E-Book, Englisch, Band 71, 393 Seiten, eBook
Reihe: Topics in Applied Physics
Binder The Monte Carlo Method in Condensed Matter Physics
1992
ISBN: 978-3-662-02855-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, Band 71, 393 Seiten, eBook
Reihe: Topics in Applied Physics
ISBN: 978-3-662-02855-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark
The Monte Carlo method is now widely used and commonly accepted as an important and useful tool in solid state physics and related fields. It is broadly recognized that the technique of "computer simulation" is complementary to both analytical theory and experiment, and can significantly contribute to ad vancing the understanding of various scientific problems. Widespread applications of the Monte Carlo method to various fields of the statistical mechanics of condensed matter physics have already been reviewed in two previously published books, namely Monte Carlo Methods in Statistical Physics (Topics Curro Phys. , Vol. 7, 1st edn. 1979, 2ndedn. 1986) and Applications of the Monte Carlo Method in Statistical Physics (Topics Curro Phys. , Vol. 36, 1st edn. 1984, 2nd edn. 1987). Meanwhile the field has continued its rapid growth and expansion, and applications to new fields have appeared that were not treated at all in the above two books (e. g. studies of irreversible growth phenomena, cellular automata, interfaces, and quantum problems on lattices). Also, new methodic aspects have emerged, such as aspects of efficient use of vector com puters or parallel computers, more efficient analysis of simulated systems con figurations, and methods to reduce critical slowing down at i>hase transitions. Taken together with the extensive activity in certain traditional areas of research (simulation of classical and quantum fluids, of macromolecular materials, of spin glasses and quadrupolar glasses, etc.
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1. Introduction.- 1.1 General Remarks.- 1.2 Progress in the Understanding of Finite Size Effects at Phase Transitions.- 1.2.1 Asymmetric First-Order Phase Transition.- 1.2.2 Coexisting Phases.- 1.2.3 Critical Phenomena Studies in the Microcanonical Ensemble.- 1.2.4 Anisotropy Effects in Finite Size Scaling.- 1.3 Statistical Errors.- 1.4 Final Remarks.- References.- 2. Vectorisation of Monte Carlo Programs for Lattice Models Using Supercomputers.- 2.1 Introduction.- 2.2 Technical Details.- 2.2.1 Basic Principles.- 2.2.2 Some “Dos” and “Don’ts” of Vectorisation.- 2.3 Simple Vectorisation Algorithms.- 2.4 Vectorised Multispin Coding Algorithms.- 2.5 Vectorised Multilattice Coding Algorithms.- 2.6 Vectorised Microcanonical Algorithms.- 2.7 Some Recent Results from Vectorised Algorithms.- 2.7.1 Ising Model Critical Behaviour.- 2.7.2 First-Order Transitions in Potts Models.- 2.7.3 Dynamic Critical Behaviour.- 2.7.4 Surface and Interface Phase Transitions.- 2.7.5 Bulk Critical Behaviour in Classical Spin Systems.- 2.7.6 Quantum Spin Systems.