {Hard Drives Methods And Materials}, + url = {https://volga.eng.yale.edu/teaching-resources/hard-drives/methods-and-materials}, + urldate = {2023-11-25} +} + diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex index 4719731..e11e630 100644 --- a/latex/sections/appendices.tex +++ b/latex/sections/appendices.tex @@ -83,7 +83,7 @@ The Boltzmann distribution is normalized using a partition function $Z$, given b Z = \sum_{\text{all possible } \mathbf{s}}^{N} \end{equation} We use the values estimated for the $2 \times 2$ case, found in \ref{tab:lattice_config}, -and find +and 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)} \\ diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index cea1be1..6644418 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -2,24 +2,32 @@ \begin{document} \section{Introduction}\label{sec:introduction} -We can study individual atoms -At a microscopic level we can study individual atoms, using microscopic theory. When a system consist of several atoms, we can study the atoms physical properties of matter using statistical physics. To determine quantities such as heat capacity and susceptibility, we have to use ... +Magnetic materials are used for their particular property, in a wide range of technological devices. It is used in compasses for navigation, in computer hard drives to store data \cite{yale:2023:hdd}, and in MRI machines to produce detailed anatomical images. Some magnetic materials, called ferromagnets, are easily magnetized. When exposed to a strong magnetic field they become saturated. And heating the ferromagnet to a critical point, will cause it to lose its ferromagnetic property. -Some uncharged materials have the ability to acquire attractive powers, that is the atomic spins align as well as the magnetic moment. Neighboring spins interact, and the spins magnetic moment can influence each other. This physical phenomenon is called ferromagnetism \cite{britannica:2023:ferromagnetism}. +The Ising model is a mathematical description of a physical system. It was developed to describe ferromagnetism, and to understand the phase transition of a system. However, it has later been used to describe other phenomena where pairwise interactions are of interest, such as networks of neurons \cite{bialek:2008:ising}. -The Ising model is a mathematical description of a physical system, of a ferromagnet. At a certain teperature, the critical point, the behavior of the ferromagnet changes. We want to study the phase transition of a ferromagnet, and find the critical temperature. +We will use the Ising model to study ferromagnet and its temperature-dependent behavior. In addition, we will numerically estimate the critical point where the system experience phase transition. -However, as the lattice size increase, so does the number of calculations. Since computational resources are finite, we use parallelization to decrease runtime and required resources. In parallelizing our implementation, we can approximate the critical temperature of a material. +In Section \ref{sec:methods}, we will present the theoretical background for this experiment, as well as the implementation and tools used. Continuing with Section \ref{sec:results}, where we present our results and discuss our findings. Lastly, we will conclude our findings in Section \ref{sec:conclusion}. -In Section \ref{sec:methods}, we will state the theoretical background for this experiment, as well as the implementation and tools used. Continuing with the results in Section \ref{sec:results}, in addition to a discussion of our results. Lastly, we will conclude our findings in Section \ref{sec:conclusion}. +% \subsection{Draft} +% We can study individual atoms +% At a microscopic level we can study individual atoms, using microscopic theory. When a system consist of several atoms, we can study the atoms physical properties of matter using statistical physics. To determine quantities such as heat capacity and susceptibility, we have to use ... -\subsection{Draft} -Study