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Diffstat (limited to 'statphys27.tex')
| -rw-r--r-- | statphys27.tex | 52 |
1 files changed, 40 insertions, 12 deletions
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} |