A guide to Monte Carlo simulations in statistical physics / David P. Landau, Center for Simulational Physics, University of Georgia, USA, Kurt Binder, Institut f�ur Physik, Johannes-Gutenberg-Universit�at, Germany.

Format:

Book

Author/Creator:

Landau, David P., author.

Binder, K. (Kurt), 1944- author.

Language:

English

Subjects (All):

Monte Carlo method.

Statistical physics.

Physical Description:

1 online resource (xviii, 564 pages) : digital, PDF file(s).

Edition:

Fifth edition.

Place of Publication:

Cambridge : Cambridge University Press, 2021.

Summary:

Dealing with all aspects of Monte Carlo simulation of complex physical systems encountered in condensed matter physics and statistical mechanics, this book provides an introduction to computer simulations in physics. The 5th edition contains extensive new material describing numerous powerful algorithms and methods that represent recent developments in the field. New topics such as active matter and machine learning are also introduced. Throughout, there are many applications, examples, recipes, case studies, and exercises to help the reader fully comprehend the material. This book is ideal for graduate students and researchers, both in academia and industry, who want to learn techniques that have become a third tool of physical science, complementing experiment and analytical theory.

Contents:

Cover

Half-title

Title page

Copyright information

Contents

Preface

1 Introduction

1.1 What is a Monte Carlo Simulation?

1.2 A Comment on the History of Monte Carlo Simulations

1.3 What Problems can We Solve with It?

1.4 What Difficulties Will We Encounter?

1.4.1 Limited computer time and memory

1.4.2 Statistical and other errors

1.4.3 Knowledge that every practitioner should have

1.5 What Strategy should We Follow in Approaching a Problem?

1.6 How do Simulations relate to Theory and Experiment?

1.7 Perspective

References

2 Some necessary background

2.1 Thermodynamics and Statistical Mechanics: A Quick Reminder

2.1.1 Basic notions

2.1.1.1 Partition function

2.1.1.2 Free energy, internal energy, and entropy

2.1.1.3 Thermodynamic potentials and corresponding ensembles

2.1.1.4 Fluctuations

2.1.2 Phase transitions

2.1.2.1 Order parameter

2.1.2.2 Correlation function

2.1.2.3 First order vs. second order

2.1.2.4 Phase diagrams

2.1.2.5 Critical behavior and exponents

2.1.2.6 Universality and scaling

2.1.2.7 Multicritical phenomena

2.1.2.8 Landau theory

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 Probability 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

2.3 The 'Art' of Random Number Generation

2.3.1 Background

2.3.2 Congruential method

2.3.3 Mixed congruential methods

2.3.4 Shift register algorithm

2.3.5 Lagged Fibonacci methods

2.3.6 Tests for quality

2.3.7 Non-uniform distributions

2.4 Non-Equilibrium and Dynamics: Some Introductory Comments

2.4.1 Physical applications of master equations.

2.4.2 Conservation laws and their consequences

2.4.3 Critical slowing down at phase transitions

2.4.4 Transport coefficients

2.4.5 Concluding comments: why bother about dynamics when doing Monte Carlo for statics?

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 transport

3.5.2 Fluid flow

3.6 The Percolation Problem

3.6.1 Site percolation

3.6.2 Cluster counting: the Hoshen-Kopelman algorithm

3.6.3 Other percolation models

3.6.4 The Lorentz gas and cherry pit models and the localization transition

3.6.5 Explosive percolation

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

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

4.2.2.1 Periodic boundary conditions

4.2.2.2 Screw periodic boundary conditions

4.2.2.3 Antiperiodic boundary conditions

4.2.2.4 Antisymmetric boundary conditions

4.2.2.5 Free edge boundary conditions

4.2.2.6 Mean-field boundary conditions

4.2.2.7 Hyperspherical boundary conditions

4.2.2.8 Generalized antiperiodic boundary conditions

4.2.2.9 Fluctuating boundary conditions

4.2.3 Finite size effects

4.2.3.1 Order of the transition

4.2.3.2 Finite size scaling and critical exponents

4.2.3.3 Finite size scaling and first order transitions.

4.2.3.4 Finite size subsystem scaling

4.2.3.5 Field mixing

4.2.3.6 Finite size effects in simulations of interfaces

4.2.3.7 Final thoughts

4.2.4 Finite sampling time effects

4.2.4.1 Statistical error

4.2.4.2 Biased sampling error: Ising criticality as an example

4.2.4.3 Cross-correlations in the scaling analysis of critical behavior

4.2.4.4 Relaxation effects

4.2.4.5 Back to finite size effects again: self-averaging

4.2.5 Critical relaxation

4.2.5.1 Non-linear relaxation

4.2.5.2 Linear relaxation

4.2.5.3 Integrated vs. asymptotic relaxation time

4.2.5.4 Dynamic finite size scaling

4.2.5.5 Final remarks

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

4.4.5 Interfaces between coexisting phases

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 Statics and Dynamics of Polymer Models on Lattices