physical processes, phase transition and their behavior near critical points. That is, study how a magnetic material changes properties when exposed to thermal energy. The Ising model was originally a mathematical description meant to simulate ferromagnets. Has found many uses like hemoglobin in absorbing oxygen. +% Some uncharged materials have the ability to acquire attractive powers, that is the atomic spins align as well as the magnetic moment. Neighboring spins interact, and the spins magnetic moment can influence each other. This physical phenomenon is called ferromagnetism \cite{britannica:2023:ferromagnetism}. -Ferromagnets exhibit long-range ordering phenomenon at atomic level, line up spins with each other by locking the magnetic moments. Thermal agitation randomizes. Critical temperature of iron is 1043 K. Ising model simply describes how a magnetic material responds to thermal energy and an external magnetic field. The direction of the spins influences the total potential energy. +% The Ising model is a mathematical description of a physical system, of a ferromagnet. At a certain teperature, the critical point, the behavior of the ferromagnet changes. We want to study the phase transition of a ferromagnet, and find the critical temperature. -Def: Ensemble is a collection of microphysics systems from which we derive expectation values and thermodynamic properties related to experiment. +% However, as the lattice size increase, so does the number of calculations. Since computational resources are finite, we use parallelization to decrease runtime and required resources. In parallelizing our implementation, we can approximate the critical temperature of a material. -Statistical physics bridge the gap between microscopic and macroscopic world, using microscopic theory we can derive thermodynamic quantities. +% In Section \ref{sec:methods}, we will state the theoretical background for this experiment, as well as the implementation and tools used. Continuing with the results in Section \ref{sec:results}, in addition to a discussion of our results. Lastly, we will conclude our findings in Section \ref{sec:conclusion}. + +% Study physical processes, phase transition and their behavior near critical points. That is, study how a magnetic material changes properties when exposed to thermal energy. The Ising model was originally a mathematical description meant to simulate ferromagnets. Has found many uses like hemoglobin in absorbing oxygen. + +% Ferromagnets exhibit long-range ordering phenomenon at atomic level, line up spins with each other by locking the magnetic moments. Thermal agitation randomizes. Critical temperature of iron is 1043 K. Ising model simply describes how a magnetic material responds to thermal energy and an external magnetic field. The direction of the spins influences the total potential energy. + +% Def: Ensemble is a collection of microphysics systems from which we derive expectation values and thermodynamic properties related to experiment. + +% Statistical physics bridge the gap between microscopic and macroscopic world, using microscopic theory we can derive thermodynamic quantities. \end{document} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 9db3ea5..31354a4 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -2,33 +2,46 @@ \begin{document} \section{Methods}\label{sec:methods} -\subsection{The Ising model}\label{sec:ising_model} -% Add definitions -The Ising model correspond to a lattice with a number of spin sites, where -each spin has the possible state up $\uparrow$ or down $\downarrow$. We will study the -two-dimensional Ising model for ferromagnets, however, the model is not resticted -to this dimentionality~\cite[p. 3]{obermeyer:2020:ising}. We will assume a square lattice, where the length $L$ of -the lattice give the number of spin sites $N = L^{2}$. When we -consider the entire 2D lattice we can study the system spin configuration, or the -microstate, given by $\mathbf{s} = (s_{1}, s_{2}, \dots, s_{N})$. The