Monte Carlo Methods in Statistical Physics / Edition 2

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Table of Contents

1. Introduction: Theory and “Technical” Aspects of Monte Carlo Simulations (With 5 Figures).- 1.1 Purpose of the Monte Carlo Method.- 1.2 Description of the Monte Carlo Technique in Classical Statistical Mechanics.- 1.2.1 Computation of Static Averages in the Canonic Ensemble.- 1.2.2 Estimation of the Free Energy. Practical Realization. Other Ensembles.- 1.2.3 Dynamic Interpretation of the Monte Carlo Process.- 1.2.4 Accuracy Considerations: Pseudorandom Numbers, Finite Time Averaging, Initial Condition, etc..- 1.2.5 Appropriate Choice of Boundary Conditions.- 1.2.6 Finite Size Problems. The Extrapolation to the Thermodynamic Limit.- 1.3 Aspects of Simulations of Kinetic Processes.- 1.3.1 Various Monte Carlo Realizations of the Same Master Equation.- 1.3.2 Computations with Conservation Laws: “Hydrodynamic” Slowing Down.- 1.3.3 Slowing Down at Phase Transitions. How to Estimate the Order of a Transition.- 1.4 Variants of the Monte Carlo Method.- 1.4.1 The Approximation of Alexandrowicz.- 1.4.2 Renormalization Group Treatments Utilizing Monte Carlo Methods.- 1.5 Conclusions.- References.- Addendum.- 2. Simulation of Classical Fluids..- 2.1 Overview.- 2.2 Hard Core and Discontinous Potentials.- 2.2.1 Hard-Sphere System.- a) The Pure Hard-Sphere System.- The Equation of State.- The Correlation Functions of the Hard-Sphere System.- b) Hard-Sphere Mixture.- Mixtures of Hard Spheres with Additive Diameter.- Mixtures of Hard Spheres with Nonadditive Diameter.- 2.2.2 Hard Spheres with Discontinuous Short-Range Potential.- a) Hard Spheres with a Square Well.- b) Hard Spheres with a Triangular Well.- c) The Hard Spheres with a Repulsive Square Well.- d) Mixtures of Hard Spheres and Hard Spheres Plus Square Well.- e) Conclusion.- 2.2.3 Two-Dimensional Systems.- a) The Hard-Disk System.- b) The Mixtures of Hard Disks.- c) System with a Repulsive “Step” Potential.- d) The Hard Parallel Square System.- 2.3 Soft Short-Range Potentials.- 2.3.1 Inverse Power Potentials.- a) The Inverse-12 Potential.- b) The Inverse-9, -6 and -4 Potentials.- 2.3.2 The Lennard-Jones (LJ) Potential.- a) The Pure Lennard-Jones (LJ) System.- The Thermodynamic Properties.- The Correlation Functions.- b) Conclusion.- c) The Lennard-Jones (LJ) Mixtures.- 2.3.3 The Two-Dimensional Lennard-Jones (LJ) System.- 2.4 Ionic Systems.- 2.4.1 Generalities.- a) Specific Problems.- b) Classes of Ionic Fluids.- 2.4.2 Fully Ionized Matter.- a) The One Component Plasma.- b) Electron Screening.- c) Charged Hard Spheres in a Uniform Background.- d) Ionic Mixtures.- e) Hydrogen Plasmas.- 2.4.3 The Primitive Model and Its Applications.- a) The Restricted Primitive Model.- b) The Dissymetric Primitive Model.- 2.4.4 Molten Salts.- a) Potentials.- b) KCl.- c) Other Alkali Halides.- d) Corresponding States Model.- e) Mixtures.- 2.4.5 Liquid Metals.- a) Generalities.- b) “Realistic” Potentials.- 2.5 Molecular Fluids.- 2.5.1 Hard Convex Bodies.- a) Hard Spherocylinders (3-dim.).- b) Hard Ellipsoids (3-dim.).- c) Hard Ellipses (2-dim.).- d) Hard Spherocylinders (2-dim.).