The Monte Carlo Method in Condensed Matter Physics
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1 The Monte Carlo Method in Condensed Matter Physics Edited by K. Binder With Contributions by A. Baumgärtner K. Binder A.N. Burkitt D.M. Ceperley H. De Raedt A.M. Ferrenberg D.W. Heermann H.J. Herrmann D.P. Landau D. Levesque W. von der Linden J.D. Reger K.E. Schmidt W. Selke D. Stauffer R.H. Swendsen J.-S. Wang J.J. Weis A.P. Young Second, Corrected and Updated Edition With 83 Figures Springer
2 1. Introduction By K. Binder (With 9 Figures) General Remarks Progress in the Understanding of Finite Size Effects at Phase Transitions Asymmetrie First-Order Phase Transition Coexisting Phases Critical Phenomena Studies in the Microcanonical Ensemble Anisotropy Effects in Finite Size Scaling Statistical Errors Final Remarks 18 References Vectorisation of Monte Carlo Programs for Lattice Models Using Supercomputers By D.P. Landau (With 11 Figures) Introduction Technical Details Basic Principles Some "Dos" and Don'ts" of Vectorisation Simple Vectorisation Algorithms Vectorised Multispin Coding Algorithms Vectorised Multilattice Coding Algorithms Vectorised Microcanonical Algorithms Some Recent Results from Vectorised Algorithms Ising Model Critical Behaviour First-Order Transitions in Potts Models Dynamic Critical Behaviour Surface and Interface Phase Transitions Bulk Critical Behaviour in Classical Spin Systems Quantum Spin Systems Spin Exchange and Diffusion Impurity Systems Other Studies 47
3 X 2.8 Conclusion 49 References Parallel Algorithms for Statistical Physics Problems By D.W. Heermann and A.N. Burkitt (With 8 Figures) Paradigms of Parallel Computing Physics-Based Description 54 (a) Event Parallelism 55 (b) Geometrie Parallelism 56 (c) Algorithmic Parallelism Machine-Based Description 57 (a) SIMD Architecture 57 (b) MIMD Architecture 58 (c) The Connectivity 58 (d) Measurements of Machine Performance Applications on Fine-Grained SIMD Machines Spin Systems Molecular Dynamics Applications on Coarse-Grained MIMD Machines Molecular Dynamics Cluster Algorithms for the Ising Model Data Parallel Algorithms 69 (a) Long-Range Interactions 70 (b) Polymers Prospects 72 References New Monte Carlo Methods for Improved Efficiency of Computer Simulations in Statistical Mechanics By R.H. Swendsen, J.-S. Wang and A.M. Ferrenberg Overview Acceleration Algorithms Critical Slowing Down and Standard Monte Carlo Method Fortuin-Kasteleyn Transformation Swendsen-Wang Algorithm Further Developments Replica Monte Carlo Method Multigrid Monte Carlo Method Histogram Methods The Single-Histogram Method The Multiple-Histogram Method History and Applications Summary 89 References 89
4 XI 5. Simulation of Random Growth Processes By HJ. Herrmann (With 17 Figures) Irreversible Growth of Clusters A Simple Example of Cluster Growth: The Eden Model Laplacian Growth 96 (a) Moving Boundary Condition Problems 96 (b) Numerical Simulation of Dielectric Breakdown. and DLA 97 (c) Fracture Reversible Probabilistic Growth Cellular Automata Damage Spreading in the Monte Carlo Method Numerical Results for the Ising Model Heat Bath Versus Glauber Dynamics in the Ising Model Relationship Between Damage and Thermodynamic Properties Damage Clusters Damage in Spin Glasses More About Damage Spreading Conclusion 117 References Recent Progress in the Simulation of Classical Fluids By D. Levesque and J.J. Weis Improvements of the Monte Carlo Method Metropolis Algorithm Monte Carlo Simulations and Statistical Ensembles 123 (a) Canonical, Grand Canonical and Semi-grand Ensembles 123 (b) Gibbs Ensemble 124 (c) MC Algorithm for "Adhesive" Particles Monte Carlo Computation of the Chemical Potential and the Free Energy 126 (a) Chemical Potential 126 (b) Free Energy Algorithms for Coulombic and Dielectric Fluids Pure Phases and Mixtures of Simple Fluids Two-Dimensional Simple Fluids Three-Dimensional Monatomic Fluids Lennard-Jones Fluids and Similar Systems Real Fluids Mixtures of Simple Fluids 140
5 XII (a) Hard Core Systems 140 (b) LJ Mixtures 141 (c) Polydisperse Fluids Coulombic and Ionic Fluids One-Component Plasma, Two-Component Plasma and Primitive Models of Electrolyte Solutions (a) OCP and TCP 144 (b) Primitive Models Realistic Ionic Systems Simulations of Inhomogeneous Simple Fluids Liquid-Vapour Interfaces Fluid-Solid Interfaces Interfaces of Charged Systems Fluids in Narrow Pores Molecular Liquids: Model Systems Two-Dimensional Systems Convex Molecules (Three-Dimensional) 158 (a) Viral Coefficients and the Equation of State (b) Pair Distribution Function 160 (c) Phase Transitions Site-Site Potentials Chain Molecules Dipolar Systems Quadrupolar Systems Polarizable Polar Fluids Molecular Liquids: Realistic Systems Nitrogen (N 2 ) Halogens (Br 2, Cl 2, I 2 ) Benzene (C 6 H 6 ) Naphthalene (C 10 H 8 ) n-alkanes: CH 3 (CH 2 ) 2CH Water (H 2 0) Methanol (CH 3 OH) Other Polar Systems Mixtures Solutions Infinite Dilution Finite Concentration Polyelectrolytes and Micelles Interfaces in Molecular Systems Polar Systems 185 (a) Model Systems 186 (b) Realistic Systems Chain Molecules Confined by Hard Plates 190 References 191