- 2.7.7 Spin Exchange and Diffusion.- 2.7.8 Impurity Systems.- 2.7.9 Other Studies.- 2.8 Conclusion.- References.- 3. Parallel Algorithms for Statistical Physics Problems.- 3.1 Paradigms of Parallel Computing.- 3.1.1 Physics-Based Description.- (a) Event Parallelism.- (b) Geometric Parallelism.- (c) Algorithmic Parallelism.- 3.1.2 Machine-Based Description.- (a) SIMD Architecture.- (b) MIMD Architecture.- (c) The Connectivity.- (d) Measurements of Machine Performance.- 3.2 Applications on Fine-Grained SIMD Machines.- 3.2.1 Spin Systems.- 3.2.2 Molecular Dynamics.- 3.3 Applications on Coarse-Grained MIMD Machines.- 3.3.1 Molecular Dynamics.- 3.3.2 Cluster Algorithms for the Ising Model.- 3.3.3 Data Parallel Algorithms.- (a) Long-Range Interactions.- (b) Polymers.- 3.4 Prospects.- References.- 4. New Monte Carlo Methods for Improved Efficiency of Computer Simulations in Statistical Mechanics.- 4.1 Overview.- 4.2 Acceleration Algorithms.- 4.2.1 Critical Slowing Down and Standard Monte Carlo Method.- 4.2.2 Fortuin—Kasteleyn Transformation.- 4.2.3 Swendsen—Wang Algorithm.- 4.2.4 Further Developments.- 4.2.5 Replica Monte Carlo Method.- 4.2.6 Multigrid Monte Carlo Method.- 4.3 Histogram Methods.- 4.3.1 The Single-Histogram Method.- 4.3.2 The Multiple-Histogram Method.- 4.3.3 History and Applications.- 4.4 Summary.- References.- 5. Simulation of Random Growth Processes.- 5.1 Irreversible Growth of Clusters.- 5.1.1 A Simple Example of Cluster Growth: The Eden Model.- 5.1.2 Laplacian Growth.- (a) Moving Boundary Condition Problems.- (b) Numerical Simulation of Dielectric Breakdown and DLA.- (c) Fracture.- 5.2 Reversible Probabilistic Growth.- 5.2.1 Cellular Automata.- 5.2.2 Damage Spreading in the Monte Carlo Method.- 5.2.3 Numerical Results for the Ising Model.- 5.2.4 Heat Bath Versus Glauber Dynamics in the Ising Model.- 5.2.5 Relationship Between Damage and Thermodynamic Properties.- 5.2.6 Damage Clusters.- 5.2.7 Damage in Spin Glasses.- 5.2.8 More About Damage Spreading.- 5.3 Conclusion.- References.- 6. Recent Progress in the Simulation of Classical Fluids.- 6.1 Improvements of the Monte Carlo Method.- 6.1.1 Metropolis Algorithm.- 6.1.2 Monte Carlo Simulations and Statistical Ensembles.- (a) Canonical, Grand Canonical and Semi-grand Ensembles.- (b) Gibbs Ensemble.- (c) MC Algorithm for “Adhesive” Particles.- 6.1.3 Monte Carlo Computation of the Chemical Potential and the Free Energy.- (a) Chemical Potential.- (b) Free Energy.- 6.1.4 Algorithms for Coulombic and Dielectric Fluids.- 6.2 Pure Phases and Mixtures of Simple Fluids.- 6.2.1 Two-Dimensional Simple Fluids.- 6.2.2 Three-Dimensional Monatomic Fluids.- 6.2.3 Lennard—Jones Fluids and Similar Systems.- 6.2.4 Real Fluids.- 6.2.5 Mixtures of Simple Fluids.- (a) Hard Core Systems.- (b) LJ Mixtures.- (c) Polydisperse Fluids.- 6.3 Coulombic and Ionic Fluids.- 6.3.1 One-Component Plasma, Two-Component Plasma and Primitive Models of Electrolyte Solutions.- (a) OCP and TCP.- (b) Primitive Models.