- 2.5.2 Atom-Atom Potentials.- a) Hard Diatomics (Fused Hard-Spheres, Dumbbells).- b) Lennard-Jones Atom-Atom Potentials.- c) Other Atom-Atom Potentials.- 2.5.3 Generalized Skmayer Potentials.- a) Dipole-Dipole Interaction.- b) Quadrupole-Quadrupole Interaction.- c) Dipole-Quadrupole Interaction.- d) Anisotropic Overlap Interaction.- 2.6 Gas-Liquid Interface.- References.- 3. Phase Diagrams of Mixtures and Magnetic Systems (With 13 Figures).- 3.1 Ordinary Phase Transitions in Magnets and Binary Alloys.- 3.1.1 Ising Model.- 3.1.2 Magnetic Systems with Isotropic Interactions.- 3.2 Multicritical Points and Crossover Behavior.- 3.2.1 Tricritical Phenomena.- 3.2.2 Bicritical and Other Multicritical Behavior.- 3.3 Phase Transitions in Miscellaneous Systems.- 3.4 Conclusions.- References.- Addendum.- 4. Quantum Many-Body Problems (With 4 Figures).- Abstract.- 4.1 Introductory Remarks.- 4.2 Variational Methods.- 4.2.1 Monte Carlo Methods with the Product Trial Function.- a) Finite System Size.- b) The Random Walk.- c) Computation of the Pseudopotential.- d) Fermion Trial Function.- e) Computing the Trial Energy.- f) The Pressure.- g) The Single Particle Density Matrix.- h) Reweighting Configurations.- 4.2.2 Application to Systems of Helium.- a) Hard-Core Boson Systems.- b) Liquid 4He.- c) Solid 4He and 3He.- d) Interatomic Helium Potential.- e) The Hard-Sphere Potential.- f) Nonuniform Helium Systems.- g) Two-Dimensional Helium.- h) Three-Body Pseudopotential.- 4.2.3 Other Bose Systems.- a) Spin-Aligned Hydrogen.- b) Soft-Core Bose Systems.- c) Bose Neutron Matter Calculations.- d) Bose One-Component Plasma.- 4.2.4 Fermi Liquids.- a) Hel ium Three.- b) Neutron Matter.- c) Yukawa Fermions.- d) The Electron Gas.- 4.2.5 Monte Carlo Techniques for Low-Temperature Excitations.- 4.3 Nearly Classical Systems.- 4.4 The Green’s Function Monte Carlo Method (GFMC).- a) Schrödinger’s Equation in Integral Form.- b) Sampling Green’s Function by Random Walks.- c) Importance Sampling.- d) Quantum Mechanical Expectations.- e) Implementation.- 4.4.1 Results.- 4.5 Virial Coefficients and Pair Correlations.- 4.6 Conclusions.- References.- 5. Simulation of Small Systems. (With 9 Figures).- 5.1 Introductory Remarks.- 5.2 Statics.- 5.2.1 Clusters in Continuous Space.- 5.2.2 Lattice Models.- a) General Remarks.- b) Finite-Size Behavior and Superparamagnetism.- c) Equilibrium Cluster Statistics in Systems with Interaction.- d) Cluster Statistics in the Percolation Problem.- 5.3 Cluster Dynamics.- 5.3.1 First-Order Phase Transitions.- 5.3.2 Second-Order Transitions.- Cluster Counting Algorithm.- References.- Addendum.- 6. Monte Carlo Studies of Relaxation Phenomena: Kinetics of Phase Changes and Critical Slowing Down. (With 15 Figures).- 6.1 Introductory Remarks.- 6.2 Kinetics of Fluctuations in Thermal Equilibrium.- 6.2.1 Dynamics of Models for Chain Molecules.- 6.2.2 Critical Slowing Down in Systems Without Conservation Laws.- 6.2.3 Relaxation in Systems with Conserved Quantities.- 6.2.4 Dynamics at a Multicritical Point.- 6.2.5 Dynamics of “Clusters”: Their Reaction Rate and Diffusion Constant.- 6.3 Kinetics of Nonlinear Relaxation.