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E-raamat: Guide to Monte Carlo Simulations in Statistical Physics

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(Johannes Gutenberg Universität Mainz, Germany), (University of Georgia)
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  • Ilmumisaeg: 29-Jul-2021
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781108809290
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Guide to Monte Carlo Simulations in Statistical PhysicsSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 29-Jul-2021
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781108809290
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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.

Arvustused

' a comprehensive guide through the multifaceted world of Monte Carlo methods in physics and related branches of science. This work can be recommended to students starting their way in statistical physics simulations as well as to established researchers who needed a fast reference to some particular issues because it comprises a sequential line of explanations with a well-organized compendium of methods and recipes supplied with lists of original papers.' Eugene Postnikov, zbMATH

Muu info

Unique coverage of Monte Carlo methods for both continuum and lattice systems, explaining particularly analysis of phase transitions.
Preface xv
1 Introduction
1(8)
1.1 What is a Monte Carlo simulation?
1(1)
1.2 A comment on the history of Monte Carlo simulations
2(1)
1.3 What problems can we solve with it?
3(1)
1.4 What difficulties will we encounter?
4(1)
1.4.1 Limited computer time and memory
4(1)
1.4.2 Statistical and other errors
4(1)
1.4.3 Knowledge that every practitioner should have
5(1)
1.5 What strategy should we follow in approaching a problem?
5(1)
1.6 How do simulations relate to theory and experiment?
6(1)
1.7 Perspective
7(1)
References
8(1)
2 Some necessary background
9(45)
2.1 Thermodynamics and statistical mechanics: a quick reminder
9(23)
2.1.1 Basic notions
9(8)
2.1.2 Phase transitions
17(12)
2.1.3 Ergodicity and broken symmetry
29(1)
2.1.4 Fluctuations and the Ginzburg criterion
30(1)
2.1.5 A standard exercise: the ferromagnetic Ising model
31(1)
2.2 Probability theory
32(6)
2.2.1 Basic notions
32(2)
2.2.2 Special probability distributions and the central limit theorem
34(1)
2.2.3 Statistical errors
35(1)
2.2.4 Markov chains and master equations
36(2)
2.3 The `art' of random number generation
38(6)
2.3.1 Background
38(1)
2.3.2 Congruential method
39(1)
2.3.3 Mixed congruential methods
40(1)
2.3.4 Shift register algorithm
40(1)
2.3.5 Lagged Fibonacci methods
41(1)
2.3.6 Tests for quality
41(3)
2.3.7 Non-uniform distributions
44(1)
2.4 Non-equilibrium and dynamics: some introductory comments
44(8)
2.4.1 Physical applications of master equations
44(2)
2.4.2 Conservation laws and their consequences
46(3)
2.4.3 Critical slowing down at phase transitions
49(2)
2.4.4 Transport coefficients
51(1)
