diff options
| author | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2017-11-07 10:43:35 -0500 |
|---|---|---|
| committer | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2017-11-07 10:43:35 -0500 |
| commit | e7220fadb6b3775e55afd9a07831e90136cfaf24 (patch) | |
| tree | 73701a2892deda076c4d60213d6da9b2d7b790a8 | |
| parent | 74da6e178889df2264486e762f088c7635704652 (diff) | |
| download | PRE_98_063306-e7220fadb6b3775e55afd9a07831e90136cfaf24.tar.gz PRE_98_063306-e7220fadb6b3775e55afd9a07831e90136cfaf24.tar.bz2 PRE_98_063306-e7220fadb6b3775e55afd9a07831e90136cfaf24.zip | |
many changes, including added references
| -rw-r--r-- | .gitignore | 3 | ||||
| -rw-r--r-- | fig_correlation.tex | 3547 | ||||
| -rw-r--r-- | fig_correlation_collapse-hL.tex | 1878 | ||||
| -rw-r--r-- | figs/fig_correlation_data_collapse-hL.dat | 106 | ||||
| -rw-r--r-- | makefile | 2 | ||||
| -rw-r--r-- | monte-carlo.bib | 153 | ||||
| -rw-r--r-- | monte-carlo.tex | 48 | ||||
| -rw-r--r-- | refs/alexandrowicz1989swendsen.pdf | bin | 0 -> 937546 bytes | |||
| -rw-r--r-- | refs/coniglio1980clusters.pdf | bin | 0 -> 442501 bytes | |||
| -rw-r--r-- | refs/coniglio1989exact.pdf | bin | 0 -> 463325 bytes | |||
| -rw-r--r-- | refs/destri1992swendsen.pdf | bin | 0 -> 438528 bytes | |||
| -rw-r--r-- | refs/dotsenko1991cluster.pdf | bin | 0 -> 172731 bytes | |||
| -rw-r--r-- | refs/du2006dynamic.pdf | bin | 0 -> 602685 bytes | |||
| -rw-r--r-- | refs/janke1998nonlocal.pdf | bin | 0 -> 269280 bytes | |||
| -rw-r--r-- | refs/lauwers1989critical.pdf | bin | 0 -> 195270 bytes | |||
| -rw-r--r-- | refs/liu2014dynamic.pdf | bin | 0 -> 2182262 bytes | |||
| -rw-r--r-- | refs/ray1990metastability.pdf | bin | 0 -> 432932 bytes | |||
| -rw-r--r-- | refs/swendsen1987nonuniversal.pdf | bin | 0 -> 136908 bytes | |||
| -rw-r--r-- | refs/wang1989clusters.pdf | bin | 0 -> 929783 bytes | |||
| -rw-r--r-- | refs/wang1990cluster.pdf | bin | 0 -> 652558 bytes | |||
| -rw-r--r-- | refs/wolff1989collective.pdf | bin | 0 -> 201013 bytes | |||
| -rw-r--r-- | refs/wolff1989comparison.pdf | bin | 0 -> 279147 bytes | |||
| -rw-r--r-- | refs/wolff1990critical.pdf | bin | 0 -> 945386 bytes |
23 files changed, 5674 insertions, 63 deletions
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${FUNCS:%=figs/%.dat} +%.tex: figs/%.gplot ${DATA:%=figs/%.dat} gnuplot $< > $@ ${DOC}.pdf: ${DOC}.tex ${DOC}.bib ${FIGS:%=%.tex} diff --git a/monte-carlo.bib b/monte-carlo.bib index 4413e2e..d527915 100644 --- a/monte-carlo.bib +++ b/monte-carlo.bib @@ -1,4 +1,59 @@ +@article{alexandrowicz1989swendsen, + title={Swendsen-Wang simulation of Ising spins and a precise definition of critical clusters}, + author={Alexandrowicz, Z}, + journal={Physica A: Statistical Mechanics and its Applications}, + volume={160}, + number={3}, + pages={310--320}, + year={1989}, + publisher={Elsevier} +} + +@article{coniglio1980clusters, + title={Clusters and Ising critical droplets: a renormalisation group approach}, + author={Coniglio, A and Klein, W}, + journal={Journal of Physics A: Mathematical and General}, + volume={13}, + number={8}, + pages={2775}, + year={1980}, + publisher={IOP Publishing} +} + +@article{coniglio1989exact, + title={Exact relations between droplets and thermal fluctuations in external field}, + author={Coniglio, A and de Liberto, F and Monroy, G and Peruggi, F}, + journal={Journal of Physics A: Mathematical and General}, + volume={22}, + number={17}, + pages={L837}, + year={1989}, + publisher={IOP Publishing} +} + +@article{destri1992swendsen, + title={Swendsen-Wang Monte Carlo study of the Ising model with external field}, + author={Destri, C and Di Renzo, F and Onofri, E and Rossi, P and Tecchiolli, GP}, + journal={Physics Letters B}, + volume={278}, + number={3}, + pages={311--316}, + year={1992}, + publisher={Elsevier} +} + +@article{dotsenko1991cluster, + title={Cluster Monte Carlo algorithms for random Ising models}, + author={Dotsenko, Vl S and Selke, W and Talapov, AL}, + journal={Physica A: Statistical