AGuideto Monte Carlo Simulations in Statistical Physics

Transcription

1 AGuideto Monte Carlo Simulations in Statistical Physics Second Edition David P. Landau Center for Simulational Physics, The University of Georgia Kurt Binder Institut für Physik, Johannes-Gutenberg-Universität Mainz,«J CAMBRIDGE 5= " UNIVERSITY PRESS

2 Preface page 1 Introduction What is a Monte Carlo Simulation? What problems can we solve with it? What difficulties will we encounter? Limited Computer time and memory Statistical and other errors What strategy should we follow in approaching a problem? How do simulations relate to theory and experiment? Perspective 6 2 Some necessary background Thermodynamics and Statistical mechanics: a quick reminder Basic notions Phase transitions Ergodicity and broken symmetry Fluctuations and the Ginzburg criterion A Standard exercise: the ferromagnetic Ising model Probability theory Basic notions Special probability distributions and the central limit theorem Statistical errors Markov chains and master equations The 'art' of random number generation Non-equilibrium and dynamics: some introductory comments Physical applications of master equations Conservation laws and their consequences Critical slowing down at phase transitions Transport coefficients Concluding comments 45 References 45 3 Simple sampling Monte Carlo methods Introduction Comparisons of methods for numerical integration of given functions 48

3 vi Simple methods Intelligent methods SO 3.3 Boundary value problems Simulation of radioactive decay Simulation of transport properties Neutron transport Fluid flow The percolation problem Site percolation Cluster counting: the Hoshen Kopelman algorithm Other percolation modeis Finding the groundstate of a Hamiltonian Generation of 'random' walks Introduction Random walks Self-avoiding walks Growing walks and other modeis Final remarks 66 References 66 4 Importance sampling Monte Carlo methods Introduction The simplest case: single spin-flip sampling for the simple Ising model Algorithm Boundary conditions Finite size effects Finite sampling time effects Critical relaxation Other discrete variable modeis Ising modeis with competing interactions state Potts modeis Baxter and Baxter-Wu modeis Clock modeis Ising spin glass modeis Complex fluid modeis Spin-exchange sampling Constant magnetization simulations Phase Separation Diffusion Hydrodynamic slowing down Microcanonical methods Demon algorithm Dynamic ensemble Q2R General remarks, choice of ensemble 122

4 vü 4.7 Stades and dynamics of polymer modeis on lattices Background Fixed bond length methods Bond fluetuation method Enhanced sampling using a fourth dimension The 'wormhole algorithm' - another method to equilibrate dense polymeric Systems Polymers in Solutions of variable quality: ö-point, collapse transition, unmixing Equilibrium polymers: a case study Some advice 133 References 134 More on importance sampling Monte Carlo methods for lattice Systems Cluster flipping methods Fortuin-Kasteleyn theorem Swendsen-Wang method Wolff method 'Improved estimators' Invaded Cluster algorithm Probability changing Cluster algorithm Specialized computational techniques Expanded ensemble methods Multispin coding iv-fold way and extensions Hybrid algorithms Multigrid algorithms Monte Carlo on vector Computers Monte Carlo on parallel Computers Classical spin modeis Introduction Simple spin-flip method Heatbath method Low temperature techniques Over-relaxation methods Wolff embedding trick and Cluster flipping Hybrid methods Monte Carlo dynamics vs. equation of motion dynamics Topological excitations and solitons Systems with quenched randomness General comments: averaging in random Systems Parallel tempering: a general method to better equilibrate Systems with complex energy landscapes Random fields and random bonds Spin glasses and optimization by simulated annealing 165