4.7.1 Background

4.7.2 Fixed bond length methods

4.7.3 Bond fluctuation method

4.7.4 Enhanced sampling using a fourth dimension

4.7.5 The 'wormhole algorithm' - another method to equilibrate dense polymeric systems

4.7.6 Polymers in solutions of variable quality: ?-point, collapse transition, unmixing

4.7.7 Equilibrium polymers: a case study

4.7.8 The pruned enriched Rosenbluth method (PERM): a biased sampling approach to simulate very long isolated chains

4.7.9 Perspective

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'

5.1.5 Invaded cluster algorithm

5.1.6 Probability changing cluster algorithm

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-tilt method

5.3.3 Heatbath method

5.3.4 Low temperature techniques

5.3.4.1 Sampling

5.3.4.2 Interpretation

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 dynamics vs. equation of motion dynamics

5.3.9 Topological excitations and solitons

5.3.10 Finite size scaling for systems with vector order parameters

5.4 Systems with Quenched Randomness

5.4.1 General comments: averaging in random systems

5.4.2 Parallel tempering: a general method to better equilibrate systems with complex energy landscapes

5.4.3 Random fields and random bonds

5.4.4 Spin glasses and optimization by simulated annealing

5.4.5 Aging in spin glasses and related systems

5.4.6 Vector spin glasses: developments and surprises

5.4.7 The ground state of the Ising spin glass on the square lattice: a case study

5.5 Models with Mixed Degrees of Freedom: Si/Ge Alloys, A Case Study

5.6 Methods for Systems with Long Range Interactions

5.7 Parallel Tempering, Simulated Tempering, and Related Methods: Accuracy Considerations

5.8 Sampling the Free Energy and Entropy

5.8.1 Thermodynamic integration.

5.8.2 Groundstate free energy determination

5.8.3 Estimation of intensive variables: the chemical potential

5.8.4 Lee-Kosterlitz method

5.8.5 Free energy from finite size dependence at Tc

5.9 Miscellaneous Topics

5.9.1 Inhomogeneous systems: surfaces, interfaces, etc.

5.9.2 Anisotropic critical phenomena: simulation boxes with arbitrary aspect ratio

5.9.3 Other Monte Carlo schemes

5.9.3.1 Damage spreading

5.9.3.2 Gaussian ensemble method

5.9.3.3 Simulations at more than one length scale

5.9.4 Inverse and reverse Monte Carlo methods

5.9.5 Finite size effects: review and summary

5.9.6 More about error estimation

5.9.7 Random number generators revisited

5.10 Summary and Perspective

6 Off-lattice models

6.1 Fluids

6.1.1 NVT ensemble and the virial theorem

6.1.2 NpT ensemble

6.1.3 'Real' microcanonical ensemble

6.1.4 Grand canonical ensemble

6.1.5 Near critical coexistence: a case study

6.1.6 Subsystems: a case study

6.1.7 Gibbs ensemble

6.1.8 Widom particle insertion method and variants

6.1.9 Monte Carlo phase switch

6.1.10 Cluster algorithm for fluids

6.1.11 Event chain algorithms

6.1.12 An extension of the 'N-fold way'-algorithm to off-lattice systems

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

6.3.4 Particle-particle particle-mesh (P3M) method

6.4 Adsorbed Monolayers

6.4.1 Smooth substrates

6.4.2 Periodic substrate potentials

6.5 Complex Fluids

6.5.1 A case study: application of the Liu-Luijten algorithm to a binary fluid mixture

6.6 Polymers: An Introduction.

6.6.1 Length scales and models.

Notes:

Title from publisher's bibliographic system (viewed on 25 Nov 2021).

ISBN:

1-108-80159-5

1-108-80929-4

1-108-78034-2

OCLC:

1288577972