- 6.3.2 Realistic Ionic Systems.- 6.4 Simulations of Inhomogeneous Simple Fluids.- 6.4.1 Liquid—Vapour Interfaces.- 6.4.2 Fluid—Solid Interfaces.- 6.4.3 Interfaces of Charged Systems.- 6.4.4 Fluids in Narrow Pores.- 6.5 Molecular Liquids: Model Systems.- 6.5.1 Two-Dimensional Systems.- 6.5.2 Convex Molecules (Three-Dimensional).- (a) Virial Coefficients and the Equation of State.- (b) Pair Distribution Function.- (c) Phase Transitions.- 6.5.3 Site—Site Potentials.- 6.5.4 Chain Molecules.- 6.5.5 Dipolar Systems.- 6.5.6 Quadrupolar Systems.- 6.5.7 Polarizable Polar Fluids.- 6.6 Molecular Liquids: Realistic Systems.- 6.6.1 Nitrogen (N2).- 6.6.2 Halogens (Br2, Cl2, I2).- 6.6.3 Benzene (C6H6).- 6.6.4 Naphthalene (C10H8).- 6.6.5 n-Alkanes: CH3(CH2)n?2CH3.- 6.6.6 Water (H2O).- 6.6.7 Methanol (CH3OH).- 6.6.8 Other Polar Systems.- 6.6.9 Mixtures.- 6.7 Solutions.- 6.7.1 Infinite Dilution.- 6.7.2 Finite Concentration.- 6.7.3 Polyelectrolytes and Micelles.- 6.8 Interfaces in Molecular Systems.- 6.8.1 Polar Systems.- (a) Model Systems.- (b) Realistic Systems.- 6.8.2 Chain Molecules Confined by Hard Plates.- References.- 7. Monte Carlo Techniques for Quantum Fluids, Solids and Droplets.- 7.1 Variational Method.- 7.1.1 Variational Wavefunctions.- 7.1.2 The Pair Product Wavefunction.- 7.1.3 Three-Body Correlations.- 7.1.4 Backflow Correlations.- 7.1.5 Pairing Correlations.- 7.1.6 Shadow Wavefunctions.- 7.1.7 Wavefunction Optimisation.- 7.2 Green’s Function Monte Carlo and Related Methods.- 7.2.1 Outline of the Method.- 7.2.2 Fermion Methods.- 7.2.3 Shadow Importance Functions.- 7.3 Path Integral Monte Carlo Method.- 7.3.1 PIMC Methodology.- 7.3.2 The High Temperature Density Matrix.- 7.3.3 Monte Carlo Algorithm.- 7.3.4 Simple Metropolis Monte Carlo Method.- 7.3.5 Normal Mode Methods.- 7.3.6 Threading Algorithm.- 7.3.7 Bisection and Staging Methods.- 7.3.8 Sampling Permutations.- 7.3.9 Calculation of the Energy.- 7.3.10 Computation of the Superfluid Density.- 7.3.11 Exchange in Quantum Crystals.- 7.3.12 Comparison of GFMC with PIMC.- 7.3.13 Applications.- 7.4 Some Results for Bulk Helium.- 7.4.1 4He Results.- 7.4.2 3He Results.- 7.4.3 Solid He.- 7.5 Momentum and Related Distributions.- 7.5.1 The Single-Particle Density Matrix.- 7.5.2 y-Scaling.- 7.5.3 Momentum Distribution Results.- 7.6 Droplets and Surfaces.- 7.6.1 Ground States of He Droplets.- 7.6.2 Excitations in Droplets.- 7.6.3 3He Droplets.- 7.6.4 Droplets at Finite Temperature.- 7.6.5 Surfaces and Interfaces.- 7.7 Future Prospects.- References.- 8. Quantum Lattice Problems.- 8.1 Overview.- 8.2 Models.- 8.3 Variational Monte Carlo Method.- 8.3.1 Method and Trial Wavefunctions.- 8.3.2 Results.- 8.4 Green’s Function Monte Carlo Method.- 8.4.1 Method.- 8.4.2 Results.- 8.5 Grand Canonical Quantum Monte Carlo Method.- 8.5.1 Method.- 8.5.2 Applications.- 8.6 Projector Quantum Monte Carlo Method.- 8.6.1 Method.- 8.6.2 Applications.- 8.7 Fundamental Difficulties.- 8.7.1 The Sign Problem.