Computational Many-Particle Physics

774 Pages·2008·8.98 MB·English

by  Ralf Schneider

Lecture Notes in Physics
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H. Fehske
R. Schneider
A. Weiße (Eds.)
Computational
Many-Particle Physics
Editors
HolgerFehske RalfSchneider
AlexanderWeiße Max-Planck-Institutfu¨rPlasmaphysik
Universita¨tGreifswald Wendelsteinstr.1
Institutfu¨rPhysik 17491Greifswald,Germany
Felix-Hausdorff-Str.6 [email protected]
17489Greifswald,
Germany
[email protected]
[email protected]
H. Fehske, R. Schneider and A. Weiße (Eds.), Computational Many-Particle Physics,
Lect.NotesPhys. 739(Springer,BerlinHeidelberg2008),DOI10.1007/978-3-540-
74686-7
LibraryofCongressControlNumber:2007936165
ISSN0075-8450
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Preface
Many-particlephysicsisatworkwheneverwedelveintotherichphenomenologyof
the realworld, or into laboratoryexperiments.Nevertheless,our physicaldescrip-
tion of nature is mostly built upon single-particle theories. For instance, Kepler’s
laws provide a basic understandingof our solar system, many features of the pe-
riodic table can be understood from the solution of a single hydrogen atom, and
even complicated microprocessors with an unbearable number of electrons float-
ingthroughmillionsof transistorscanbe developedbasedontheeffectivesingle-
particlemodelsofsemiconductorphysics.Theseapproachesaresuccessfulbecause
quiteofteninteractionsaffectphysicalsystemsinaperturbativeway.Classicalper-
turbationtheoryyieldscorrectionstoaplanet’sorbitduetootherplanets,quantum
chemistryreliesonvariousapproximationschemestodealwithcomplicatedatoms
and small molecules, and solid state theory uses weakly interacting quasiparticles
as elementaryexcitations.This fortunatesituation changes,however,when we try
tounderstandmorecomplexorstronglyinteractingsystems,orwhenwetrytoex-
plorethe nature of matteritself. Condensatesof cold bosonicatoms, for example,
show subtle many-particle effects, strongly correlated fermions may give rise to
high-temperaturesuperconductivity,and the way quarks build up elementary par-
ticles (hadronization)is a highly non-trivial few-body problem. Another example
are quantum computers, which many scientist envision as a replacement for our
present-daymicroprocessors,andwhichexploittheentanglementpropertyofquan-
tum many-particlestates. Last but not least, we mention the complexity of fusion
plasmas,whichsomedaymayhelpfeedingourever-growinghungerfornewenergy
resources.Unfortunately,eventhemostsophisticatedanalyticalapproacheslargely
failtodescribesuchsystems.Hence,atpresent,unbiasednumericalinvestigations
providethemostreliabletooltoaddresstheseproblems.Thisisthepointwherethe
expertuseoflarge-scalecomputerscomesintoplay.
The increasing importance of computational many-particle physics calls for a
comprehensiveintroductionintothisrapidlydevelopingfieldsuitableforgraduate
studentsandyoungresearchers.Therefore,wedecidedtoorganizeasummerschool
on “Computational Many-Particle Physics” in September 2006, during the 550th
anniversary of the University Greifswald. Generously sponsored by the Wilhelm
and Else Heraeus Foundation and hosted by the Max-Planck-Institute for Plasma
Physics and the Institute for Physics, we brought together more than 40 students
and 20 distinguished scientists working on such diverse fields as fusion plasmas,
VI Preface
statisticalphysics,solidstatetheoryandhighperformancecomputing.Thepresent
LectureNotessummarizeandextendthematerialshowcasedovera2-weekperiod
oftightlyscheduledtutorials,seminarsandexercises.Theemphasisisonaveryped-
agogicalandsystematicintroductiontovariousnumericalconceptsandtechniques,
withthehopethatthereadermayquicklystarttoprogramhimself.Thespectrumof
thenumericalmethodspresentedisverybroad,coveringclassicalaswellasquan-
tumfew-andmany-particlesystems.Thetrade-offbetweenthenumberofparticles,
thecomplexityoftheunderlyingmicroscopicmodelsandtheimportanceofthein-
teractionsdeterminethe choice of the appropriatenumericalapproach.Therefore,
wearrangedthebookalongthealgorithmsandtechniquesemployed,ratherthanon
the physics applications, which we think is more naturalfor a book on numerical
methods.
