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A Guide to Monte Carlo Simulations in Statistical Physics.

By: Contributor(s): Publisher: Cambridge : Cambridge University Press, 2000Copyright date: ©2000Description: 1 online resource (398 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9780511151224
Subject(s): Genre/Form: Additional physical formats: Print version:: A Guide to Monte Carlo Simulations in Statistical PhysicsDDC classification:
  • 530.13
LOC classification:
  • QC174.85.M64 -- L36 2000eb
Online resources:
Contents:
Intro -- Contents -- Preface -- 1 Introduction -- 1.1 What is a Monte Carlo simulation -- 1.2 What problems can we solve with it? -- 1.3 What difficulties will we encounter? -- 1.3.1 Limited computer time and memory -- 1.3.2 Statistical and other errors -- 1.4 What strategy should we follw in approaching a problem? -- 1.5 How do simulations relate to theory and experiment? -- 2 Some necessary background -- 2.1 Thermodynamics and statistical mechanics: a quick reminder -- 2.1.1 Basic notions -- 2.1.2 Phase transitions -- 2.1.3 Ergodicity and broken symmetry -- 2.1.4 Fluctuations and the Ginzburg criterion -- 2.1.5 A standard exercise: the ferromagnetic Ising model -- 2.2 Probabilty theory -- 2.2.1 Basic notions -- 2.2.2 Special probability distributions and the central limit theorem -- 2.2.3 Statistical errors -- 2.2.4 Markov chains and master equations -- 2.2.5 The 'art' of random number generation -- 2.3 Non-equilibrium and dynamics: some introductory comments -- 2.3.1 Physical applications of master equations -- 2.3.2 Conservation laws and their consequences -- 2.3.3 Critical slowing down at phase transitions -- 2.3.4 Transport coefficients -- 2.3.5 Concluding comments: why bother about dynamics whendoing Monte Carlo for statics? -- References -- 3 Simple sampling Monte Carlo methods -- 3.1 Introduction -- 3.2 Comparisons of methods for numerical integration of given functions -- 3.2.1 Simple methods -- 3.2.2 Intelligent methods -- 3.3 Boundary value problems -- 3.4 Simulation of radioactive decay -- 3.5 Simulation of transport properties -- 3.5.1 Neutron support -- 3.5.2 Fluid flow -- 3.6 The percolation problem -- 3.61 Site percolation -- 3.6.2 Cluster counting: the Hoshen-Kopelman alogorithm -- 3.6.3 Other percolation models -- 3.7 Finding the groundstate of a Hamiltonian -- 3.8 Generation of 'random' walks -- 3.8.1 Introduction.
3.8.2 Random walks -- 3.8.3 Self-avoiding walks -- 3.8.4 Growing walks and other models -- 3.9 Final remarks -- References -- 4 Importance sampling Monte Carlo methods -- 4.1 Introduction -- 4.2 The simplest case: single spin-flip sampling for the simple Ising model -- 4.2.1 Algorithm -- 4.2.2 Boundary conditions -- 4.2.3 Finite size effects -- 4.2.4 Finite sampling time effects -- 4.2.5 Critical relaxation -- 4.3 Other discrete variable models -- 4.3.1 Ising models with competing interactions -- 4.3.2 q-state Potts models -- 4.3.3 Baxter and Baxter-Wu models -- 4.3.4. Clock models -- 4.3.5 Ising spin glass models -- 4.3.6 Complex fluid models -- 4.4 Spin-exchange sampling -- 4.4.1 Constant magnetization simulations -- 4.4.2 Phase separation -- 4.4.3 Diffusion -- 4.4.4 Hydrodynamic slowing down -- 4.5 Microcanonical methods -- 4.5.1 Demon algorithm -- 4.5.2 Dynamic ensemble -- 4.5.3 Q2R -- 4.6 General remarks, choice of ensemble -- 4.7 Staticsand dynamics of polymer models on lattices -- 4.7.1 Background -- 4.7.2 Fixed length bond methods -- 4.7.3 Bond fluctuation method -- 4.7.4 Polymers in solutions of variable quality: θ-point, collapse transition, unmixing -- 4.7.5 Equilibrium polymers: a case study -- 4.8 Some advice -- References -- 5 More on importance sampling Monte Carlo methods for lattice systems -- 5.1 Cluster flipping methods -- 5.1.1 Fortuin-Kasteleyn theorem -- 5.1.2 Swendsen-Wang method -- 5.1.3 Wolff method -- 5.1.4 'Improved estimators' -- 5.2 Specialized computational techniques -- 5.2.1 Expanded ensemble methods -- 5.2.2 Multispin coding -- 5.2.3 N-fold way and extensions -- 5.2.4 Hybrid algorithms -- 5.2.5 Multigrid algorithms -- 5.2.6 Monte Carlo on vector computers -- 5.2.7 Monte Carlo on parallel computers -- 5.3 Classical spin models -- 5.3.1 Introduction -- 5.3.2 Simple spin-flip method -- 5.3.3 Heatbath method.
