Monte Carlo Methods in Statistical Physics

In the seven years since this volume first appeared. there has been an enormous expansion of the range of problems to which Monte Carlo computer simulation methods have been applied. This fact has already led to the addition of a companion volume ("Applications of the Monte Carlo Method in Statistical Physics", Topics in Current Physics. Vol . 36), edited in 1984, to this book. But the field continues to develop further; rapid progress is being made with respect to the implementation of Monte Carlo algorithms, the construction of special-purpose computers dedicated to exe­ cute Monte Carlo programs, and new methods to analyze the "data" generated by these programs. Brief descriptions of these and other developments, together with numerous addi­ tional references, are included in a new chapter , "Recent Trends in Monte Carlo Simulations" , which has been written for this second edition. Typographical correc­ tions have been made and fuller references given where appropriate, but otherwise the layout and contents of the other chapters are left unchanged. Thus this book, together with its companion volume mentioned above, gives a fairly complete and up­ to-date review of the field. It is hoped that the reduced price of this paperback edition will make it accessible to a wide range of scientists and students in the fields to which it is relevant: theoretical phYSics and physical chemistry , con­ densed-matter physics and materials science, computational physics and applied mathematics, etc.

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 Stockmayer 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.