diff options
| -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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454.70001389847135 0.39891341844240713 0.18170524445924247 16 720.6509566396612 0.21741361507934745 0.00010145294308730456 @@ -43,18 +43,18 @@ 32 1.0523403171229262 0.6146304699498576 0.0029411128080925717 32 1.667847004760328 0.5757182738289438 0.0017891042515727899 32 2.643359363911546 0.519746074483992 0.00044408860844279895 -32 4.18944226109172 0.4670204427314064 0.001365894277698315 -32 6.639818519813105 0.42560291165019704 0.0015942680639849103 -32 10.523403171229258 0.38949748338986645 0.00240252809173605 -32 16.678470047604744 0.3542902285727013 0.0006252639838920104 -32 26.433593639115465 0.32260194466630127 0.00013593524267008217 -32 41.8944226109172 0.29316056163502646 0.0008711389856035645 -32 66.39818519813163 0.2673108060826328 0.0002561260604034554 -32 105.23403171229113 0.2439722389470263 0.00041353736622222465 -32 166.78470047604742 0.22408499741937016 0.0009748993257848032 -32 264.33593639115344 0.20622295754570164 0.0003192570981769976 -32 418.9442261091729 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0.00014040830435756158 32 2643.359363911534 0.17612308355378462 0.00008859092338167617 @@ -73,18 +73,18 @@ 64 2.435496171525549 0.5305996387485918 0.0014959230318421645 64 3.8600013025160367 0.48557922109035934 0.0036751832028803612 64 6.117689787248085 0.44044962601219084 0.0018857447024898469 -64 9.695884897399262 0.4056083464205059 0.0030207911414076583 -64 15.36694196877451 0.37034496808040485 0.0011285047217174937 -64 24.35496171525549 0.3410597472226905 0.001984517358535637 -64 38.60001302516036 0.3119763635141043 0.0015139424073597196 -64 61.176897872486215 0.28648946213147886 0.0010216188795957037 -64 96.95884897399264 0.2606451714862178 0.0006902931938541912 -64 153.6694196877451 0.23837531396287037 0.0007131188988310638 -64 243.54961715255703 0.2166054845624757 0.00014709214343654187 -64 386.00013025159825 0.1984726135248987 0.0006330601894378868 -64 611.7689787248622 0.18185027101891627 0.0001649096826504964 -64 969.588489739922 0.16742136264578164 0.0002487946153808817 -64 1536.6941968774543 0.1565974673693383 0.00004431211808114676 +64 9.695884897399262 0.40563766234685533 0.0017443009479081834 +64 15.36694196877451 0.37085340478832524 0.0008264475310672823 +64 24.35496171525549 0.34124393616706267 0.001160472008568937 +64 38.60001302516036 0.3119031918247171 0.0008771324303633754 +64 61.176897872486215 0.28632943948682943 0.0006111537928509062 +64 96.95884897399264 0.2605250393511181 0.0004162530813228257 +64 153.6694196877451 0.23782958962548653 0.0006836138581365798 +64 243.54961715255703 0.21664193300132426 0.00009241494241417804 +64 386.00013025159825 0.19947059323368097 0.0010628037920008965 +64 611.7689787248622 0.1817052030857358 0.00017352167889887098 +64 969.588489739922 0.1674466007578578 0.0001458419778916332 +64 1536.6941968774543 0.15661382019143888 0.00003036340001161581 64 2435.496171525572 0.15118568965616075 0.001975073557340154 64 3860.001302515986 0.1448337200806049 0.0002595070942839617 64 6117.68978724862 0.367371300110232 0.22380440827218762 @@ -100,28 +100,28 @@ 128 1.415854720473395 0.5725642835937848 0.00014026941966266887 128 2.2439785079905805 0.5272959382383 0.0025756573485712577 128 3.5564662613452867 0.48174955931471536 0.0005001598678584231 -128 5.636619166821931 0.44491238376473485 0.0003661776347952949 -128 8.933439345993705 0.4113572807767111 0.0024356478537859434 -128 14.158547204730013 0.3817803172768068 0.00008389740373021536 -128 22.439785079913676 0.353218055274667 0.0015682350992511638 -128 35.56466261344893 0.3254983052094862 0.0011331832832819908 -128 56.36619166823111 0.3003941905165709 0.0008062274638504784 -128 89.33439345993705 0.27427695641426886 0.001109371145218918 -128 141.5854720473001 0.25338014351418314 0.0012608266730329276 -128 224.39785079915646 0.2312919586007066 0.0005096147290300223 -128 355.6466261344894 