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diff --git a/aps_mm_2017.tex b/aps_mm_2017.tex index e85f277..89f54ed 100644 --- a/aps_mm_2017.tex +++ b/aps_mm_2017.tex @@ -5,7 +5,7 @@ % Copyright (c) 2012 pants productions. All rights reserved. % -\documentclass[fleqn]{beamer} +\documentclass[fleqn,aspectratio=169]{beamer} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,latexsym,graphicx} @@ -19,9 +19,9 @@ \setbeamertemplate{navigation symbols}{} \title{Universal scaling and the essential singularity at the abrupt Ising transition} -\author{ Jaron~Kent-Dobias\inst{1} \and James~Sethna\inst{1}} -\institute{\inst{1}Cornell University} -\date{16 March 2016} +\author{ Jaron~Kent-Dobias \and James~Sethna} +\institute{Cornell University} +\date{16 March 2017} \begin{document} @@ -35,107 +35,116 @@ \end{frame} \begin{frame} - \frametitle{Parametric Ising Models} + \frametitle{Outline} + \begin{itemize} + \item Renormalization and the Ising model + \pause + \item Metastability and complex free energy + \pause + \item Analytic constraints on the stable free energy + \pause + \item Closed-form results for the {\sc 2d} Ising susceptibility + \end{itemize} + \vfill +\end{frame} + +\begin{frame} + \frametitle{Renormalization and the Ising Model} \begin{columns} \begin{column}{0.4\textwidth} - \includegraphics[width=\textwidth]{figs/fig7} - - \vspace{1em} - - \includegraphics[width=\textwidth,height=0.5\textwidth]{susceptibility.jpg} - - \tiny\texttt{a plot of susceptibility with precision parametric fit and it - isn't - very good at the abrupt transition} + \centering + \includegraphics[width=\textwidth]{figs/fig2}\\ + \tiny + From \emph{Scaling and Renormalization in Statistical Physics} by John + Cardy \end{column} \begin{column}{0.6\textwidth} - Scaling forms of Ising variables that do well globally. - - \pause\vspace{1em} - - Incorporate the critical point in a natural way: - \begin{itemize} - \item singular scaling with the ``radial coordinate'' - \item analytic scaling with the ``angular coordinate'' - \end{itemize} - - \vspace{1em} \pause - - Typically do a very poor job near the abrupt transition at $H=0$. + {\sc Rg} analytically maps system space onto itself. + \vspace{1em}\pause\\ + Fixed points correspond to phases, criticality. + \vspace{1em}\pause\\ + Nonanalytic behavior is preserved by {\sc rg}. \end{column} \end{columns} \end{frame} \begin{frame} - \frametitle{Renormalization and Universality} - + \frametitle{Renormalization and the Ising Model} \begin{columns} \begin{column}{0.4\textwidth} - \centering - \includegraphics[width=\textwidth]{figs/fig2} - - \tiny - From \emph{Scaling and Renormalization in Statistical Physics} by John - Cardy + \includegraphics{figs/fig3} \end{column} \begin{column}{0.6\textwidth} - Renormalization is an analytic scaling transformation that acts on - system space. - - \vspace{1em}\pause - - Fixed points are scale invariant, corresponding to systems representing - idealized phases or critical behavior. - - \vspace{1em}\pause - - Nonanalytic behavior---like power laws and logarithms---are preserved - under {\sc rg} and shared by \emph{any} system that flows to the same - point. - - \vspace{1em}\pause - - Not all nonanalytic behavior are power laws! + Ising critical point has power laws, logarithms in thermodynamic + variables. + \vspace{1em}\pause\\ + Connected to line of abrupt transitions. + \vspace{1em}\pause\\ + We've identified nonanalytic behavior along the abrupt transition line. \end{column} \end{columns} \end{frame} \begin{frame} - \frametitle{The Metastable Ising Model} - + \frametitle{Metastability \& Complex Free Energy} \begin{columns} - \begin{column}{0.3\textwidth} - \only<1-1>{\includegraphics[width=\textwidth]{figs/fig3}} - \only<2->{\includegraphics[width=\textwidth]{figs/fig4}} - - \vspace{1em} - - \includegraphics[width=\textwidth]{figs/fig8} - + \begin{column}{0.4\textwidth} + \includegraphics{figs/fig11} \end{column} - \begin{column}{0.7\textwidth} - Consider an Ising-class model brought into a metastable state. - - \vspace{1em} \pause\pause - - A domain of $N$ spins entering the stable phase causes a free energy - change - \[ - \Delta F=\Sigma N^\sigma-MHN - \] + \begin{column}{0.6\textwidth} + \includegraphics{figs/fig12} + \end{column} + \end{columns} +\end{frame} - \pause +\begin{frame} + \frametitle{Metastability \& Complex Free Energy} + \begin{columns} + \begin{column}{0.4\textwidth} + \includegraphics{figs/fig13} + \end{column} + \begin{column}{0.6\textwidth} + \includegraphics{figs/fig14} + \end{column} + \end{columns} +\end{frame} - The metastable phase is stable to domains smaller than - \[ - N_\crit=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)} - \] - but those larger will grow to occupy the entire system. +\begin{frame} + \frametitle{Metastability \& Complex Free Energy} + \begin{columns} + \begin{column}{0.4\textwidth} + \includegraphics{figs/fig4} + \end{column} + \begin{column}{0.6\textwidth} + Thermodynamics can be continued into metastable phase. + \vspace{1em}\pause\\ + Decay rate related to imaginary free energy, + $\Gamma\propto\frac1{kT}\im F$ (Langer 1969). + \vspace{1em}\pause\\ + Ising metastable decay somewhat well studied (G\"unther 1980, Houghton + 1980) \end{column} \end{columns} \end{frame} \begin{frame} + \frametitle{Metastability \& Complex Free Energy} + Decay of metastable phase occurs when domain of critical size forms. + \vspace{1em}\pause\\ + Domain of $N$ spins entering the stable phase causes a free energy + change + \[ + \Delta F=\Sigma N^\sigma-MHN + \] + \pause + Metastable phase is stable to domains smaller than + \[ + N_\crit=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)} + \] + but larger will grow to occupy the entire system, decay to stable phase. +\end{frame} + +\begin{frame} \frametitle{The Metastable Ising Model} The formation of a critical domain has energy cost |