value of -each spin $s_{i}$, where $i \in [1, N]$, is given by the direction of the spin -$\uparrow = +1$ and $\downarrow = -1$. +\subsection{The Ising Model}\label{subsec:ising_model} +The Ising model in two dimensions consist of a lattice of spins, and can be though of as atoms in a grid. The length of the lattice is given by $L$, and the number of spins within a lattice is given by $N = L^{2}$. When we consider the entire lattice, the spin configuration, is given by $\mathbf{s} = [s_{1}, s_{2}, \dots, s_{N}]$. The total number of spin configurations, or microstates, is $2^{N}$. A given spin $i$ can take one of two possible discrete values $s_{i} \in \{-1, +1\}$, either spin down or spin up. -We find the total energy of the system using -\begin{equation}\label{eq:energy} - E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} +The spins interact with its nearest neighbors, resulting in a total system energy given by Eq. \eqref{eq:energy_total} +\begin{equation}\label{eq:energy_total} + E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} \ . \end{equation} -where $\langle k l \rangle$ denotes the neighboring spins. +$\langle k l \rangle$ denote a pair of spins, and $J$ is the coupling constant. In a two-dimensional lattice, a spin can have 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, such that all spins have exactly four nearest neighbors. We count the neighboring spins as visualized in Fig. \ref{fig:tikz_boundary}, also for lattices where $L > 2$. +\begin{figure} + \centering + \begin{tikzpicture} + \draw[help lines] (0, 0) grid (2, 2); + % \node[inner] (s1) at (0.5, 0.5) {s}; + \node (s1) at (0.5, 1.5) {$s_1$}; + \node (s2) at (1.5, 1.5) {$s_2$}; + \node (s3) at (0.5, 0.5) {$s_3$}; + \node (s4) at (1.5, 0.5) {$s_4$}; + \node[gray] (s12) at (2.5, 1.5) {$s_1$}; + \node[gray] (s34) at (2.5, 0.5) {$s_3$}; + \node[gray] (s13) at (0.5, -0.5) {$s_1$}; + \node[gray] (s24) at (1.5, -0.5) {$s_2$}; + + \draw[red, ->] (s1.east) -- (s2.west); + \draw[red, ->] (s2.east) -- (s12.west); + \draw[red, ->] (s1.south) -- (s3.north); + \draw[red, ->] (s2.south) -- (s4.north); + \draw[red, ->] (s3.east) -- (s4.west); + \draw[red, ->] (s4.east) -- (s34.west); + \draw[red, ->] (s3.south) -- (s13.north); + \draw[red, ->] (s4.south) -- (s24.north); + \end{tikzpicture} + \caption{Visualization of counting rules for pairs of spins in a $2 \times 2$ lattice, where periodic boundary conditions are applied.} + \label{fig:tikz_boundary} +\end{figure} -The total magnetization of the system is given by -\begin{equation}\label{eq:magnetization} - M(\mathbf{s}) = \sum_{i}^{N} s_{i}, +In addition, we consider the total magnetization of the system, which is given by Eq. \eqref{eq:magnetization_total} +\begin{equation}\label{eq:magnetization_total} + M(\mathbf{s}) = \sum_{i}^{N} s_{i} \ . \end{equation} -with degeneracy denoting how many states have the same values. We will consider -a 2D lattice with $L = 2$, with periodic boundary conditions, following the -counting pattern shown in figure~\ref{fig:tikz_counting}. The resulting values for -the $2 \times 2$ lattice can be found in table \ref{tab:lattice_config}. - +We also consider the state degeneracy, that is the number of different microstates with the same value of total magnetization. To find the analytical expressions necessary for validating our model implementation, we will assume a $2 \times 2$ lattice. Using the possible system states we find the values for the lattice, which can be found in table \ref{tab:lattice_config}. \begin{table}[H] \centering \begin{tabular}{cccc} % @{\extracolsep{\fill}} @@ -43,130 +56,276 @@ the $2 \times 2$ lattice can be found in table \ref{tab:lattice_config}. 0 & -8J & -4 & 1 \\ \hline \end{tabular} - \caption{Values} + \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.