2.4.5 Concluding comments: why bother about dynamics when doing Monte Carlo for statics?
51(1)
References
52(2)
3 Simple sampling Monte Carlo methods
54(26)
3.1 Introduction
54(1)
3.2 Comparisons of methods for numerical integration of given functions
54(3)
3.2.1 Simple methods
54(2)
3.2.2 Intelligent methods
56(1)
3.3 Boundary value problems
57(2)
3.4 Simulation of radioactive decay
59(1)
3.5 Simulation of transport properties
60(2)
3.5.1 Neutron transport
60(1)
3.5.2 Fluid flow
61(1)
3.6 The percolation problem
62(10)
3.6.1 Site percolation
62(5)
3.6.2 Cluster counting: the Hoshen-Kopelman algorithm
67(1)
3.6.3 Other percolation models
68(1)
3.6.4 The Lorentz gas and cherry pit models and the localization transition
69(1)
3.6.5 Explosive percolation
70(2)
3.7 Finding the groundstate of a Hamiltonian
72(1)
3.8 Generation of `random' walks
73(5)
3.8.1 Introduction
73(1)
3.8.2 Random walks
74(1)
3.8.3 Self-avoiding walks
75(2)
3.8.4 Growing walks and other models
77(1)
3.9 Final remarks
78(1)
References
78(2)
4 Importance sampling Monte Carlo methods
80(81)
4.1 Introduction
80(1)
4.2 The simplest case: single spin-flip sampling for the simple Ising model
81(42)
4.2.1 Algorithm
82(3)
4.2.2 Boundary conditions
85(6)
4.2.3 Finite size effects
91(14)
4.2.4 Finite sampling time effects
105(10)
4.2.5 Critical relaxation
115(8)
4.3 Other discrete variable models
123(9)
4.3.1 Ising models with competing interactions
123(4)
4.3.2 q-state Potts models
127(1)
4.3.3 Baxter and Baxter-Wu models
128(1)
4.3.4 Clock models
129(1)
4.3.5 Ising spin glass models
130(1)
4.3.6 Complex fluid models
131(1)
4.4 Spin-exchange sampling
132(8)
4.4.1 Constant magnetization simulations
132(1)
4.4.2 Phase separation
133(2)
4.4.3 Diffusion
135(3)
4.4.4 Hydrodynamic slowing down
138(1)
4.4.5 Interface between coexisting phases
139(1)
4.5 Microcanonical methods
140(1)
4.5.1 Demon algorithm
140(1)
4.5.2 Dynamic ensemble
140(1)
4.5.3 Q2R
141(1)
4.6 General remarks, choice of ensemble
141(2)
4.7 Statics and dynamics of polymer models on lattices
143(13)
4.7.1 Background
143(1)
4.7.2 Fixed bond length methods
143(2)
4.7.3 Bond fluctuation method
145(1)
4.7.4 Enhanced sampling using a fourth dimension
145(2)
4.7.5 The `wormhole algorithm' - another method to equilibrate dense polymeric systems
147(1)
4.7.6 Polymers in solutions of variable quality: #-point, collapse transition, unmixing
147(3)
4.7.7 Equilibrium polymers: a case study
150(3)
4.7.8 The pruned enriched Rosenbluth method (PERM): a biased sampling approach to simulate very long isolated chains
153(3)
4.7.9 Perspective
156(1)
4.8 Some advice
156(1)
References
157(4)
5 More on importance sampling Monte Carlo methods for lattice systems
161(82)
5.1 Cluster flipping methods
161(7)
5.1.1 Fortuin-Kasteleyn theorem
161(1)
5.1.2 Swendsen-Wang method
162(3)
5.1.3 Wolffmethod
165(1)
5.1.4 `Improved estimators'
166(1)
5.1.5 Invaded cluster algorithm
167(1)
5.1.6 Probability changing cluster algorithm
167(1)
5.2 Specialized computational techniques
168(7)
5.2.1 Expanded ensemble methods
168(1)
5.2.2 Multispin coding
168(1)
5.2.3 iV-fold way and extensions
169(3)
5.2.4 Hybrid algorithms
172(1)
5.2.5 Multigrid algorithms
172(1)
5.2.6 Monte Carlo on vector computers
173(1)
5.2.7 Monte Carlo on parallel computers
174(1)
5.3 Classical spin models
175(15)
5.3.1 Introduction
175(1)
5.3.2 Simple spin-tilt method
176(2)
5.3.3 Heatbath method
178(1)
5.3.4 Low temperature techniques
179(1)
5.3.5 Over-relaxation methods
179(1)
5.3.6 Wolff embedding trick and cluster flipping
180(1)
5.3.7 Hybrid methods
181(1)
5.3.8 Monte Carlo dynamics vs. equation of motion dynamics
182(1)
5.3.9 Topological excitations and solitons
182(4)
5.3.10 Finite size scaling for systems with vector order parameters
186(4)
5.4 Systems with quenched randomness
190(17)
5.4.1 General comments: averaging in random systems
190(4)
5.4.2 Parallel tempering: a general method to better equilibrate systems with complex energy landscapes
194(1)
5.4.3 Random fields and random bonds
195(3)
5.4.4 Spin glasses and optimization by simulated annealing
198(5)
5.4.5 Aging in spin glasses and related systems
203(1)
5.4.6 Vector spin glasses: developments and surprises
204(1)
5.4.7 The ground state of the Ising spin glass on the square lattice: a case study
204(3)
5.5 Models with mixed degrees of freedom: Si/Ge alloys, a case study
207(2)
5.6 Methods for systems with long range interactions
209(2)
5.7 Parallel tempering, simulated tempering, and related methods: accuracy considerations
211(3)
5.8 Sampling the free energy and entropy
214(4)
5.8.1 Thermodynamic integration
214(2)
5.8.2 Groundstate free energy determination
216(1)
5.8.3 Estimation of intensive variables: the chemical potential
216(1)
5.8.4 Lee-Kosterlitz method
217(1)
5.8.5 Free energy from finite size dependence at Te
218(1)
5.9 Miscellaneous topics
218(19)
