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-rw-r--r--.gitignore2
-rw-r--r--figs/timescales.pngbin0 -> 122732 bytes
-rw-r--r--statphys27.tex52
3 files changed, 42 insertions, 12 deletions
diff --git a/.gitignore b/.gitignore
index 45ac685..2c3e9a5 100644
--- a/.gitignore
+++ b/.gitignore
@@ -5,4 +5,6 @@
*.snm
*.synctex.gz
*.toc
+*.fdb_latexmk
+*.fls
statphys27.pdf
diff --git a/figs/timescales.png b/figs/timescales.png
new file mode 100644
index 0000000..c8af389
--- /dev/null
+++ b/figs/timescales.png
Binary files differ
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}{}
@@ -24,16 +23,20 @@
\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}