} \label{tab:lattice_config} \end{table} +We will use energy per spin given by Eq. \eqref{eq:energy_spin} and magnetization per spin given by Eq. \eqref{eq:magnetization_spin}, to compare the quantities for different lattice sizes. +\begin{align} + \epsilon (\mathbf{s}) &\eqiv \frac{E(\mathbf{s})}{N} \ , + \label{eq:energy_spin} \\ + m (\mathbf{s}) &\eqiv \frac{M(\mathbf{s})}{N} \ , + \label{eq:magnetization_spin} +\end{align} +for a $N \times N$ lattice. -Probability stuff, for a system with a given temperature $T$. We use the Boltzmann -distribution in the analytical expression, to compare with our results. This is -a probability distribution from the family of exponential -\begin{equation}\label{eq:boltzmann_distribution} +When we study the phase transitions of ferromagnets, we have to consider the probability of a microstate $\mathbf{s}$, at a fixed temperature $T$, is given by +\begin{equation} p(\mathbf{s} \ | \ T) = \frac{1}{Z} \exp^{-\beta E(\mathbf{s})}, + \label{eq:boltzmann_distribution} \end{equation} -where $Z$ is the partition function $\beta$ is the related to the inverse of the -temperature as -\begin{equation}\label{eq:beta} - \beta = \frac{1}{k_{\text{B}} T}. -\end{equation} -We derive an analytical expression for $Z$ in appendix \ref{sec:analytical_expressions} - -We will assume a $2 \times 2$ lattice to study the possible system states, and -to find analytical expressions necessary in the Markov Chain Monte Carlo method. -These results are used to test our code during implementation. - -We study the 2D lattice and find the total energy and the total magnetization of -of the system, in addition to the degeneracy. In addition, we want to work with -unitless spins. To obtain this, we introduce the base unit for energy $J$, and -with the Boltzmann constant we derive other units necessary. The resulting units -can be found in table \ref{tab:units}. - -% Problem 1 - -\begin{equation}\label{eq:partition_function} - Z = 4 \cosh (8 \beta J) + 12, -\end{equation} -% -\begin{equation}\label{eq:energy_spin_first} - \langle \epsilon \rangle = \frac{-2J \sinh(8 \beta J)}{ \cosh(8 \beta J) + 3} -\end{equation} -% -\begin{equation}\label{eq:energy_spin_second} - \langle \epsilon^{2} \rangle = \frac{4 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} -\end{equation} -% -\begin{equation}\label{eq:magnetization_spin_first} - \langle |m| \rangle = \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} -\end{equation} -% -\begin{equation}\label{eq:magnetization_spin_second} - \langle |m|^{2} \rangle = \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} -\end{equation} -% -\begin{equation}\label{eq:energy_total_first} - \langle E \rangle = -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} -\end{equation} -% -\begin{equation}\label{eq:energy_total_first_squared} - \langle E \rangle^{2} = \frac{64J^{2} \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} -\end{equation} -% -\begin{equation}\label{eq:energy_total_second} - \langle E^{2} \rangle = \frac{64J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} -\end{equation} -% -\begin{equation}\label{eq:magnetization_total_first} - \langle |M| \rangle = \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} -\end{equation} -% -\begin{equation}\label{eq:magnetization_total_first_squared} - \langle |M| \rangle^{2} = \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} -\end{equation} -% -\begin{equation}\label{eq:magnetization_total_second} - \langle M^{2} \rangle = \frac{8(e^{8 \beta J} + 1)}{\cosh(8 \beta J) + 3} -\end{equation} -% -\begin{equation}\label{eq:specific_heat_capacity} - C_{V} = \frac{64J^{2} }{N k_{\text{B}} T^{2}} \frac{(3\cosh(8 \beta J) + 1)}{(\cosh(8 \beta J) + 3)^{2}} -\end{equation} -% -\begin{equation}\label{eq:sesceptibility} - \chi = \frac{4}{N k_{\text{B}} T} \frac{(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}} -\end{equation} -The derivation of analytical expressions can be found in appendix \ref{sec:analytical_expressions}. - -\subsection{Markov Chain Monte Carlo methods}\label{sec:mcmc_methods} - - -\subsection{Algorithm and implementation}\label{sec:algo_implementation} - - -\subsection{Tools}\label{sec:tools} -The Ising model and MCMC methods are implemented