5.3.4 Low temperature techniques -- 5.3.5 Over-relaxation methods -- 5.3.6 Wolff embedding trick and cluster flipping -- 5.3.7 Hybrid methods -- 5.3.8 Monte Carlo dnamics vs. equation of motion dynamics -- 5.3.9 Topological excitations and solitons -- 5.4 Systems with quenched randomness -- 5.4.1 General comments: averaging in random systems -- 5.4.2 Random fields and random bonds -- 5.4.3 Spin glasses and optimization by simulated annealing -- 5.5 Models with mixed degrees of freedom: Si/Ge alloys, a case study -- 5.6 Sampling the free energy and entropy -- 5.6.1 Thermodynamic integration -- 5.6.2 Groundstate free energy determination -- 5.6.3 Estimation of intensive variables: the chemical potential -- 5.6.4 Lee-Kosterlitz method -- 5.6.5 Free energy from finite size dependence at Tc -- 5.7 Miscellaneous topics -- 5.7.1 Inhomogeneous systems: surfaces, interfaces, etc. -- 5.7.2 Other Monte Carlo schemes -- 5.7.3 Finite size effects: a review and summary -- 5.7.4 More about error estimation -- 5.7.5 Random number generators revisited -- 5.8 Summary and perspective -- References -- 6 Off-lattice models -- 6.1 Fluids -- 6.1.1 NVT ensemble and the virial theorem -- 6.1.2 NρT ensemble -- 6.1.3 Grand canonical ensemble -- 6.1.4 Subsystems: a case study -- 6.1.5 Gibbs ensemble -- 6.1.6 Widom article insertion method and variants -- 6.2 'Short range' interactions -- 6.2.1 Cutoffs -- 6.2.2 Verlet tables and cell structure -- 6.2.3 Minimum image convention -- 6.2.4 Mixed degrees of freedom reconsidered -- 6.3 Treatment of long range forces -- 6.3.1 Reaction field method -- 6.3.2 Ewald method -- 6.3.3 Fast multipole method -- 6.4 Absorbed monolayers -- 6.4.1 Smooth substrates -- 6.4.2 Periodic substrate potentials -- 6.5 Complex fluids -- 6.6 Polymers: an introduction -- 6.6.1 Length scales and models -- 6.6.2 Asymmetric polymer mixtures: a case study.
6.6.3 Applications: dynamics of polymer melts -- thin adsorbed polymeric fields -- 6.7 Conifgurational bias and 'smart Monte Carlo' -- References -- 7 Reweighting methods -- 7.1 Background -- 7.1.1 Distribution functions -- 7.1.2 Umbrella sampling -- 7.2 Single histogram method: the Ising model as a case study -- 7.3 Multi-histogram method -- 7.4 Broad histogram method -- 7.5 Multicanonical sampling -- 7.5.1 The multicanonical approach and its relationship to canonical sampling -- 7.5.2 Near first order transitions -- 7.5.3 Groundstates in complicated energy landscapes -- 7.5.4 Interface free energy estimation -- 7.6 A Case study: the Casimir effect in critical systems -- References -- 8 Quantum Monte Carlo methods -- 8.1 Introduction -- 8.2 Feynman path integral formulation -- 8.2.1 Off-lattice problems: low-temperature properties of crystals -- 8.2.2 Bose statistics and superfluidity -- 8.2.3 Path integral formulation for rotational degrees of freedom -- 8.3 Lattice problems -- 8.3.1 The Ising model in a transverse field -- 8.3.2 Anisotropic Heisenberg chain -- 8.3.3 Fermions on a lattice -- 8.3.4 An intermezzo: the minus sign problem -- 8.3.5 Spinless fermions revisited -- 8.3.6 Cluster methods for quantum lattice models -- 8.3.7 Decoupled cell method -- 8.3.8 Handscomb's method -- 8.3.9 Fermion determinants -- 8.4 Monte Carlo methods for the study of groundstate properties -- 8.4.1 Variational Monte Carlo (VMC) -- 8.4.2 Green's function Monte Carlo methods (GFMC) -- 8.5 Concluding remarks -- References -- 9 Monte Carlo renormalization group methods -- 9.1 Introduction to renormalization group theory -- 9.2 Real space renormalization group -- 9.3 Monte Carlo renormalization group -- 9.3.1 Large cell renormalization -- 9.3.2 Ma's methods: finding critical exponents and the fixed point Hamiltonian -- 9.3.3 Swendsen's method.