5.9.1 Inhomogeneous systems: surfaces, interfaces, etc.
218(7)
5.9.2 Anisotropic critical phenomena: simulation boxes with arbitrary aspect ratio
225(2)
5.9.3 Other Monte Carlo schemes
227(2)
5.9.4 Inverse and reverse Monte Carlo methods
229(2)
5.9.5 Finite size effects: review and summary
231(1)
5.9.6 More about error estimation
231(2)
5.9.7 Random number generators revisited
233(4)
5.10 Summary and perspective
237(1)
References
237(6)
6 Off-lattice models
243(83)
6.1 Fluids
243(35)
6.1.1 NVT ensemble and the virial theorem
243(4)
6.1.2 NpT ensemble
247(4)
6.1.3 `Real' microcanonical ensemble
251(1)
6.1.4 Grand canonical ensemble
252(4)
6.1.5 Near critical coexistence: a case study
256(2)
6.1.6 Subsystems: a case study
258(6)
6.1.7 Gibbs ensemble
264(2)
6.1.8 Widom particle insertion method and variants
266(3)
6.1.9 Monte Carlo phase switch
269(4)
6.1.10 Cluster algorithm for fluids
273(1)
6.1.11 Event chain algorithms
274(3)
6.1.12 An extension of the `N-fold way'-algorithm to off-lattice systems
277(1)
6.2 `Short range' interactions
278(2)
6.2.1 Cutoffs
278(1)
6.2.2 Verlet tables and cell structure
278(1)
6.2.3 Minimum image convention
279(1)
6.2.4 Mixed degrees of freedom reconsidered
279(1)
6.3 Treatment of long range forces
280(3)
6.3.1 Reaction field method
280(1)
6.3.2 Ewald method
280(1)
6.3.3 Fast multipole method
281(1)
6.3.4 Particle-particle particle-mesh (P'M) method
282(1)
6.4 Adsorbed monolayers
283(2)
6.4.1 Smooth substrates
283(1)
6.4.2 Periodic substrate potentials
283(2)
6.5 Complex fluids
285(4)
6.5.1 A case study: application of the Liu-Luijten algorithm to a binary fluid mixture
288(1)
6.6 Polymers: an introduction
289(16)
6.6.1 Length scales and models
289(7)
6.6.2 Asymmetric polymer mixtures: a case study
296(2)
6.6.3 Applications: dynamics of polymer melts; thin adsorbed polymeric films
298(5)
6.6.4 Polymer melts: speeding up bond fluctuation model simulations
303(2)
6.7 Liquid crystals; an introduction
305(4)
6.8 Configurational bias and `smart Monte Carlo'
309(4)
6.9 Estimation of excess free energies due to walls for fluids and solids
313(3)
6.10 A symmetric, Lennard-Jones mixture: a case study
316(1)
6.11 Finite size effects on interfacial properties: a case study
317(4)
6.12 Outlook
321(1)
References
321(5)
7 Reweighting methods
326(39)
7.1 Background
326(3)
7.1.1 Distribution functions
326(1)
7.1.2 Umbrella sampling
326(3)
7.2 Single histogram method
329(12)
7.2.1 The Ising model as a case study
330(8)
7.2.2 The surface-bulk multicritical point: another case study
338(3)
7.3 Multihistogram method
341(1)
7.4 Broad histogram method
341(1)
7.5 Transition matrix Monte Carlo
342(1)
7.6 Multicanonical sampling
342(6)
7.6.1 The multicanonical approach and its relationship to canonical sampling
342(2)
7.6.2 Near first order transitions
344(2)
7.6.3 Groundstates in complicated energy landscapes
346(1)
7.6.4 Interface free energy estimation
347(1)
7.7 A case study: the Casimir effect in critical systems
348(1)
7.8 Wang-Landau sampling
348(11)
7.8.1 Basic algorithm
348(5)
7.8.2 Applications to models with continuous variables