Mechanics and its Applications}, + volume={170}, + number={2}, + pages={278--281}, + year={1991}, + publisher={Elsevier} +} + @article{du2006dynamic, title={Dynamic critical exponents for Swendsen--Wang and Wolff algorithms obtained by a nonequilibrium relaxation method}, author={Du, Jianqing and Zheng, Bo and Wang, Jian-Sheng}, @@ -10,6 +65,39 @@ publisher={IOP Publishing} } +@article{hastings1970monte, + title={Monte Carlo sampling methods using Markov chains and their applications}, + author={Hastings, W Keith}, + journal={Biometrika}, + volume={57}, + number={1}, + pages={97--109}, + year={1970}, + publisher={Biometrika Trust} +} + +@article{janke1998nonlocal, + title={Nonlocal Monte Carlo algorithms for statistical physics applications}, + author={Janke, Wolfhard}, + journal={Mathematics and computers in simulation}, + volume={47}, + number={2}, + pages={329--346}, + year={1998}, + publisher={Elsevier} +} + +@article{lauwers1989critical, + title={The critical 2D Ising model in a magnetic field. A Monte Carlo study using a Swendesen-Wang algorithm}, + author={Lauwers, Paul G and Rittenberg, Vladimir}, + journal={Physics Letters B}, + volume={233}, + number={1-2}, + pages={197--200}, + year={1989}, + publisher={Elsevier} +} + @article{liu2014dynamic, title={Dynamic scaling at classical phase transitions approached through nonequilibrium quenching}, author={Liu, Cheng-Wei and Polkovnikov, Anatoli and Sandvik, Anders W}, @@ -21,6 +109,61 @@ publisher={APS} } +@article{metropolis1953equation, + title={Equation of state calculations by fast computing machines}, + author={Metropolis, Nicholas and Rosenbluth, Arianna W and Rosenbluth, Marshall N and Teller, Augusta H and Teller, Edward}, + journal={The journal of chemical physics}, + volume={21}, + number={6}, + pages={1087--1092}, + year={1953}, + publisher={AIP} +} + +@article{ray1990metastability, + title={Metastability and nucleation in Ising models with Swendsen-Wang dynamics}, + author={Ray, TS and Wang, Jian-Sheng}, + journal={Physica A: Statistical Mechanics and its Applications}, + volume={167}, + number={3}, + pages={580--588}, + year={1990}, + publisher={Elsevier} +} + +@article{swendsen1987nonuniversal, + title={Nonuniversal critical dynamics in Monte Carlo simulations}, + author={Swendsen, Robert H and Wang, Jian-Sheng}, + journal={Physical review letters}, + volume={58}, + number={2}, + pages={86}, + year={1987}, + publisher={APS} +} + +@article{wang1989clusters, + title={Clusters in the three-dimensional Ising model with a magnetic field}, + author={Wang, Jian-Sheng}, + journal={Physica A: Statistical Mechanics and its Applications}, + volume={161}, + number={2}, + pages={249--268}, + year={1989}, + publisher={Elsevier} +} + +@article{wang1990cluster, + title={Cluster monte carlo algorithms}, + author={Wang, Jian-Sheng and Swendsen, Robert H}, + journal={Physica A: Statistical Mechanics and its Applications}, + volume={167}, + number={3}, + pages={565--579}, + year={1990}, + publisher={Elsevier} +} + @article{wolff1989collective, title={Collective Monte Carlo updating for spin systems}, author={Wolff, Ulli}, @@ -43,3 +186,13 @@ publisher={Elsevier} } +@article{wolff1990critical, + title={Critical slowing down}, + author={Wolff, Ulli}, + journal={Nuclear Physics B-Proceedings Supplements}, + volume={17}, + pages={93--102}, + year={1990}, + publisher={Elsevier} +} + diff --git a/monte-carlo.tex b/monte-carlo.tex index fe2a315..2490b31 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -82,7 +82,7 @@ \begin{document} -\title{An efficient cluster algorithm for the Ising model in an external field} +\title{Efficiently sampling Ising states in an external field} \author{Jaron Kent-Dobias} \author{James P.