Westartwithmethodsforclassicalmany-particlesystems.Here,moleculardy-
namicsapproachestracethemotionofindividualparticles,kineticapproacheswork
withthe distributionfunctionsofparticlesandmomenta,while hybridapproaches
combine both concepts. A prominent example is the particle-in-cell method typi-
callyappliedtomodelplasmas,wherethetimeevolutionofdistributionfunctionsis
approximatedbythedynamicsofpseudo-particles,representingthousandsormil-
lions of real particles. Of course, at a certain length scale the quantum nature of
the particles becomes important. As an attempt to close the gap between classi-
calandquantumsystems,weoutlineanumberofsemi-classical(Wigner-function,
Boltzmann- and Vlasov-equation based) approaches, which in particular address
transport properties. The concept of Monte Carlo sampling is equally important
for classical, statistical and quantum physicalproblems. The correspondingchap-
ters therefore account for a substantial part of the book and introduce the major
stochasticapproachesinapplicationtoverydifferentphysicalsituations.Focussing
on solids and their properties, we continue with ab initio approaches to the elec-
tronic structure problem,where band structure effects are taken into accountwith
full detail, but Coulomb interactions and the resulting correlations are treated ap-
proximately.Dynamicalmeanfieldtheoriesandclusterapproachesaimatimprov-
ingthedescriptionofcorrelationsandbridgethegaptoanexactnumericaltreatment
of basic microscopic models. Exact diagonalizationof finite systems givesaccess
totheirground-state,spectralandthermodynamicproperties.Sincethesemethods
workwiththefullmany-particleHamiltonian,thestudyofadecentnumberofpar-
ticles or larger system sizes is a challenging task, and there is a strong demand
tocircumventthese limitations.Alongthislinethedensitymatrixrenormalization
group represents a clever technique to restrict the many-particle Hilbert space to
the physically most important subset. Finally, all the discussed methods heavily
relyontheuseofpowerfulcomputers,andthebookwouldbeincompletewithout
twodetailedchaptersonparallelprogrammingandoptimizationtechniquesforhigh
performancecomputing.
Ofcourse,thepreparationofsuchacomprehensivebookwouldhavebeenim-
possiblewithoutsupportfrommanycolleaguesandsponsors.Firstofall,wethank
the lecturers and authors for their engagement, enthusiasm and patience. We are
Preface VII
greatly indebted to Milena Pfafferottand Andrea Pulss for their assistance during
theeditorialworkandthefine-tuningofthearticles.JuttaGauger,BeateKemnitz,
ThomasMeyerandGeraldSchubertdidaninvaluablejobintheorganizationofthe
summer school. Finally, we acknowledgefinancial supportfrom the Wilhelm and
Else Heraeusfoundation,the Deutsche ForschungsgemeinschaftthroughSFB 652
andTR24andtheHelmholtz-GemeinschaftthroughCOMAS.
Greifswald, HolgerFehske
July2007 RalfSchneider
AlexanderWeiße
Contents
PartI MolecularDynamics
1 IntroductiontoMolecularDynamics
RalfSchneider,AmitRajSharma,andAbhaRai ........................ 3
1.1 BasicApproach ............................................... 3
1.2 MacroscopicParameters........................................ 6
1.3 Inter-AtomicPotentials......................................... 8
1.4 NumericalIntegrationTechniques................................ 14
1.5 AnalysisofMDRuns .......................................... 18
1.6 FromClassicaltoQuantum-MechanicalMD....................... 23
1.7 AbInitioMD ................................................. 24
1.8 Car-ParrinelloMolecularDynamics .............................. 25
1.9 PotentialEnergySurface ....................................... 28
1.10 AdvancedNumericalMethods................................... 29
References ......................................................... 37
2 WignerFunctionQuantumMolecularDynamics
V.S.Filinov,M.Bonitz,A.Filinov,andV.O.Golubnychiy ................ 41
2.1 QuantumDistributionFunctions ................................. 41
2.2 SemiclassicalMolecularDynamics............................... 43
2.3 QuantumDynamics............................................ 50
2.4 TimeCorrelationFunctionsintheCanonicalEnsemble.............. 54
2.5 Discussion ................................................... 58
References ......................................................... 59
PartII ClassicalMonteCarlo
3 TheMonteCarloMethod,anIntroduction
DetlevReiter .................................................... 63
3.1 WhatisaMonteCarloCalculation? .............................. 63
3.2 RandomNumberGeneration .................................... 67
3.3 IntegrationbyMonteCarlo ..................................... 71
3.4 Summary .................................................... 77
References ......................................................... 78
X Contents
4 MonteCarloMethodsinClassicalStatisticalPhysics