353(1)
7.8.3 Two-dimensional Wang-Landau sampling: a case study
354(1)
7.8.4 Microcanonical entropy inflection points
354(2)
7.8.5 Back to numerical integration
356(1)
7.8.6 Replica exchange Wang-Landau sampling
357(2)
7.9 A case study: evaporation/condensation transition of droplets
359(3)
References
362(3)
8 Quantum Monte Carlo methods
365(51)
8.1 Introduction
365(2)
8.2 Feynman path integral formulation
367(10)
8.2.1 Off-lattice problems: low temperature properties of crystals
367(6)
8.2.2 Bose statistics and superfluidity
373(2)
8.2.3 Path integral formulation for rotational degrees of freedom
375(2)
8.3 Lattice problems
377(21)
8.3.1 The Ising model in a transverse field
377(1)
8.3.2 Anisotropic Heisenberg chain: an early case study
378(4)
8.3.3 Fermions on a lattice
382(3)
8.3.4 An intermezzo: the minus sign problem
385(2)
8.3.5 Spinless fermions revisited
387(2)
8.3.6 Cluster methods for quantum lattice models
389(2)
8.3.7 Continuous time simulations
391(1)
8.3.8 Decoupled cell method
392(1)
8.3.9 Handscomb's method and the stochastic series expansion (SSE) approach
393(1)
8.3.10 Wang-Landau sampling for quantum models
394(2)
8.3.11 Fermion determinants
396(2)
8.4 Monte Carlo methods for the study of groundstate properties
398(4)
8.4.1 Variational Monte Carlo (VMC)
398(2)
8.4.2 Green's function Monte Carlo methods (GFMC)
400(2)
8.5 Quantum Monte Carlo in nuclear physics
402(2)
8.6 Towards constructing the nodal surface of off-lattice, many-Fermion systems: the `survival of the fittest algorithm'
404(4)
8.7 Bypassing the minus sign problem: phase transitions in antiferromagnetic metals
408(3)
8.8 Concluding remarks
411(1)
References
412(4)
9 Monte Carlo renormalization group methods
416(14)
9.1 Introduction to renormalization group theory
416(4)
9.2 Real space renormalization group
420(1)
9.3 Monte Carlo renormalization group
421(8)
9.3.1 Large cell renormalization
421(2)
9.3.2 Ma's method: finding critical exponents and the fixed point Hamiltonian
423(1)
9.3.3 Swendsen's method
424(2)
9.3.4 Location of phase boundaries
426(2)
9.3.5 Dynamic problems: matching time-dependent correlation functions
428(1)
9.3.6 Inverse Monte Carlo renormalization group transformations
428(1)
References
429(1)
10 Non-equilibrium and irreversible processes
430(35)
10.1 Introduction and perspective
430(1)
10.2 Driven diffusive systems (driven lattice gases)
431(3)
10.3 Crystal growth
434(3)
10.4 Domain growth
437(6)
10.4.1 Phase separation in mixtures
437(3)
10.4.2 A case study: growth of domains and aging phenomena in spin glasses
440(3)
10.5 Polymer growth
443(2)
10.5.1 Linear polymers
443(1)
10.5.2 Kinetic gelation: a case study
443(2)
10.6 Growth of structures and patterns
445(4)
10.6.1 Eden model of cluster growth
445(1)
10.6.2 Diffusion limited aggregation
445(3)
10.6.3 Cluster-cluster aggregation
448(1)
10.6.4 Cellular automata
448(1)
10.7 Models for film growth
449(5)
10.7.1 Background
449(1)
10.7.2 Ballistic deposition
450(1)
10.7.3 Sedimentation
451(1)
10.7.4 Kinetic Monte Carlo and MBE growth
452(2)
10.8 Transition path sampling
454(3)