9.3.4 Location of phase boundaries -- 9.3.5 Dynamic problems: matching time-dependent correlation functions -- References -- 10 Non-equilibrium and irreversible processes -- 10.1 Introduction and perspective -- 10.2 Driven diffusive systems (driven lattice gases) -- 10.3 Crystal growth -- 10.4 Domain growth -- 10.5 Polymer growth -- 10.5.1 Linear polymers -- 10.5.2 Gelation -- 10.6 Growth of structures and patterns -- 10.6.1 Eden model of cluster growth -- 10.6.2 Diffusion limited aggregation -- 10.6.3 Cluster-cluster aggregation -- 10.6.4 Cellular automata -- 10.7 Models for film growth -- 10.7.1 Background -- 10.7.2 Ballistic deposition -- 10.7.3 Sedimentation -- 10.7.4 Kinetic Monte Carlo and MBE growth -- 10.8 Outlook: variations on a theme -- References -- 11 Lattice gauge models: a brief introduction -- 11.1 Introduction: gauge invariance and lattice gauge theory -- 11.2 Some technical matters -- 11.3 Results for Z(N) lattice gauge models -- 11.4 Compact U(I) gauge theory -- 11.5 SU(2) lattice gauge theory -- 11.6 Introduction: quantum chomodynamics (QCD) and phase transitions of nuclear matter -- 11.7 The deconfinement transition of QCD -- References -- 12 A brief view of other methods of computer simulation -- 12.1 Introduction -- 12.2 Molecular dynamics -- 12.2.1 Integration methods (microcannonical ensemble) -- 12.2.2 Other ensembles (constant temperature, constant pressure, etc.) -- 12.2.3 Non-equilibrium molecular dynamics -- 12.2.4 Hybrid methods (MD + MC) -- 12.2.5 Ab initio molecular dynamics -- 12.3 Quasi-classical spin dynamics -- 12.4 Langevin equations and variations (cell dynamics) -- 12.5 Lattice gas cellular automata -- References -- 13 Outlook -- Appendix -- Index.
Summary: Graduate textbook on simulation methods in statistical physics.
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Intro -- Contents -- Preface -- 1 Introduction -- 1.1 What is a Monte Carlo simulation -- 1.2 What problems can we solve with it? -- 1.3 What difficulties will we encounter? -- 1.3.1 Limited computer time and memory -- 1.3.2 Statistical and other errors -- 1.4 What strategy should we follw in approaching a problem? -- 1.5 How do simulations relate to theory and experiment? -- 2 Some necessary background -- 2.1 Thermodynamics and statistical mechanics: a quick reminder -- 2.1.1 Basic notions -- 2.1.2 Phase transitions -- 2.1.3 Ergodicity and broken symmetry -- 2.1.4 Fluctuations and the Ginzburg criterion -- 2.1.5 A standard exercise: the ferromagnetic Ising model -- 2.2 Probabilty theory -- 2.2.1 Basic notions -- 2.2.2 Special probability distributions and the central limit theorem -- 2.2.3 Statistical errors -- 2.2.4 Markov chains and master equations -- 2.2.5 The 'art' of random number generation -- 2.3 Non-equilibrium and dynamics: some introductory comments -- 2.3.1 Physical applications of master equations -- 2.3.2 Conservation laws and their consequences -- 2.3.3 Critical slowing down at phase transitions -- 2.3.4 Transport coefficients -- 2.3.5 Concluding comments: why bother about dynamics whendoing Monte Carlo for statics? -- References -- 3 Simple sampling Monte Carlo methods -- 3.1 Introduction -- 3.2 Comparisons of methods for numerical integration of given functions -- 3.2.1 Simple methods -- 3.2.2 Intelligent methods -- 3.3 Boundary value problems -- 3.4 Simulation of radioactive decay -- 3.5 Simulation of transport properties -- 3.5.1 Neutron support -- 3.5.2 Fluid flow -- 3.6 The percolation problem -- 3.61 Site percolation -- 3.6.2 Cluster counting: the Hoshen-Kopelman alogorithm -- 3.6.3 Other percolation models -- 3.7 Finding the groundstate of a Hamiltonian -- 3.8 Generation of 'random' walks -- 3.8.1 Introduction.