in C++, and parallelized using -\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. - -\subsection{Draft} -Def: Correlation length $\xi$ of phase transition, and is related to the spin-correlation function which defines the magnetic susceptibility. The spin-correlation function is the covariance of the magnetization, expressing the degree of correlation between spins. $\xi$ is normally finite at critical points. Proportional to the lattice size at the critical point, can be used to relate behavior for infinitly large lattice $T_{C}(L) - T_{C}(L = \infty) = \propto \alpha T^{-1}$ - -Implementation: The metropolis algo only consider ratio, not necessary to calculate partition function. - -Metropolis algo: -\begin{algorithm}[H] - \caption{Metropolis algorithm p. 435 in lecture notes} - \label{algo:metropolis_algo} - \begin{algorithmic} - \Procedure{Metropolis algorithm}{$args$} - \State $E_{\text{before}} \leftarrow \text{Set initial random state}$ - \State $ \text{Flip spin}$ - \State $E_{\text{after}} \text{Calculate energy}$ - \State $\Delta E \leftarrow E_{\text{after}} - E_{\text{before}}$ - \If{$\Delta E \leq 0$} Accept new config if - \State $\text{Update expectation values}$ - \ElsIf{$\Delta E > 0$} - \State $\text{Calculate}$ - \EndIf - \EndProcedure - \end{algorithmic} -\end{algorithm} - -Probability: Using the expectation value require a stochastic random variable X (stoch being that it is tied to an event), a function from the "utfallsrommet" to the real numbers. - -Understanding a systems properties close to a critical point. For the 2D ising model, where L is large, we can observe discontinuity in $C_{V}$ and $\chi$. They diverge at the critical point in the thermosynamic limit, meaning the variance in energy and magnetization diverge. For lattice of finite size the variance scale as $1 / \sqrt{M}$ where $M = 2^{N}$ is the number of microstates. +known as the Boltzmann distribution. This is an exponential distribution, where $\beta$ and $Z$ is given by \begin{align*} - e^{-\beta \Delta E} &= \frac{p_{after}}{p_{before}} = \frac{e^{-\beta E_{after}}}{e^{-\beta E_{before}}} + \beta =& \frac{1}{k_{B}} \ , & + Z &= \sum_{\text{all possible } \mathbf{s}} e^{-\beta E(\mathbf{s}} \ , \\ \end{align*} +and $k_{B}$ is the Boltzmann constant. $Z$ is also called the partition function, and is the normalizing factor of the pdf. We derive the analytical expression for $Z$ in Appendix \ref{sec:analytical_expressions}, +\begin{align*} + z &= 4 \cosh (8 \beta J) + 12 \ . +\end{align*} +Using this partition function, we can derive analytical expressions for expectation values. 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}} +\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 $E$. + +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 use the expression for $\langle E \rangle$ and $\langle M \rangle$ to find the expectation values per spin. For energy per spin +\begin{align*} + \langle \epsilon \rangle &= \sum_{i=1}^{N} \epsilon(s_{i}) p(s_{i} \ | \ T) \\ + &= \sum_{i=1}^{N} \frac{E(\mathbf{s})}{N} p(s_{i} \ | \ T) \\ + &= \frac{1}{N} \sum_{i=1}^{N} E(\mathbf{s}) p(s_{i} \ | \ T) +\end{align*} +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*} +We find the expected total energy to be +\begin{equation*} %\label{eq:energy_total_first} + \langle E \rangle = -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} \ , +\end{equation*} +and the expected energy per spin +\begin{equation*} %\label{eq:energy_spin_first} + \langle \epsilon \rangle = \frac{-2J \sinh(8 \beta J)}{ \cosh(8 \beta J) + 3} \ . +\end{equation*} +We find the expected absolute total magnetization to be +\begin{equation*} %\label{eq:magnetization_total_first} + \langle |M| \rangle = \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} \ , +\end{equation*} +and the expected magnetization per spin +\begin{equation*}\label{eq:magnetization_spin_first} + \langle |m| \rangle = \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} \ . +\end{equation*} +We will determine the heat