10.8.1 What is transition path sampling?
454(1)
10.8.2 A case study: the disk to slab transition in the two-dimensional Ising model
455(2)
10.9 Forced polymer pore translocation: a case study
457(3)
10.10 The Jarzynski non-equilibrium work theorem and its application to obtain free energy differences from trajectories
460(2)
10.11 Outlook: variations on a theme
462(1)
References
462(3)
11 Lattice gauge models: a brief introduction
465(19)
11.1 Introduction: gauge invariance and lattice gauge theory
465(2)
11.2 Some technical matters
467(1)
11.3 Results for Z(N) lattice gauge models
467(1)
11.4 Compact U(l) gauge theory
468(1)
11.5 SU(2) lattice gauge theory
469(1)
11.6 Introduction: quantum chromodynamics (QCD) and phase transitions of nuclear matter
470(3)
11.7 The deconfinement transition of QCD
473(2)
11.8 Finite size scaling based on Polyakov loops: a case study
475(3)
11.9 Towards quantitative predictions
478(3)
11.10 Density of states in gauge theories
481(1)
11.11 Perspective
481(1)
References
482(2)
12 A brief review of other methods of computer simulation
484(35)
12.1 Introduction
484(1)
12.2 Molecular dynamics
484(11)
12.2.1 Integration methods (microcanonical ensemble)
484(4)
12.2.2 Other ensembles (constant temperature, constant pressure, etc.)
488(3)
12.2.3 Non-equilibrium molecular dynamics
491(1)
12.2.4 Hybrid methods (MD + MC)
491(1)
12.2.5 Ab initio molecular dynamics
492(1)
12.2.6 Hyperdynamics and metadynamics
493(1)
12.2.7 Molecular dynamics for systems with open boundaries
494(1)
12.3 Quasi-classical spin dynamics
495(5)
12.3.1 Combined molecular dynamics-spin dynamics (MD-SD) simulations
499(1)
12.4 Langevin equations and variations (cell dynamics)
500(1)
12.5 Micromagnetics
501(1)
12.6 Dissipative particle dynamics (DPD)
502(1)
12.7 Lattice gas cellular automata
503(1)
12.8 Lattice Boltzmann equation
504(1)
12.9 Multiscale simulation
505(2)
12.10 Multiparticle collision dynamics
507(2)
12.11 Active matter
509(3)
12.12 Machine learning
512(3)
References
515(4)
13 Monte Carlo simulations at the periphery of physics and beyond
519(21)
13.1 Commentary
519(1)
13.2 Astrophysics
519(1)
13.3 Materials science
520(2)
13.4 Chemistry
522(2)
13.5 `Biologically inspired' physics
524(3)
13.5.1 Commentary and perspective
524(1)
13.5.2 Lattice proteins
524(3)
13.5.3 Cell sorting
527(1)
13.6 Biology
527(2)
13.7 Mathematics/statistics/computer science
529(1)
13.8 Sociophysics
530(1)
13.9 Econophysics
530(1)
13.10 `Traffic' simulations
531(2)
13.11 Medicine
533(1)
13.12 Networks: what connections really matter?
534(1)
13.13 Finance
535(1)
References
536(4)
14 Monte Carlo studies of biological molecules
540(14)
14.1 Introduction
540(1)
14.2 Protein folding
541(7)
14.2.1 Introduction
541(1)
14.2.2 How to best simulate proteins: Monte Carlo or molecular dynamics
542(1)
14.2.3 Generalized ensemble methods
543(2)
14.2.4 Globular proteins: a case study
545(1)
14.2.5 Simulations of membrane proteins
545(3)
14.3 Monte Carlo simulations of RNA structures
548(1)
14.4 Monte Carlo simulations of carbohydrates
549(2)
14.5 Determining macromolecular structures
551(1)
14.6 Outlook
551(1)
References
552(2)
15 Emerging trends
554(2)
Index 556
David P. Landau is the Distinguished Research Professor of Physics and founding Director of the Center for Simulational Physics at the University of Georgia, USA. Kurt Binder is Professor Emeritus of Theoretical Physics and Gutenberg Fellow at the Institut für Physik, Johannes Gutenberg Universität, Mainz, Germany.
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