From eee8a108b8592b23778b642d5785a8ba0773e3bb Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 5 Jul 2019 16:49:15 -0400
Subject: laptop changes

---
 statphys27.tex | 52 ++++++++++++++++++++++++++++++++++++++++------------
 1 file changed, 40 insertions(+), 12 deletions(-)

(limited to 'statphys27.tex')

diff --git a/statphys27.tex b/statphys27.tex
index 6230887..4c2387b 100644
--- a/statphys27.tex
+++ b/statphys27.tex
@@ -1,7 +1,6 @@
 
 \documentclass[fleqn,aspectratio=169]{beamer}
 
-
 \setbeamerfont{frametitle}{family=\bf}
 \setbeamerfont{normal text}{family=\rm}
 \setbeamertemplate{navigation symbols}{}
@@ -22,18 +21,22 @@
   \maketitle
 \end{frame}
 
+\begin{frame}
+  \frametitle{Monte Carlo is too slow}
+\end{frame}
+
 \begin{frame}
   \frametitle{Monte Carlo is too slow}
 
-  Critical phenomena are often studied on lattice models using Monte Carlo, but near critical points it suffers from \emph{critical slowing down}, power-law divergence of timescales.
+  Monte Carlo useful for lattice models, but near critical points suffers from \emph{critical slowing down}, power-law divergence of timescales.
 
   \vspace{1em}
 
-  Slowing down has been alleviated in many models using cluster algorithms and their derivatives, but many applications lack a clean solution.
+  Often alleviated with cluster algorithms, but many applications lack a clean solution.
 
   \vspace{1em}
 
-  We introduce a generic, natural, and efficient way to extend models with existing cluster algorithms to operate in arbitrary external fields.
+  We introduce a generic, natural, efficient way to extend models with existing cluster algorithms to operate in arbitrary external fields.
 
   \vspace{1em}
 
@@ -58,14 +61,18 @@
   \framesubtitle{The Fortuin--Kasteleyn representation}
 
   The Ising model
-  \[
+  $
     \mathcal H=-\sum_{\langle ij\rangle}J_{ij}s_is_j
-  \]
-  for $s_i=\pm1$ on the lattice sites has a representation
+  $
+  for $s_i=\pm1$ can be written
   \[
     Z=\tr_se^{-\beta\mathcal H}\propto\tr_f\tr_s\prod_{\langle ij\rangle}\big[\delta_{f_{ij},0}(1-p_{ij})+\delta_{f_{ij},1}\delta_{s_i,s_j}p_{ij}\big]
   \]
-  for $f_{ij}\in\{0,1\}$ on the lattice bonds and $p_{ij}=1-e^{-2\beta J_{ij}}$. This gives joint probability distributions
+  for $f_{ij}\in\{0,1\}$ on the bonds and $p_{ij}=1-e^{-2\beta J_{ij}}$.
+
+  \vspace{1em}
+
+  This gives conditional probabilities 
   \begin{align*}
     P(f_{ij}=1\mid s_i,s_j)=\begin{cases}p_{ij} & s_i=s_j \\ 0 & s_i\neq s_j\end{cases}
     &&
@@ -271,6 +278,7 @@
       \end{overprint}
     \end{column}
     \begin{column}{0.45\textwidth}
+      Example: 5-spin clock model with a field favoring the two states to the bottom right.
       \begin{enumerate}
         \item\alert<2>{Take a spin configuration.}
         \item\alert<3>{Draw a self-inverse $r\in G$.}
@@ -284,17 +292,37 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{Summary}
+  \frametitle{Other lattice models}
+  \framesubtitle{The method is good}
+
+  Results generalize to arbitrary bond and site dependence.
+
+  \vspace{0.5em}
+
+  Models already efficient at zero field are more efficient with a field.
+
+  \vspace{0.5em}
+
+  Extension appears natural in the scaling sense.
+
+  \centering
+
+  \includegraphics[width=0.85\textwidth]{figs/timescales}
+  
+\end{frame}
+
+\begin{frame}
+  \frametitle{Summary \& Extensions}
 
   Introduced a generic method for running cluster Monte Carlo on lattice systems with any external field.
-
+s-
   \vspace{1em}
 
-  Results generalize to arbitrary bond, site dependence.
+  Already used to efficiently show relevance/irrelevance of various harmonic perturbations to the XY model.
 
   \vspace{1em}
 
-  Dynamic scaling works as expected with Wolff or Swendsen--Wang exponents: models efficient at zero field are more efficient with a field, extension appears natural in the scaling sense.
+  Presently being used to model novel lattice models with coupled spins on sites and bonds which act as effective fields for each other.
 
   \vspace{1em}
 
-- 
cgit v1.2.3-54-g00ecf