- 8.7.2 Numerical Instabilities.- 8.7.3 Dynamic Susceptibilities.- 8.7.4 Applicability.- 8.8 Concluding Remarks.- 8.A Appendix.- References.- 9. Simulations of Macromolecules.- 9.1 Techniques and Models.- 9.1.1 Polymer Models.- (a) Lattice Models.- (b) Off-Lattice Models.- 9.1.2 Monte Carlo Techniques.- (a) Kink-Jump and Crankshaft Algorithm.- (b) Reptation Algorithm.- (c) General Reptation Algorithm.- (d) Grand Canonical Reptation Algorithm.- (e) Collective Reptation Method.- (f) Pivot Algorithm.- (g) Growth and Scanning Algorithms.- 9.2 Amorphous Systems.- 9.2.1 Dynamics of Polymers.- (a) Polymer Melts.- (b) Polymers in Flow.- (c) Gel Electrophoresis.- 9.2.2 The Glassy State.- 9.2.3 Equation of State.- 9.3 Disorder Effects.- 9.3.1 Polymer Chains in Random Media.- 9.3.2 Effect of Disorder on Phase Transitions.- 9.3.3 Diffusion in Disordered Media.- 9.4 Mesomorphic Systems.- 9.4.1 Hard Rods.- 9.4.2 Semirigid Chains.- 9.4.3 Anisotropic Interactions.- 9.5 Networks.- 9.5.1 Tethered Membranes.- 9.5.2 Branched Polymers and Random Networks.- 9.6 Segregation.- 9.6.1 Collapse Transition.- 9.6.2 Polymer Mixtures.- 9.6.3 Dynamics of Decomposition.- 9.7 Surfaces and Interfaces.- 9.7.1 Adsorption on Rough Surfaces.- 9.7.2 Entropic Repulsion.- 9.7.3 Confined Polymer Melts.- 9.8 Special Polymers.- 9.8.1 Polyelectrolytes.- 9.8.2 Proteins.- (a) Protein Folding.- (b) Protein Dynamics.- References.- 10. Percolation, Critical Phenomena in Dilute Magnets, Cellular Automata and Related Problems.- 10.1 Percolation.- 10.2 Dilute Ferromagnets.- 10.3 Cellular Automata.- 10.4 Multispin Programming of Cellular Automata.- 10.5 Kauffman Model and da Silva—Herrmann Algorithm.- References.- 11. Interfaces, Wetting Phenomena, Incommensurate Phases.- 11.1 Interfaces in Ising Models.- 11.1.1 The Three-Dimensional Nearest-Neighbour Ising Model.- 11.1.2 Alloys and Microemulsions.- 11.1.3 Adsorbates and Two-Dimensional Systems.- 11.2 Interfaces in Multistate Models.- 11.3 Dynamical Aspects.- 11.3.1 Growth of Wetting Layers and Interfaces.- 11.3.2 Domain Growth.- 11.4 Spatially Modulated Structures.- 11.5 Conclusions.- References.- 12. Spin Glasses, Orientational Glasses and Random Field Systems.- 12.1 Spin Glasses.- 12.1.1 The Spin Glass Transition.- 12.1.2 The Edwards Anderson Model.- 12.1.3 Phase Transitions.- 12.1.4 The Low Temperature State.- 12.1.5 The Vortex Glass.- 12.2 Potts Glasses.- 12.2.1 Introduction to Potts Glasses.- 12.2.2 Mean-Field Theory.- 12.2.3 The Critical Dimensions.- 12.2.4 The Short-Range Potts Model.- (a) Phenomenological T = 0 Scaling.- (b) Monte Carlo Simulations.- (c) Transfer Matrix Calculations.- (d) High-Temperature Series Expansions.- 12.3 Orientational Glasses.- 12.3.1 Introduction to Orientational Glasses.- 12.3.2 Static and Dynamic Properties of the Isotropic Orientational Glass (m = 3) in Two and Three Dimensions.- 12.3.3 More Realistic Models.- 12.4 The Random-Field Ising Model.- 12.5 Concluding Remarks and Outlook.- References.