capacity and the susceptibility of our system, which require an expression of the variance of the energy and magnetization. We can use the definition of variance, given by +\begin{align*} + \mathbb{V}(X) &= \sum_{x \in D} [(x - \mathbb{E}(x))^2 \cdot p(x)] & \text{\cite[p. 132]{springer:2012:modernstat}} \\ + &= \mathbb{E}(X^{2}) - [\mathbb{E}(X)]^{2} \ , +\end{align*} +for a stochastic variable $X$. We use this to find the heat capacity +\begin{equation*} %\label{eq:specific_heat_capacity} + C_{V} = \frac{64J^{2} }{N k_{\text{B}} T^{2}} \frac{(3\cosh(8 \beta J) + 1)}{(\cosh(8 \beta J) + 3)^{2}} \ , +\end{equation*} +and the susceptibility +\begin{equation*} %\label{eq:susceptibility} + \chi = \frac{4}{N k_{\text{B}} T} \frac{(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}} +\end{equation*} +The derivation of all analytical expressions can be found in Appendix \ref{sec:analytical_expressions}. + +For scalability, we want to work with unitless spins. To obtain this, we introduce coupling constant $J$ as the base unit for energy. With the Boltzmann constant we derive the remaining units, which can be found in Table \ref{tab:units}. +\begin{table}[H] + \centering + \begin{tabular}{cccc} % @{\extracolsep{\fill}} + \hline + Value & Unit \\ + \hline + $[E]$ & $J$ \\ + $[T]$ & $J / k_{B}$ \\ + $[C_{V}]$ & $k_{B}$ \\ + $[\chi]$ & $1 / J$ \\ + \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.} + \label{tab:units} +\end{table} + +% \subsection{Phase transition and critical temperature}\label{subsec:phase_temp} + + +\subsection{Sampling approach}\label{subsec:sampling} +We will generate samples from the Ising model, to approximate the probability distribution of the sample set. We sample using the Markov chain Monte Carlo method. 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 method, introduces randomness to the sampling to determine statistical quantities \cite{tds:2021:mcmc}. + +The Ising model is initialized in a random state, and the chain is evolved until it reaches an equilibrium. At this point we start sampling, as the distribution of the set of samples will tend toward the actual distribution. At each step of flipping a spin, the change in energy will be evaluated as +\begin{align*} + \Delta E &= E_{\text{after}} - E_{\text{before}} \ . +\end{align*} +We have to evaluate +% The samples generated are microstates, each of which occur with some probability. Since the dependency between samples only apply to the neighboring samples, and the Markov chain is ergodic + +% Since we are generating samples from a stationary probability distribution, with a in +% We will use use In addition, we Since t + +\subsection{Implementation}\label{subsec:algo_implementation} + + +\subsection{Tools}\label{subsec:tools} +The Ising model and MCMC methods are implemented in C++, and parallelized using \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. + +% \subsection{Draft} +% The Ising model correspond to a lattice with a number of spin sites, where +% each spin has the possible state up $\uparrow$ or down $\downarrow$. We will study the +% two-dimensional Ising model for ferromagnets, however, the model is not resticted +% to this dimentionality~\cite[p. 3]{obermeyer:2020:ising}. We will assume a square lattice, where the length $L$ of +% the lattice give the number of spin sites $N = L^{2}$. When we +% consider the entire 2D lattice we can study the system spin configuration, or the +% microstate, given by $\mathbf{s} = (s_{1}, s_{2}, \dots, s_{N})$. The value of +% each spin $s_{i}$, where $i \in [1, N]$, is given by the direction of the spin +% $\uparrow = +1$ and $\downarrow = -1$. + +% We find the total energy of the system using +% \begin{equation}\label{eq:energy} +% E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} +% \end{equation} +% where $\langle k l \rangle$ denotes the neighboring spins. + +% The total magnetization of the system is given by +% \begin{equation}\label{eq:magnetization} +% M(\mathbf{s}) = \sum_{i}^{N} s_{i}, +% \end{equation} +% with