- 6.3.1 Nonlinear Critical Slowing Down.- 6.3.2 Nucleation Kinetics at First-Order Phase Transitions.- 6.3.3 Kinetics of Spinodal Decomposition and Grain Growth in Alloys.- 6.4 Conclusions and Outlook.- References.- 7. Monte Carlo Simulation of Crystal Growth (With 17 Figures).- 7.1 Introductory Remarks.- 7.2 Crystal Surfaces at Equilibrium.- 7.2.1 Singular Faces.- a) Kossel Model (Simple Cubic-(100) Face), Two-Dimensional Spin-s-Ising Model, Thin Films.- b) Other Low-Index Faces.- 7.2.2 Surface Steps.- 7.2.3 Roughening Transition.- 7.3 Growth Kinetics.- 7.3.1 General Aspects of Kinetic Simulations in Crystal Growth.- 7.3.2 Low-Index Faces.- a) Two-Dimensional Ising Model.- b) Kossel Model (100) Face.- c) Orientational Dependence of the Growth Rate (sc-, fcc-Lattices).- d) Adsorption on a Substrate.- e) Surface Diffusion.- 7.3.3 Surface Steps.- a) Straight Steps.- b) Spiral Growth and Spreading of Two-Dimensional Nuclei.- 7.3.4 Roughening Transition.- 7.3.5 More Component Crystals and Segregation of Impurities.- 7.4 Outlook.- References.- 8. Monte Carlo Studies of Systems with Disorder (With 17 Figures).- 8.1 Dilute Impurities in Magnets.- 8.2 Dilute Ferromagnets and the Percolation Problem.- 8.2.1 Thermodynamic Properties at Nonzero Temperatures.- 8.2.2 Percolation Cluster Numbers at Zero Temperatures.- 8.2.3 Cluster Surfaces and Correlations.- 8.2.4 Conductivity and Spin Waves.- 8.2.5 Miscellaneous Topics.- 8.3 Spin Glasses.- 8.3.1 Physical Properties of Spin Glass Systems.- 8.3.2 Distribution of Interactions and Effective Fields.- 8.3.3 Susceptibility and Specific Heat.- 8.3.4 Magnetization and Order Parameters.- 8.3.5 Kinetic Phenomena.- 8.3.6 Ground-State Properties.- 8.4 Disordered Heteropolymers and Their Helix-Coil Transition.- 8.5 Structurally Disordered Solids (Glasses etc.).- 8.6 Conclusions and Outlook.- References.- 9. Applications in Surface Physics. (With 11 Figures).- 9.1 Introductory Remarks.- 9.2 Critical Behavior of Magnetic Systems with Surfaces.- 9.3 Surface Effects in Binary Alloys.- 9.3.1 Surface Enrichment.- 9.3.2 Surface Critical Phenomena.- 9.4 Phase Transitions in Adsorbed Surface Layers.- 9.4.1 Lattice Gas Models.- 9.4.2 “Continuum” Models.- 9.5 Kinetic Phenomena at Surfaces.- 9.6 Conclusions.- References.- Addendum.- 10. Recent Trends in the Development and Application of the Monte Carlo Method. (With 6 Figures).- 10.1 Performance of Monte Carlo Programs.- 10.1.1 General Comments.- 10.1.2 Monte Carlo Calculations on Special-Purpose Computers.- 10.1.3 Speeding-up of Monte Carlo Programs on Vector Processors.- 10.2 Some Comments on Finite-Size Effects.- 10.2.1 Statement of the Problem.- 10.2.2 Standard Finite-Size Scaling Theory.- 10.2.3 Phenomenological Renormalization of Monte Carlo Data.- 10.2.4 Finite-Size Effects at First-Order Phase Transitions.- 10.3 New Directions for the Application of the Monte Carlo Method.- 10.3.1 Monte Carlo Renormalization Group (MCRG).- 10.3.2 Applications to Lattice Gauge Theory.- 10.3.3 Kinetics of Aggregation and Growth Phenomena.- 10.4 Quantum Statistical Mechanics on Lattices.- 10.5 Concluding Remarks.- References.- Addendum.

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