WolfhardJanke .................................................. 79
4.1 Introduction .................................................. 79
4.2 StatisticalPhysicsPrimer ....................................... 80
4.3 TheMonteCarloMethod....................................... 85
4.4 ClusterAlgorithms ............................................ 93
4.5 StatisticalAnalysisofMonteCarloData .......................... 99
4.6 ReweightingTechniques........................................108
4.7 Finite-SizeScalingAnalysis ....................................114
4.8 GeneralizedEnsembleMethods..................................129
4.9 ConcludingRemarks...........................................135
References .........................................................135
5 TheMonteCarloMethodforParticleTransportProblems
DetlevReiter .................................................... 141
5.1 TransportProblemsandStochasticProcesses ......................141
5.2 TheTransportEquation:FredholmIntegral
EquationofSecondKind .......................................143
5.3 TheBoltzmannEquation .......................................144
5.4 TheLinearIntegralEquationfortheCollisionDensity ..............147
5.5 MonteCarloSolution ..........................................150
5.6 SomeSpecialSamplingTechniques ..............................154
5.7 AnIllustrativeExample ........................................156
References .........................................................158
PartIII KineticModelling
6 TheParticle-in-CellMethod
DavidTskhakaya................................................. 161
6.1 GeneralRemarks..............................................161
6.2 IntegrationofEquationsofParticleMotion ........................163
6.3 PlasmaSourceandBoundaryEffects .............................166
6.4 CalculationofPlasmaParametersandFields
ActingonParticles ............................................170
6.5 SolutionofMaxwell’sEquations.................................175
6.6 ParticleCollisions .............................................183
6.7 FinalRemarks ................................................188
References .........................................................188
7 Gyrokinetic and Gyrofluid Theory and Simulation
ofMagnetizedPlasmas
RichardD.Sydora................................................ 191
7.1 Introduction ..................................................191
7.2 SingleParticleDynamics .......................................193
7.3 ContinuumGyrokinetics........................................200
Contents XI
7.4 GyrofluidModel ..............................................204
7.5 GyrokineticParticleSimulationModel............................207
7.6 GyrokineticParticleSimulationModelApplications ................210
7.7 Summary ....................................................217
References .........................................................218
PartIV SemiclassicalApproaches
8 BoltzmannTransportinCondensedMatter
FranzXaverBronold.............................................. 223
8.1 BoltzmannEquationforQuasiparticles ...........................223
8.2 TechniquesfortheSolutionoftheBoltzmannEquation..............230
8.3 Conclusions ..................................................252
References .........................................................253
9 SemiclassicalDescription of QuantumMany-ParticleDynamics
inStrongLaserFields
ThomasFennelandJo¨rgKo¨hn...................................... 255
9.1 SemiclassicalMany-ParticleDynamics
inMean-FieldApproximation ...................................255
9.2 SemiclassicalGroundState .....................................261
9.3 ApplicationtoSimple-MetalClusters.............................265
References .........................................................272
PartV QuantumMonteCarlo
10 World-line andDeterminantalQuantumMonte CarloMethods
forSpins,PhononsandElectrons
F.F.AssaadandH.G.Evertz........................................ 277
10.1 Introduction ..................................................277
10.2 DiscreteImaginaryTimeWorldLines
fortheXXZSpinChain ........................................278
10.3 World-LineRepresentationswithoutDiscretizationError ............299
10.4 LoopOperatorRepresentation
oftheHeisenbergModel .......................................303
10.5 Spin-PhononSimulations.......................................308
10.6 AuxiliaryFieldQuantumMonteCarloMethods....................312
10.7 NumericalStabilizationSchemesforLatticeModels ................325
10.8 TheHirsch-FyeImpurityAlgorithm..............................337
10.9 SelectedApplicationsoftheAuxiliaryFieldMethod ................344
10.10 Conclusion...................................................345
10.A TheTrotterDecomposition .....................................345