From 05725b9fa4843d19561b026567979709cc6b62a2 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 21 Feb 2018 11:40:28 -0500 Subject: initial commit --- .gitignore | 7 ++ .gitmodules | 3 + aps_mm_2018.html | 194 +++++++++++++++++++++++++++++++++++++++ aps_mm_2018.tex | 166 +++++++++++++++++++++++++++++++++ figs/test.webm | Bin 0 -> 231287 bytes figs/wolff-scoop_explanation.png | Bin 0 -> 98441 bytes figs/wolff-scoop_title.png | Bin 0 -> 139907 bytes main.css | 32 +++++++ vissense | 1 + 9 files changed, 403 insertions(+) create mode 100644 .gitignore create mode 100644 .gitmodules create mode 100644 aps_mm_2018.html create mode 100644 aps_mm_2018.tex create mode 100644 figs/test.webm create mode 100644 figs/wolff-scoop_explanation.png create mode 100644 figs/wolff-scoop_title.png create mode 100644 main.css create mode 160000 vissense diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..203e646 --- /dev/null +++ b/.gitignore @@ -0,0 +1,7 @@ +aps_mm_2018.aux +aps_mm_2018.log +aps_mm_2018.nav +aps_mm_2018.out +aps_mm_2018.pdf +aps_mm_2018.snm +aps_mm_2018.toc diff --git a/.gitmodules b/.gitmodules new file mode 100644 index 0000000..071b2fe --- /dev/null +++ b/.gitmodules @@ -0,0 +1,3 @@ +[submodule "vissense"] + path = vissense + url = https://github.com/vissense/vissense.git diff --git a/aps_mm_2018.html b/aps_mm_2018.html new file mode 100644 index 0000000..afa519b --- /dev/null +++ b/aps_mm_2018.html @@ -0,0 +1,194 @@ + + + + Title + + + + + + + + + + + + + + diff --git a/aps_mm_2018.tex b/aps_mm_2018.tex new file mode 100644 index 0000000..34e9114 --- /dev/null +++ b/aps_mm_2018.tex @@ -0,0 +1,166 @@ +% +% research_midsummer.tex - Research Presentation for the Topaz lab. +% +% Created by Jaron Kent-Dobias on Tue Mar 20 20:57:40 PDT 2012. +% Copyright (c) 2012 pants productions. All rights reserved. +% + +\documentclass[fleqn,aspectratio=169]{beamer} + +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,latexsym,graphicx,multimedia} +\usepackage{concmath} +\usepackage[T1]{fontenc} + +\usecolortheme{beaver} +\usefonttheme{serif} +\setbeamertemplate{navigation symbols}{} + +\title{Direct Measurement of Metastable Properties Near Critical Points} +\author{ Jaron~Kent-Dobias \and James~Sethna} +\institute{Cornell University} +\date{9 March 2018} + +\begin{document} + +\def\H{\mathcal H} +\def\Z{\mathbb Z} + +\begin{frame} + \titlepage +\end{frame} + +\begin{frame} + \frametitle{Outline} + \begin{itemize} + \item Simulating equilibrium spin systems + \item Local updates: it's got problems + \item Solution: cluster flips! + \item Cluster flips\dots in an external field??? + \item \dots{}yes! + \item Analysis of runtime, efficiency + \item Formal redefinition of magnetization + \item Use: direct measurement of the metastable state + \end{itemize} + \vfill +\end{frame} + +\begin{frame} + \frametitle{Spin systems: we love them} + + Described by Hamiltonians + \[ + \H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iH(s_i) + \] + for $Z$ invariant under rotations $R$: $Z(R(s),R(t))=Z(s,t)$ + + \begin{table} + \renewcommand{\tabcolsep}{7pt} + \begin{tabular}{l||cccc} + & $s$ & $R$ & $Z(s_i,s_j)$ & $H(s)$ \\ + \hline\hline + Ising model & $\{-1,1\}$ & $s\mapsto-s$ & $s_is_j$ & $Hs$ \\ + Order-$n$ model & $S^n$ & $\mathop{\mathrm{SO}}(n)$ (rotation) & $s_i\cdot s_j$ & $H\cdot s$ \\ + Potts model & $\Z/q\Z$ & addition mod $q$ & $\delta(s_i,s_j)$ & + $\sum_iH_i\delta(i,s)$ \\ + Clock model & $\Z/q\Z$ & addition mod $q$ & $\cos(2\pi\frac{s_i-s_j}q)$ & + $\sum_iH_i\cos(2\pi\frac{s-i}q)$ + \end{tabular} + \end{table} + + Relatively simple with extremely rich behavior, phase transitions galore! +\end{frame} + +\begin{frame} + \frametitle{Local Monte Carlo: Not Great} + + Standard approach to modelling