~Sethna} \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} @@ -90,15 +90,43 @@ \date\today \begin{abstract} - An abstract. + We introduce an extension of the Wolff algorithm that preforms efficiently + in an external magnetic field. Near the Ising critical point, the + correlation time of our algorithm has a conventional scaling form that + reduces to that of the Wolff algorithm at zero field. As an application, we + directly measure scaling functions of observables in the metastable state of + the 2D Ising model. \end{abstract} \maketitle -\section{Introduction} - The Ising model is a simple model of a magnet comprised of locally interacting -spins. +spins. Like most large thermal systems, computation of its properties cannot +be carried out explicitly and is preformed using Monte Carlo methods. Near its +continuous phase transition, divergent correlation length leads to divergent +correlation time in any locally-updating algorithm, hampering computation. +At zero external field, this was largely alleviated by cluster algorithms, +like the Wolff algorithm, whose dynamics are nonlocal and each step flips +groups of spins whose size diverges with the correlation length. However, the +Wolff algorithm only works at zero field. We describe an extension of this +algorithm that works in arbitrary external field while preserving the Wolff +algorithm's small dynamic exponent. + +The Wolff algorithm works by first choosing a random spin and adding it to an +empty cluster. Every neighbor of that spin pointed in the same direction as +the spin is added to the cluster with probability $1-e^{-2\beta J}$, where +$\beta=1/T$ and $J$ is the coupling between sites. This process is iterated +again for neighbors of every spin added to the cluster. When all sites +surrounding the cluster have been exhausted, the cluster is flipped. Our +algorithm is a simple extension of this. An extra spin is introduced (often +referred to as a ``ghost spin'') that couples to all others with coupling $H$. +The traditional Wolff algorithm is then preformed on this larger lattice exactly as described above, +with the extra spin treated no differently from any others. Observables in the +original system can be exactly estimated on the new one using a simple +mapping. As an application, we use our algorithm to measure critical scaling functions +of the 2D Ising model in its metastable phase. + +\section{Introduction} Consider an undirected graph $G=(V,E)$ describing a system of interacting spins. The set of vertices $V=\{1,\ldots,N\}$ enumerates the sites of the network, and the @@ -147,7 +175,8 @@ $S^n$ according to the Boltzmann distribution $e^{-\beta\H(s)}$, so that averages of observables made using their samples asymptotically approach the true expected value. -The Metropolis--Hastings algorithm is very popular for systems in statistical +The Metropolis--Hastings algorithm +\cite{metropolis1953equation,hastings1970monte} is very popular for systems in statistical physics. A starting state $s\in S^n$ is randomly perturbed to the state $s'$, usually by flipping one spin. The change in energy $\Delta\H=\H(s')-\H(s)$ due to the perturbation is then computed. If the change is negative the perturbed @@ -177,7 +206,7 @@ take many perturbations to move between in configuration space. \end{algorithm} \end{figure} -The Wolff algorithm solves many of these problems, but only at zero external +The Wolff algorithm \cite{wolff1989collective} solves many of these problems, but only at zero external field, $H=0$. This algorithm solves the problem of critical slowing-down by flipping carefully-constructed clusters of spins at once in a way that samples high-correlated states quickly while also always accepting prospective states. @@ -196,6 +225,9 @@ cluster acceptance or rejection across the spins in the cluster as they are added. In this version, the cluster is abandoned with probability $1-e^{-\beta H}$ every time a spin is added to it. +$z=0.29(1)$ \cite{wolff1989comparison,liu2014dynamic} $z=0.35(1)$ for +Swendsen--Wang \cite{swendsen1987nonuniversal} + \begin{figure} \begin{algorithm}[H] \begin{algorithmic} @@ -491,7 +523,7 @@ a given average if the associated state is in the reduced space of interest. 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