6 XIII 7. Monte Carlo Techniques for Quantum Fluids, Solids and Droplets By K.E. Schmidt and D.M. Ceperley (With 12 Figures) Variational Method Variational Wavefunctions The Pair Product Wavefunction Three-Body Correlations Backflow Correlations Pairing Correlations Shadow Wavefunctions Wavefunction Optimisation Green's Function Monte Carlo and Related Methods Outline of the Method Fermion Methods Shadow Importance Functions Path Integral Monte Carlo Method PIMC Methodology The High Temperature Density Matrix Monte Carlo Algorithm Simple Metropolis Monte Carlo Method Normal Mode Methods Threading Algorithm Bisection and Staging Methods Sampling Permutations Calculation of the Energy Computation of the Superfluid Density Exchange in Quantum Crystals Comparison of GFMC with PIMC Applications Some Results for Bulk Helium He Results He Results Solid He Momentum and Related Distributions The Single-Particle Density Matrix y-scaling Momentum Distribution Results Droplets and Surfaces Ground States of He Droplets Excitations in Droplets He Droplets Droplets at Finite Temperature Surfaces and Interfaces Future Prospects 244 References 245
7 XIV 8. Quantum Lattice Problems By H. De Raedt and W. von der Linden (With 1 Figure) Overview Models Variational Monte Carlo Method Method and Trial Wavefunctions Results Green's Function Monte Carlo Method Method Results Grand Canonical Quantum Monte Carlo Method Method Applications Projector Quantum Monte Carlo Method Method Applications Fundamental Difficulties The Sign Problem Numerical Instabilities Dynamic Susceptibilities Applicability Concluding Remarks A Appendix 276 References Simulations of Macromolecules By A. Baumgärtner Techniques and Models Polymer Models 285 (a) Lattice Models 285 (b) Off-Lattice Models Monte Carlo Techniques 286 (a) Kink-Jump and Crankshaft Algorithm 286 (b) Reptation Algorithm 287 (c) General Reptation Algorithm 287 (d) Grand Canonical Reptation Algorithm 287 (e) Collective Reptation Method 287 (f) Pivot Algorithm 288 (g) Growth and Scanning Algorithms Amorphous Systems Dynamics of Polymers 289 (a) Polymer Melts 289 (b) Polymers in Flow 290 (c) Gel Electrophoresis The Glassy State 291
8 XV Equation of State Disorder Effects Polymer Chains in Random Media Effect of Disorder on Phase Transitions Diffusion in Disordered Media Mesomorphic Systems Hard Rods Semirigid Chains Anisotropie Interactions Networks Tethered Membranes Branched Polymers and Random Networks Segregation Collapse Transition Polymer Mixtures Dynamics of Decomposition Surfaces and Interfaces Adsorption on Rough Surfaces Entropie Repulsion Confined Polymer Melts Special Polymers Polyelectrolytes Proteins 309 (a) Protein Folding 309 (b) Protein Dynamics 310 References Percolation, Critical Phenomena in Dilute Magnets, Cellular Automata and Related Problems By D. Stauner (With 2 Figures) Percolation Dilute Ferromagnets Cellular Automata Multispin Programming of Cellular Automata Kauffman Model and da Silva-Herrmann Algorithm References Interfaces, Wetting Phenomena, Incommensurate Phases By W. Selke (With 12 Figures) Interfaces in Ising Models The Three-Dimensional Nearest-Neighbour Ising Model Alloys and Microemulsions Adsorbates and Two-Dimensional Systems Interfaces in Multistate Models 341
9 XVI 11.3 Dynamical Aspects Growth of Wetting Layers and Interfaces Domain Growth Spatially Modulated Structures Conclusions 351 References Spin Glasses, Orientational Glasses and Random Field Systems By A.P. Young, J.D. Reger and K. Binder (With 11 Figures) Spin Glasses The Spin Glass Transition The Edwards Anderson Model Phase Transitions The Low Temperature State The Vortex Glass Potts Glasses Introduction to Potts Glasses Mean-Field Theory The Critical Dimensions The Short-Range Potts Model 367 (a) Phenomenological T=0 Scaling 368 (b) Monte Carlo Simulations 368 (c) Transfer Matrix Calculations 369 (d) High-Temperature Series Expansions Orientational Glasses Introduction to Orientational Glasses Static and Dynamic Properties of the Isotropie Orientational Glass (m = 3) in Two and Three Dimensions More Realistic Models The Random-Field Ising Model Concluding Remarks and Outlook 379 References Recent Developments in the Monte Carlo Simulation of Condensed Matter By K. Binder, A. Baumgärtner, A.N. Burkitt, D. Ceperley, A.M. Ferrenberg, D.W. Heermann, H.J. Herrmann, D.P. Landau, W. von der Linden, H. De Raedt, K.E. Schmidt, W. Selke, D. Stauffer and A.P. Young Progress in the Understanding of Finite Size Effects Vectorization of Monte Carlo Programs for Lattice Models Using Supercomputers: New Results Parallel Algorithms for Statistical-Physics Problems: A Recent Perspective 391
10 XVII 13.4 New Monte Carlo Methods for Improved Efficiency: Recent Aspects Quantum Fluids, Solids and Droplets: Some Recent Results Quantum Lattice Problems: Recent Results Simulations of Polymers: Some Recent Results Percolation, Diluted Magnets, Cellular Automata: Some Recent Developments Interfaces, Wetting Phenomena, Incommensurate Phases: Some Recent Results Spin Glasses, Orientational Glasses, and Random Field Systems: Some Recent Results Simulation of Random Growth Processes: Some Comments on Recent Results Concluding Remarks 403 References 404 Subject Index 411
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