5 viii Ageing in spin glasses and related Systems Vector spin glasses: developments and surprises Models with mixed degrees of freedom: Si/Ge alloys, a case study Sampling the free energy and entropy Thermodynamic Integration Groundstate free energy determination Estimation of intensive variables: the chemical potential Lee-Kosterlitz method Free energy from finite size dependence at T c Miscellaneous topics Inhomogeneous Systems: surfaces, interfaces, etc Other Monte Carlo schemes Inverse Monte Carlo methods Finite size effects: a review and summary More about error estimation Random number generators revisited Summary and perspective 189 References Off-lattice modeis Fluids NVT ensemble and the virial theorem NpT ensemble Grand canonical ensemble Near critical coexistence: a case study Subsystems: a case study Gibbs ensemble Widom particle insertion method and variants Monte Carlo Phase Switch Cluster algorithm for fluids 'Short ränge' interactions Cutoffs Verlet tables and cell structure Minimum image Convention Mixed degrees of freedom reconsidered Treatment of long ränge forces Reaction field method Ewald method Fast multipole method Adsorbed monolayers Smooth Substrates Periodic Substrate potentials Complex fluids Application of the Liu-Luijten algorithm to a binary fluid mixture 230

6 ix 6.6 Polymers: an introduction Length scales and modeis Asymmetrie polymer mixtures: a case study Applications: dynamics of polymer melts; thin adsorbed Polymerie films Configurational bias and 'smart Monte Carlo' 245 References Reweighting methods Background Distribution funetions Umbrella sampling Single histogram method: the Ising model as a case study Multi-histogram method Broad histogram method Transition matrix Monte Carlo Multicanonical sampling The multicanonical approach and its relationship to canonical sampling Near first order transitions Groundstates in complicated energy landscapes Interface free energy estimation A case study: the Casimir effect in critical Systems 'Wang-Landau sampling' A case study: evaporation/condensation transition of droplets 273 References Quantum Monte Carlo methods Introduction Feynman path integral formulation Off-lattice problems: low-temperature properties of crystals Böse statistics and superfluidity Path integral formulation for rotational degrees of freedom Lattice problems The Ising model in a transverse field Anisotropie Heisenberg chain Fermions on a lattice An intermezzo: the minus sign problem Spinless fermions revisited Cluster methods for quantum lattice modeis Continuous time simulations Decoupled cell method Handscomb's method Wang-Landau sampling for quantum modeis Fermion determinants Monte Carlo methods for the study of groundstate properties Variational Monte Carlo (VMC) 308

7 x Green's function Monte Carlo methods (GFMC) Concluding remarks 311 References Monte Carlo renormalization group methods Introduction to renormalization group theory Real space renormalization group Monte Carlo renormalization group Large cell renormalization Ma's method: finding critical exponents and the fixed point Hamiltonian Swendsen's method Location of phase boundaries Dynamic problems: matching time-dependent correlation functions Inverse Monte Carlo renormalization group transformations 327 References Non-equilibrium and irreversible processes Introduction and perspective Driven diffusive Systems (driven lattice gases) Crystal growth Domain growth Polymer growth Linear polymers Gelation Growth of structures and patterns Eden model of Cluster growth Diffusion limited aggregation Cluster-cluster aggregation Cellular automata Models for film growth Background Ballistic deposition Sedimentation Kinetic Monte Carlo and MBE growth Transition path sampling Outlook: variations on a theme 348 References Lattice gauge modeis: a brief introduction Introduction: gauge invariance and lattice gauge theory Some technical matters Results for Z(N) lattice gauge modeis Compact U(l) gauge theory SU(2) lattice gauge theory 354

8 xi 11.6 Introduction: quantum chromodynamics (QCD) and phase transitions of nuclear matter The deconfinement transition of QCD Where are we now? 360 References A brief review of other methods of Computer Simulation Introduction Molecular dynamics Integration methods (microcanonical ensemble) Other ensembles (constant temperature, constant pressure, etc.) Non-equilibrium molecular dynamics Hybrid methods (MD + MC) Abinitio molecular dynamics Quasi-classical spin dynamics Langevin equations and variations (cell dynamics) Micromagnetics Dissipative particle dynamics (DPPD) Lattice gas cellular automata Lattice Boltzmann Equation Multiscale Simulation 379 References Monte Carlo methods outside of physics Commentary Protein folding Introduction Generalized ensemble methods Globular proteins: a case study 'Biologically inspired physics' Mathematics/statistics Sociophysics Econophysics 'Traffic' simulations Medicine 391 References Outlook 393 Appendix: listing of programs mentioned in the text 395 Index 427