arbitrary stat mech system: metropolis. + + \begin{enumerate} + \item Pick random spin. + \item Pick random rotation $R$. + \item Compute change in energy $\Delta\H$ resulting from taking $s$ to + $R(s)$. + \item Take $s$ to $R(s)$ with probability $\max\{1,e^{-\beta\Delta\H}\}$. + \end{enumerate} + + Problem: Scales very poorly near phase transitions. + + Correlation time $\tau\sim L^z$ at critical point, $\tau\sim t^{-z/\nu}$ + approaching it. + + $z$ takes large integer values for Ising, order-$n$, Potts model critical + points. +\end{frame} + +\begin{frame} + \frametitle{Wolff: wow, what a solution} + + \begin{enumerate} + \item Pick random spin, add to cluster. + \item Pick random rotation $R$. + \item For every neighboring spin, add to cluster with probability + $\min\{0,1-e^{-\beta(Z(R(s),t)-Z(R(s),R(t)))}\}$. + \item Repeat 3 for every spin added to cluster. + \item Transform entire cluster with rotation $R$. + \end{enumerate} + Relies on symmetry of $Z$ + + Fast near the critical point: early studies thought $z$ was zero, actually + $0.1$--$0.4$. +\end{frame} + +\begin{frame} + \frametitle{We want to apply an external field, though} + + The external field $H$ is not invariant under global rotations! + + Let's make it that way: $R_s$ is the rotation that takes $s$ to the + identity + (1, first basis vector, etc) + \[ + \tilde Z(s_i,s_j)= + \begin{cases} + Z(s_i,s_j) & \text{if $i,j\neq N$}\\ + H(R_{s_0}s_i) & \text{if $j=0$}\\ + H(R_{s_0}s_j) & \text{if $i=0$} + \end{cases} + \] + Exact correspondence between expectation values of operators in old and new + models: if $A(s)$ is an observable on old model, $\tilde + A(s_0,s)=A(R_{s_0}s)$ has the property + \[ + \langle\tilde + A\rangle=\mathop{\mathrm{Tr}}\nolimits_s\mathop{\mathrm{Tr}}\nolimits_{s_0}\tilde + A(s_0,s)=\mathop{\mathrm{Tr}}\nolimits_sA(s)=\langle A\rangle + \] +\end{frame} + +\begin{frame} + \centering + \includegraphics[height=0.8\textheight]{figs/wolff-scoop_title} +\end{frame} + +\begin{frame} + \centering + \includegraphics[height=0.8\textheight]{figs/wolff-scoop_explanation} +\end{frame} + +\begin{frame} + \frametitle{But does it actually work well? (yes)} + \movie[height=.4\textwidth,width=0.4\textwidth]{}{figs/test.avi} +\end{frame} + +\begin{frame} + \frametitle{Measuring direction-dependant quantities} +\end{frame} + +\begin{frame} + \frametitle{Metastable state!} +\end{frame} + +\begin{frame} +\end{frame} + +\end{document} + diff --git a/figs/test.webm b/figs/test.webm new file mode 100644 index 0000000..811be0e Binary files /dev/null and b/figs/test.webm differ diff --git a/figs/wolff-scoop_explanation.png b/figs/wolff-scoop_explanation.png new file mode 100644 index 0000000..4c2e891 Binary files /dev/null and b/figs/wolff-scoop_explanation.png differ diff --git a/figs/wolff-scoop_title.png b/figs/wolff-scoop_title.png new file mode 100644 index 0000000..8bbb0c2 Binary files /dev/null and b/figs/wolff-scoop_title.png differ diff --git a/main.css b/main.css new file mode 100644 index 0000000..f16954f --- /dev/null +++ b/main.css @@ -0,0 +1,32 @@ +td { + padding-bottom: 5px; + padding-top: 5px; +} + +td.first { + border-right: 2px solid #000; + padding-right: 5px; +} + +td:not(.first) { + text-align: center; +} + + +.column:first-of-type {float:left} +.column:last-of-type {float:right} + +.split-40 .column:first-of-type {width: 60%} +.split-40 .column:last-of-type {width: 40%} + +.remark-slide-number { + opacity: 0.0; +} + +img { + display: block; + margin-left: auto; + margin-right: auto; + width: 80%; +} + diff --git a/vissense b/vissense new file mode 160000 index 0000000..f169c93 --- /dev/null +++ b/vissense @@ -0,0 +1 @@ +Subproject commit f169c9347dc98d3895de65fbb6bcd725351d6248 -- cgit v1.2.3-70-g09d2