degeneracy denoting how many states have the same values. We will consider +% a 2D lattice with $L = 2$, with periodic boundary conditions, following the +% counting pattern shown in figure~\ref{fig:tikz_counting}. The resulting values for +% the $2 \times 2$ lattice can be found in table \ref{tab:lattice_config}. + +% \begin{table}[H] +% \centering +% \begin{tabular}{cccc} % @{\extracolsep{\fill}} +% \hline +% Spins up & $E(\mathbf{s})$ & $M(\mathbf{s})$ & Degeneracy \\ +% \hline +% 4 & -8J & 4 & 1 \\ +% 3 & 0 & 2 & 4 \\ +% \multirow{2}*{2} & 0 & 0 & 4 \\ +% & 8J & 0 & 2 \\ +% 1 & 0 & -2 & 4 \\ +% 0 & -8J & -4 & 1 \\ +% \hline +% \end{tabular} +% \caption{Values} +% \label{tab:lattice_config} +% \end{table} + +% Probability stuff, for a system with a given temperature $T$. We use the Boltzmann +% distribution in the analytical expression, to compare with our results. This is +% a probability distribution from the family of exponential +% \begin{equation}\label{eq:boltzmann_distribution} +% p(\mathbf{s} \ | \ T) = \frac{1}{Z} \exp^{-\beta E(\mathbf{s})}, +% \end{equation} +% where $Z$ is the partition function $\beta$ is the related to the inverse of the +% temperature as +% \begin{equation}\label{eq:beta} +% \beta = \frac{1}{k_{\text{B}} T}. +% \end{equation} +% We derive an analytical expression for $Z$ in appendix \ref{sec:analytical_expressions} + +% We will assume a $2 \times 2$ lattice to study the possible system states, and +% to find analytical expressions necessary in the Markov Chain Monte Carlo method. +% These results are used to test our code during implementation. + +% We study the 2D lattice and find the total energy and the total magnetization of +% of the system, in addition to the degeneracy. In addition, we want to work with +% unitless spins. To obtain this, we introduce the base unit for energy $J$, and +% with the Boltzmann constant we derive other units necessary. The resulting units +% can be found in table \ref{tab:units}. + +% % Problem 1 + +% \begin{equation}\label{eq:partition_function} +% Z = 4 \cosh (8 \beta J) + 12, +% \end{equation} +% % +% \begin{equation}\label{eq:energy_spin_first} +% \langle \epsilon \rangle = \frac{-2J \sinh(8 \beta J)}{ \cosh(8 \beta J) + 3} +% \end{equation} +% % +% \begin{equation}\label{eq:energy_spin_second} +% \langle \epsilon^{2} \rangle = \frac{4 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} +% \end{equation} +% % +% \begin{equation}\label{eq:magnetization_spin_first} +% \langle |m| \rangle = \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} +% \end{equation} +% % +% \begin{equation}\label{eq:magnetization_spin_second} +% \langle |m|^{2} \rangle = \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} +% \end{equation} +% % +% \begin{equation}\label{eq:energy_total_first} +% \langle E \rangle = -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} +% \end{equation} +% % +% \begin{equation}\label{eq:energy_total_first_squared} +% \langle E \rangle^{2} = \frac{64J^{2} \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} +% \end{equation} +% % +% \begin{equation}\label{eq:energy_total_second} +% \langle E^{2} \rangle = \frac{64J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} +% \end{equation} +% % +% \begin{equation}\label{eq:magnetization_total_first} +% \langle |M| \rangle = \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} +% \end{equation} +% % +% \begin{equation}\label{eq:magnetization_total_first_squared} +% \langle |M| \rangle^{2} = \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} +% \end{equation} +% % +% \begin{equation}\label{eq:magnetization_total_second} +% \langle M^{2} \rangle = \frac{8(e^{8 \beta J} + 1)}{\cosh(8 \beta J) + 3} +% \end{equation} +% % +% \begin{equation}\label{eq:specific_heat_capacity} +% C_{V} = \frac{64J^{2} }{N k_{\text{B}} T^{2}} \frac{(3\cosh(8 \beta J) + 1)}{(\cosh(8 \beta J) + 3)^{2}} +% \end{equation} +% % +% \begin{equation}\label{eq:sesceptibility} +% \chi = \frac{4}{N