0.21077098134013394 0.00039002269998274214 -128 563.6619166823111 0.1927527400715816 0.00027052134638859415 -128 893.3439345993783 0.17525595209598482 0.00013628284241916366 +128 5.636619166821931 0.4438388073898844 0.0010941944918729596 +128 8.933439345993705 0.41092720925776854 0.0014705174828234917 +128 14.158547204730013 0.38121192764431805 0.0005704498509406503 +128 22.439785079913676 0.3529764733474239 0.0009370960123763205 +128 35.56466261344893 0.32488640337259517 0.0008958005595521174 +128 56.36619166823111 0.3006193023569792 0.0005170521397121607 +128 89.33439345993705 0.27456708026595844 0.0007031405468330241 +128 141.5854720473001 0.2536271681032865 0.0007687104661479951 +128 224.39785079915646 0.23100431012855005 0.000411473815548584 +128 355.6466261344894 0.21147268527569854 0.00073694933024878 +128 563.6619166823111 0.1931657687811568 0.0004415729247632602 +128 893.3439345993783 0.17567152403394282 0.00042295512764053896 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 @@ -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} diff --git a/refs/alexandrowicz1989swendsen.pdf b/refs/alexandrowicz1989swendsen.pdf Binary files differnew file mode 100644 index 0000000..08c1c5e --- /dev/null +++ b/refs/alexandrowicz1989swendsen.pdf diff --git a/refs/coniglio1980clusters.pdf b/refs/coniglio1980clusters.pdf Binary files differnew file mode 100644 index 0000000..9ab9c4a --- /dev/null +++ b/refs/coniglio1980clusters.pdf diff --git a/refs/coniglio1989exact.pdf b/refs/coniglio1989exact.pdf Binary files differnew file mode 100644 index 0000000..1b145fa --- /dev/null +++ b/refs/coniglio1989exact.pdf diff --git a/refs/destri1992swendsen.pdf b/refs/destri1992swendsen.pdf Binary files differnew file mode 100644 index 0000000..591c8f6 --- /dev/null +++ b/refs/destri1992swendsen.pdf diff --git a/refs/dotsenko1991cluster.pdf b/refs/dotsenko1991cluster.pdf Binary files differnew file mode 100644 index 0000000..599984e --- /dev/null +++ b/refs/dotsenko1991cluster.pdf diff --git a/refs/du2006dynamic.pdf b/refs/du2006dynamic.pdf Binary files differnew file mode 100644 index 0000000..685ba66 --- /dev/null +++ b/refs/du2006dynamic.pdf diff --git a/refs/janke1998nonlocal.pdf b/refs/janke1998nonlocal.pdf Binary files differnew file mode 100644 index 0000000..f2666e0 --- /dev/null +++ b/refs/janke1998nonlocal.pdf diff --git a/refs/lauwers1989critical.pdf b/refs/lauwers1989critical.pdf Binary files differnew file mode 100644 index 0000000..09063c5 --- /dev/null +++ b/refs/lauwers1989critical.pdf diff --git a/refs/liu2014dynamic.pdf b/refs/liu2014dynamic.pdf Binary files differnew file mode 100644 index 0000000..af17e6a --- /dev/null +++ b/refs/liu2014dynamic.pdf diff --git a/refs/ray1990metastability.pdf b/refs/ray1990metastability.pdf Binary files differnew file mode 100644 index 0000000..f7f1e2f --- /dev/null +++ b/refs/ray1990metastability.pdf diff --git a/refs/swendsen1987nonuniversal.pdf b/refs/swendsen1987nonuniversal.pdf Binary files differnew file mode 100644 index 0000000..6c359eb --- /dev/null +++ b/refs/swendsen1987nonuniversal.pdf diff --git a/refs/wang1989clusters.pdf b/refs/wang1989clusters.pdf Binary files differnew file mode 100644 index 0000000..c06de0b --- /dev/null +++ b/refs/wang1989clusters.pdf diff --git a/refs/wang1990cluster.pdf b/refs/wang1990cluster.pdf Binary files differnew file mode 100644 index 0000000..d308d39 --- /dev/null +++ b/refs/wang1990cluster.pdf diff --git a/refs/wolff1989collective.pdf b/refs/wolff1989collective.pdf Binary files differnew file mode 100644 index 0000000..6407658 --- /dev/null +++ b/refs/wolff1989collective.pdf diff --git a/refs/wolff1989comparison.pdf b/refs/wolff1989comparison.pdf Binary files differnew file mode 100644 index 0000000..4465d2e --- /dev/null +++ b/refs/wolff1989comparison.pdf diff --git a/refs/wolff1990critical.pdf b/refs/wolff1990critical.pdf Binary files differnew file mode 100644 index 0000000..6526c0f --- /dev/null +++ b/refs/wolff1990critical.pdf |