3.8.2 Random walks -- 3.8.3 Self-avoiding walks -- 3.8.4 Growing walks and other models -- 3.9 Final remarks -- References -- 4 Importance sampling Monte Carlo methods -- 4.1 Introduction -- 4.2 The simplest case: single spin-flip sampling for the simple Ising model -- 4.2.1 Algorithm -- 4.2.2 Boundary conditions -- 4.2.3 Finite size effects -- 4.2.4 Finite sampling time effects -- 4.2.5 Critical relaxation -- 4.3 Other discrete variable models -- 4.3.1 Ising models with competing interactions -- 4.3.2 q-state Potts models -- 4.3.3 Baxter and Baxter-Wu models -- 4.3.4. Clock models -- 4.3.5 Ising spin glass models -- 4.3.6 Complex fluid models -- 4.4 Spin-exchange sampling -- 4.4.1 Constant magnetization simulations -- 4.4.2 Phase separation -- 4.4.3 Diffusion -- 4.4.4 Hydrodynamic slowing down -- 4.5 Microcanonical methods -- 4.5.1 Demon algorithm -- 4.5.2 Dynamic ensemble -- 4.5.3 Q2R -- 4.6 General remarks, choice of ensemble -- 4.7 Staticsand dynamics of polymer models on lattices -- 4.7.1 Background -- 4.7.2 Fixed length bond methods -- 4.7.3 Bond fluctuation method -- 4.7.4 Polymers in solutions of variable quality: θ-point, collapse transition, unmixing -- 4.7.5 Equilibrium polymers: a case study -- 4.8 Some advice -- References -- 5 More on importance sampling Monte Carlo methods for lattice systems -- 5.1 Cluster flipping methods -- 5.1.1 Fortuin-Kasteleyn theorem -- 5.1.2 Swendsen-Wang method -- 5.1.3 Wolff method -- 5.1.4 'Improved estimators' -- 5.2 Specialized computational techniques -- 5.2.1 Expanded ensemble methods -- 5.2.2 Multispin coding -- 5.2.3 N-fold way and extensions -- 5.2.4 Hybrid algorithms -- 5.2.5 Multigrid algorithms -- 5.2.6 Monte Carlo on vector computers -- 5.2.7 Monte Carlo on parallel computers -- 5.3 Classical spin models -- 5.3.1 Introduction -- 5.3.2 Simple spin-flip method -- 5.3.3 Heatbath method.

5.3.4 Low temperature techniques -- 5.3.5 Over-relaxation methods -- 5.3.6 Wolff embedding trick and cluster flipping -- 5.3.7 Hybrid methods -- 5.3.8 Monte Carlo dnamics vs. equation of motion dynamics -- 5.3.9 Topological excitations and solitons -- 5.4 Systems with quenched randomness -- 5.4.1 General comments: averaging in random systems -- 5.4.2 Random fields and random bonds -- 5.4.3 Spin glasses and optimization by simulated annealing -- 5.5 Models with mixed degrees of freedom: Si/Ge alloys, a case study -- 5.6 Sampling the free energy and entropy -- 5.6.1 Thermodynamic integration -- 5.6.2 Groundstate free energy determination -- 5.6.3 Estimation of intensive variables: the chemical potential -- 5.6.4 Lee-Kosterlitz method -- 5.6.5 Free energy from finite size dependence at Tc -- 5.7 Miscellaneous topics -- 5.7.1 Inhomogeneous systems: surfaces, interfaces, etc. -- 5.7.2 Other Monte Carlo schemes -- 5.7.3 Finite size effects: a review and summary -- 5.7.4 More about error estimation -- 5.7.5 Random number generators revisited -- 5.8 Summary and perspective -- References -- 6 Off-lattice models -- 6.1 Fluids -- 6.1.1 NVT ensemble and the virial theorem -- 6.1.2 NρT ensemble -- 6.1.3 Grand canonical ensemble -- 6.1.4 Subsystems: a case study -- 6.1.5 Gibbs ensemble -- 6.1.6 Widom article insertion method and variants -- 6.2 'Short range' interactions -- 6.2.1 Cutoffs -- 6.2.2 Verlet tables and cell structure -- 6.2.3 Minimum image convention -- 6.2.4 Mixed degrees of freedom reconsidered -- 6.3 Treatment of long range forces -- 6.3.1 Reaction field method -- 6.3.2 Ewald method -- 6.3.3 Fast multipole method -- 6.4 Absorbed monolayers -- 6.4.1 Smooth substrates -- 6.4.2 Periodic substrate potentials -- 6.5 Complex fluids -- 6.6 Polymers: an introduction -- 6.6.1 Length scales and models -- 6.6.2 Asymmetric polymer mixtures: a case study.