k_{\text{B}} T} \frac{(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}} +% \end{equation} +% The derivation of analytical expressions can be found in appendix \ref{sec:analytical_expressions}. + + +% The Ising model and MCMC methods are implemented in C++, and parallelized using +% \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. + + +% Def: Correlation length $\xi$ of phase transition, and is related to the spin-correlation function which defines the magnetic susceptibility. The spin-correlation function is the covariance of the magnetization, expressing the degree of correlation between spins. $\xi$ is normally finite at critical points. Proportional to the lattice size at the critical point, can be used to relate behavior for infinitly large lattice $T_{C}(L) - T_{C}(L = \infty) = \propto \alpha T^{-1}$ + +% Implementation: The metropolis algo only consider ratio, not necessary to calculate partition function. + +% Metropolis algo: +% \begin{algorithm}[H] +% \caption{Metropolis algorithm p. 435 in lecture notes} +% \label{algo:metropolis_algo} +% \begin{algorithmic} +% \Procedure{Metropolis algorithm}{$args$} +% \State $E_{\text{before}} \leftarrow \text{Set initial random state}$ +% \State $ \text{Flip spin}$ +% \State $E_{\text{after}} \text{Calculate energy}$ +% \State $\Delta E \leftarrow E_{\text{after}} - E_{\text{before}}$ +% \If{$\Delta E \leq 0$} Accept new config if +% \State $\text{Update expectation values}$ +% \ElsIf{$\Delta E > 0$} +% \State $\text{Calculate}$ +% \EndIf +% \EndProcedure +% \end{algorithmic} +% \end{algorithm} + +% Probability: Using the expectation value require a stochastic random variable X (stoch being that it is tied to an event), a function from the "utfallsrommet" to the real numbers. + +% Understanding a systems properties close to a critical point. For the 2D ising model, where L is large, we can observe discontinuity in $C_{V}$ and $\chi$. They diverge at the critical point in the thermosynamic limit, meaning the variance in energy and magnetization diverge. For lattice of finite size the variance scale as $1 / \sqrt{M}$ where $M = 2^{N}$ is the number of microstates. +% \begin{align*} +% e^{-\beta \Delta E} &= \frac{p_{after}}{p_{before}} = \frac{e^{-\beta E_{after}}}{e^{-\beta E_{before}}} +% \end{align*} \end{document} diff --git a/src/scripts/burn_in_time.py b/src/scripts/burn_in_time.py index e4010ee..6ec2229 100644 --- a/src/scripts/burn_in_time.py +++ b/src/scripts/burn_in_time.py @@ -16,22 +16,22 @@ plt.rcParams.update(params) def plot_from_file(): files = [ - # "output/burn_in_time/unordered_1_0_1421110368.txt", - # "output/burn_in_time/ordered_1_0_611577739.txt", + "output/burn_in_time/unordered_1_0_1421110368.txt", + "output/burn_in_time/ordered_1_0_611577739.txt", "output/burn_in_time/unordered_2_4_1212892317.txt", "output/burn_in_time/ordered_2_4_2408603856.txt", ] labels = [ - "Unordered", - "Ordered", - # "2.4, unordered", - # "2.4, ordered" + "1.0, unordered", + "1.0, urdered", + "2.4, unordered", + "2.4, ordered" ] colors = [ "seagreen", "darkred", - # "darkgoldenrod", - # "steelblue" + "darkgoldenrod", + "steelblue" ] fig1, ax1 = plt.subplots() @@ -54,13 +54,13 @@ def plot_from_file(): ax2.plot(t, magnetization, label=f"{label}", color=color) ax1.set_xlabel("t") - ax1.set_ylabel(r"$\langle \epsilon \rangle$") - ax1.legend(title="Initial state") - fig1.savefig("../latex/images/burn_in_time_energy_2_4.pdf", bbox_inches="tight") + ax1.set_ylabel(r"$\langle \epsilon \rangle \ [J]$") + ax1.legend(title="Initial state", loc="upper right") + fig1.savefig("../latex/images/burn_in_time_energy.pdf", bbox_inches="tight") ax2.set_ylabel(r"$\langle |m| \rangle$") ax2.set_xlabel("t") - ax2.legend(title="Initial state") - fig2.savefig("../latex/images/burn_in_time_magnetization_2_4.pdf", bbox_inches="tight") + ax2.legend(title="Initial state", loc="upper right") + fig2.savefig("../latex/images/burn_in_time_magnetization.pdf", bbox_inches="tight") def main(): plot_from_file() From 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