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-rw-r--r--.gitignore3
-rw-r--r--fig_correlation.tex3547
-rw-r--r--fig_correlation_collapse-hL.tex1878
-rw-r--r--figs/fig_correlation_data_collapse-hL.dat106
-rw-r--r--makefile2
-rw-r--r--monte-carlo.bib153
-rw-r--r--monte-carlo.tex48
-rw-r--r--refs/alexandrowicz1989swendsen.pdfbin0 -> 937546 bytes
-rw-r--r--refs/coniglio1980clusters.pdfbin0 -> 442501 bytes
-rw-r--r--refs/coniglio1989exact.pdfbin0 -> 463325 bytes
-rw-r--r--refs/destri1992swendsen.pdfbin0 -> 438528 bytes
-rw-r--r--refs/dotsenko1991cluster.pdfbin0 -> 172731 bytes
-rw-r--r--refs/du2006dynamic.pdfbin0 -> 602685 bytes
-rw-r--r--refs/janke1998nonlocal.pdfbin0 -> 269280 bytes
-rw-r--r--refs/lauwers1989critical.pdfbin0 -> 195270 bytes
-rw-r--r--refs/liu2014dynamic.pdfbin0 -> 2182262 bytes
-rw-r--r--refs/ray1990metastability.pdfbin0 -> 432932 bytes
-rw-r--r--refs/swendsen1987nonuniversal.pdfbin0 -> 136908 bytes
-rw-r--r--refs/wang1989clusters.pdfbin0 -> 929783 bytes
-rw-r--r--refs/wang1990cluster.pdfbin0 -> 652558 bytes
-rw-r--r--refs/wolff1989collective.pdfbin0 -> 201013 bytes
-rw-r--r--refs/wolff1989comparison.pdfbin0 -> 279147 bytes
-rw-r--r--refs/wolff1990critical.pdfbin0 -> 945386 bytes
23 files changed, 5674 insertions, 63 deletions
diff --git a/.gitignore b/.gitignore
index 7156c49..9325c3c 100644
--- a/.gitignore
+++ b/.gitignore
@@ -4,6 +4,7 @@
*.blg
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*.dvi
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diff --git a/fig_correlation.tex b/fig_correlation.tex
new file mode 100644
index 0000000..eb09910
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index edeac8e..fd6cc85 100644
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128 1415.8547204729814 0.16083831181683386 0.000038393852729745597
128 2243.9785079915646 0.14755136241680297 0.000010427848840532938
128 3556.4662613448777 0.13613167291690653 0.00003824559764080415
128 5636.619166823122 0.12770211002072657 0.000480391007107131
128 8933.43934599379 0.12140474759127326 0.000021786704140921986
-128 14158.547204729826 0.1304089776503813 0.
-128 22439.785079915637 0.11590410749694247 0.
-128 35564.66261344877 0.11053344320283695 0.
-128 56366.191668231244 0.11290716591905191 0.
-128 89334.39345993791 42.3724331482014 0.
+128 14158.547204729826 0.12450333905336947 0.005905638597011847
+128 22439.785079915637 0.11616358912822208 0.0002594816312796177
+128 35564.66261344877 0.11074819112465964 0.00021474792182268887
+128 56366.191668231244 0.11173511904556105 0.0011720468734908519
+128 89334.39345993791 43.80504345251086 1.432610304309456
256 0.32767999999811953 0.6075086695911571 0.002178147172579081
256 0.519337801301477 0.5847006386014801 0.0013928429465341339
256 0.8230949458839193 0.5812979760560055 0.0009786732427118916
diff --git a/makefile b/makefile
index 2a9a11a..364b659 100644
--- a/makefile
+++ b/makefile
@@ -5,7 +5,7 @@ DATA=fig_correlation_data_collapse-hL
all: ${DOC}.pdf
-%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${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.
Thanks!
\end{acknowledgments}
-%\bibliography{monte-carlo}
+\bibliography{monte-carlo}
\end{document}
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