6.6.3 Applications: dynamics of polymer melts -- thin adsorbed polymeric fields -- 6.7 Conifgurational bias and 'smart Monte Carlo' -- References -- 7 Reweighting methods -- 7.1 Background -- 7.1.1 Distribution functions -- 7.1.2 Umbrella sampling -- 7.2 Single histogram method: the Ising model as a case study -- 7.3 Multi-histogram method -- 7.4 Broad histogram method -- 7.5 Multicanonical sampling -- 7.5.1 The multicanonical approach and its relationship to canonical sampling -- 7.5.2 Near first order transitions -- 7.5.3 Groundstates in complicated energy landscapes -- 7.5.4 Interface free energy estimation -- 7.6 A Case study: the Casimir effect in critical systems -- References -- 8 Quantum Monte Carlo methods -- 8.1 Introduction -- 8.2 Feynman path integral formulation -- 8.2.1 Off-lattice problems: low-temperature properties of crystals -- 8.2.2 Bose statistics and superfluidity -- 8.2.3 Path integral formulation for rotational degrees of freedom -- 8.3 Lattice problems -- 8.3.1 The Ising model in a transverse field -- 8.3.2 Anisotropic Heisenberg chain -- 8.3.3 Fermions on a lattice -- 8.3.4 An intermezzo: the minus sign problem -- 8.3.5 Spinless fermions revisited -- 8.3.6 Cluster methods for quantum lattice models -- 8.3.7 Decoupled cell method -- 8.3.8 Handscomb's method -- 8.3.9 Fermion determinants -- 8.4 Monte Carlo methods for the study of groundstate properties -- 8.4.1 Variational Monte Carlo (VMC) -- 8.4.2 Green's function Monte Carlo methods (GFMC) -- 8.5 Concluding remarks -- References -- 9 Monte Carlo renormalization group methods -- 9.1 Introduction to renormalization group theory -- 9.2 Real space renormalization group -- 9.3 Monte Carlo renormalization group -- 9.3.1 Large cell renormalization -- 9.3.2 Ma's methods: finding critical exponents and the fixed point Hamiltonian -- 9.3.3 Swendsen's method.

9.3.4 Location of phase boundaries -- 9.3.5 Dynamic problems: matching time-dependent correlation functions -- References -- 10 Non-equilibrium and irreversible processes -- 10.1 Introduction and perspective -- 10.2 Driven diffusive systems (driven lattice gases) -- 10.3 Crystal growth -- 10.4 Domain growth -- 10.5 Polymer growth -- 10.5.1 Linear polymers -- 10.5.2 Gelation -- 10.6 Growth of structures and patterns -- 10.6.1 Eden model of cluster growth -- 10.6.2 Diffusion limited aggregation -- 10.6.3 Cluster-cluster aggregation -- 10.6.4 Cellular automata -- 10.7 Models for film growth -- 10.7.1 Background -- 10.7.2 Ballistic deposition -- 10.7.3 Sedimentation -- 10.7.4 Kinetic Monte Carlo and MBE growth -- 10.8 Outlook: variations on a theme -- References -- 11 Lattice gauge models: a brief introduction -- 11.1 Introduction: gauge invariance and lattice gauge theory -- 11.2 Some technical matters -- 11.3 Results for Z(N) lattice gauge models -- 11.4 Compact U(I) gauge theory -- 11.5 SU(2) lattice gauge theory -- 11.6 Introduction: quantum chomodynamics (QCD) and phase transitions of nuclear matter -- 11.7 The deconfinement transition of QCD -- References -- 12 A brief view of other methods of computer simulation -- 12.1 Introduction -- 12.2 Molecular dynamics -- 12.2.1 Integration methods (microcannonical ensemble) -- 12.2.2 Other ensembles (constant temperature, constant pressure, etc.) -- 12.2.3 Non-equilibrium molecular dynamics -- 12.2.4 Hybrid methods (MD + MC) -- 12.2.5 Ab initio molecular dynamics -- 12.3 Quasi-classical spin dynamics -- 12.4 Langevin equations and variations (cell dynamics) -- 12.5 Lattice gas cellular automata -- References -- 13 Outlook -- Appendix -- Index.

Graduate textbook on simulation methods in statistical physics.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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