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| -rw-r--r-- | aps_mm_2017.aux | 37 | ||||
| -rw-r--r-- | aps_mm_2017.log | 1349 | ||||
| -rw-r--r-- | aps_mm_2017.nav | 18 | ||||
| -rw-r--r-- | aps_mm_2017.out | 0 | ||||
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| -rw-r--r-- | aps_mm_2017.snm | 0 | ||||
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| -rw-r--r-- | aps_mm_2017.tex | 91 | ||||
| -rw-r--r-- | aps_mm_2017.toc | 1 | ||||
| -rw-r--r-- | figs/fig1.pdf | bin | 0 -> 7095 bytes | |||
| -rw-r--r-- | gen_figs.nb | 2706 |
11 files changed, 4194 insertions, 8 deletions
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char U+0164 (decimal 356) + defining Unicode char U+0165 (decimal 357) + defining Unicode char U+0168 (decimal 360) + defining Unicode char U+0169 (decimal 361) + defining Unicode char U+016A (decimal 362) + defining Unicode char U+016B (decimal 363) + defining Unicode char U+016C (decimal 364) + defining Unicode char U+016D (decimal 365) + defining Unicode char U+016E (decimal 366) + defining Unicode char U+016F (decimal 367) + defining Unicode char U+0170 (decimal 368) + defining Unicode char U+0171 (decimal 369) + defining Unicode char U+0172 (decimal 370) + defining Unicode char U+0173 (decimal 371) + defining Unicode char U+0174 (decimal 372) + defining Unicode char U+0175 (decimal 373) + defining Unicode char U+0176 (decimal 374) + defining Unicode char U+0177 (decimal 375) + defining Unicode char U+0178 (decimal 376) + defining Unicode char U+0179 (decimal 377) + defining Unicode char U+017A (decimal 378) + defining Unicode char U+017B (decimal 379) + defining Unicode char 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index 0000000..e69de29 --- /dev/null +++ b/aps_mm_2017.snm diff --git a/aps_mm_2017.synctex.gz b/aps_mm_2017.synctex.gz Binary files differnew file mode 100644 index 0000000..7a7b570 --- /dev/null +++ b/aps_mm_2017.synctex.gz diff --git a/aps_mm_2017.tex b/aps_mm_2017.tex index 47f5e77..eda12da 100644 --- a/aps_mm_2017.tex +++ b/aps_mm_2017.tex @@ -5,33 +5,108 @@ % Copyright (c) 2012 pants productions. All rights reserved. % -\pdfminorversion=4 \documentclass[fleqn]{beamer} \usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,latexsym,graphicx} +\usepackage{concmath} +%\usepackage{bera} +%\usepackage{merriweather} \usepackage[T1]{fontenc} -\usepackage[scaled]{helvet} -\usepackage{amsmath,amssymb,latexsym,graphicx,soul,amsthm,rotating,etex,sansmath,listings,tabularx,tikz} -\usepackage[export]{adjustbox} -\usecolortheme{beaver} -\usetikzlibrary{decorations.pathreplacing} +\usecolortheme{beaver} +\usefonttheme{serif} \title{Universal scaling and the essential singularity at the Ising first-order transition} \author{ Jaron~Kent-Dobias\inst{1} \and James~Sethna\inst{1}} \institute{\inst{1}Cornell University} \date{16 March 2016} -\usepackage[normalem]{ulem} \begin{document} \def\dd{\mathrm d} +\def\im{\mathop{\mathrm{Im}}} \begin{frame} - \usefont{T1}{phv}{b}{n} \titlepage \end{frame} +\begin{frame} + \frametitle{Renormalization and free energy} + Rescale a system by a factor $b$, with couplings $K\to K'$. + From John Cardy's \emph{Scaling and Renormalization in Statistical Physics}, + free energy per site $f$ + \[ + f(\{K\})=g(\{K\})+b^{-d}f(\{K'\}) + \] + \begin{quote} + However, if we are interested in extracting only the \emph{singular} + behavior of $f$, \dots we may obtain a \emph{homogeneous} transformation law for the + \emph{singular part} of the free energy $f_s$ + \[ + f_s(\{K\})=b^{-d}f_s(\{K'\}) + \] + \end{quote} + Defense: $g(\{K\})$ is an analytic function of $\{K\}$, while the singular + part is nonanalytic +\end{frame} + +\begin{frame} + Follow thermodynamic functions onto metastable branch. +\end{frame} + +\begin{frame} + \[\Delta f\sim\Sigma\gamma(N)-HMN\] + Near the critical point, $\gamma(N)\sim N^{\frac{d-1}d}$ + \[ + M=|t|^\beta\mathcal M(h/{t^{\beta\delta}}) + \] + \[ + N_{crit}\sim\bigg(\frac{\Sigma}{HM}\Big(1-\tfrac1d\Big)\bigg)^d + \] + \[ + \Delta f_{crit}\sim\Sigma\bigg(\frac{\Sigma}{HM}\bigg)^{d-1} + \sim X^{-(d-1)}\frac{\mathcal + S^d(X)}{\mathcal M^{d-1}(X)} + \] + $X=h/t^{\beta\delta}$ + The probability that such a domain forms and the metastable state decays is given by the Boltzmann factor, + so that + $\Sigma\sim|t|^\mu$, $\mu=-\nu+\gamma+2\beta$ + \[ + \im f\sim e^{-\beta\, \Delta f_{crit}} + \sim\mathcal F(X)e^{-1/X^{d-1}} + \] +\end{frame} + +\begin{frame} + $e^{-1/x}$ is nonanalytic at $x=0$: all derivatives vanish, means that free + energy (which has no imaginary part in stable phase) is smooth + + \centering + \includegraphics[width=0.7\textwidth]{figs/fig1} +\end{frame} +\begin{frame} + Analyticity of $F$ means that the imaginary + + \[ + f(h)=\sum_n A_nh^N + \] + \[ + A_n=(-B)^{1-n}\Gamma(n-1) + \] + + \[ + f(h)=\frac1\pi\int_{h'<0}\frac{\dd h'\,\im f(h')}{h'-h} + \] +\end{frame} + +\begin{frame} + Field theorists (Lubensky, blah blah blah) + \[ + \mathcal + \] +\end{frame} \end{document} diff --git a/aps_mm_2017.toc b/aps_mm_2017.toc new file mode 100644 index 0000000..ae73954 --- /dev/null +++ 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