Computational Physics: An Introduction To Monte Carlo Simulations Of Matrix Field Theory

by Badis Ydri

Overview

This book is divided into two parts. In the first part we give an elementary introduction to computational physics consisting of 21 simulations which originated from a formal course of lectures and laboratory simulations delivered since 2010 to physics students at Annaba University. The second part is much more advanced and deals with the problem of how to set up working Monte Carlo simulations of matrix field theories which involve finite dimensional matrix regularizations of noncommutative and fuzzy field theories, fuzzy spaces and matrix geometry. The study of matrix field theory in its own right has also become very important to the proper understanding of all noncommutative, fuzzy and matrix phenomena. The second part, which consists of 9 simulations, was delivered informally to doctoral students who were working on various problems in matrix field theory. Sample codes as well as sample key solutions are also provided for convenience and completeness.

Product Details

ISBN-13:	9789813200210
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	04/10/2017
Pages:	312
Product dimensions:	6.90(w) x 9.70(h) x 0.70(d)

Table of Contents

Preface vii

Introductory Remarks xv

Introduction to Computational Physics 1

1 Euler Algorithm 3

1.1 Euler Algorithm 3

1.2 First Example and Sample Code 4

1.2.1 Radioactive Decay 4

1.2.2 A Sample Fortran Code 6

1.3 More Examples 7

1.3.1 Air Resistance 7

1.3.2 Projectile Motion 9

1.4 Periodic Motions and Euler-Cromer and Verlet Algorithms 10

1.4.1 Harmonic Oscillator 10

1.1.1 Euler Algorithm 11

1.4.1 Euler-Cromer Algorithm 12

1.4.2 Verlet Algorithm 13

1.5 Exercises 13

1.6 Simulation 1: Euler Algorithm - Air Resistance 14

1.7 Simulation 2: Euler Algorithm - Projectile Motion 15

1.8 Simulation 3: Euler, Fuler-Cromer and Verlet Algorithms 16

2 Classical Numerical Integration 17

2.1 Rectangular Approximation 17

2.2 Trapezoidal Approximation 18

2.3 Parabolic Approximation or Simpson's Rule 18

2.4 Errors 20

2.5 Simulation 4: Numerical Integrals 21

3 Newton-Raphson Algorithms and Interpolation 23

3.1 Bisection Algorithm 23

3.2 Newton-Raphson Algorithm 23

3.3 Hybrid Method 24

3.4 Lagrange Interpolation 25

3.5 Cubic Spline Interpolation 26

3.6 The Method of Least Squares 28

3.7 Simulation 5: Newton Raphson Algorithm 29

4 The Solar System: The Runge-Kutta Methods 31

4.1 The Solar System 31

4.1.1 Newton's Second Law 31

4.1.2 Astronomical Units and Initial Conditions 32

4.1.3 Kepler's Laws 32

4.1.4 The Inverse-Square Law and Stability of Orbits 34

4.2 Euler Cromer Algorithm 35

4.3 The Runge-Kutta Algorithm 36

4.3.1 The Method 36

4.3.2 Example 1: The Harmonic Oscillator 37

4.3.3 Example 2: The Solar System 37

4.4 Precession of the Perihelion of Mercury 39

4.5 Exercises 39

4.6 Simulation 6: Runge-Kutta Algorithm: Solar System 40

4.7 Simulation 7: Precession of the perihelion of Mercury 41

5 Chaotic Pendulum 43

5.1 Equation of Motion 43

5.2 Numerical Algoritlmis 45

5.2.1 Euler Cromer Algorithm 46

5.2.2 Runge-Kutta Algorithm 46

5.3 Elements of Chaos 47

5.3.1 Butterfly Effect: Sensitivity to Initial Conditions 47

5.3.2 Poincaré Section and Attractors 48

5.3.3 Period-Doubling Bifurcations 48

5.3.4 Feigenbaum Ratio 49

5.3.5 Spontaneous Symmetry Breaking 49

5.4 Simulation 8: The Butterfly Effect 50

5.5 Simulation 9: Poincare Sections 50

5.6 Simulation 10: Period Doubling 52

5.7 Simulation 11: Bifurcation Diagrams 53

6 Molecular Dynamics 55

6.1 Introduction 55

6.2 The Lennard-Jones Potential 56

6.3 Units, Boundary Conditions and Verlet Algorithm 57

6.4 Some Physical Applications 59

6.4.1 Dilute Gas and Maxwell Distribution 59

6.4.2 The Melting Transition 60

6.5 Simulation 12: Maxwell Distribution 60

6.6 Simulation 13: Melting Transition 61

7 Pseudo Random Numbers and Random Walks 63

7.1 Random Numbers 63

7.1.1 Linear Congruent or Power Residue Method 63

7.1.2 Statistical Tests of Randomness 64

7.2 Random Systems 66

7.2.1 Random Walks 66

7.2.2 Diffusion Equation 67

7.3 The Random Number Generators RAN 0,1,2 69

7.4 Simulation 14: Random Numbers

7.5 Simulation 15: Random Walks 73

8 Monte Carlo Integration 75

8.1 Numerical Integration 75

8.1.1 Rectangular Approximation Revisited 75

8.1.2 Midpoint Approximation of Multidimensional Integrals 76

8.1.3 Spheres and Balls in d Dimensions 78

8.2 Monte Carlo Integration: Simple Sampling 78

8.2.1 Sampling (Hit or Miss) Method 79

8.2.2 Sample Mean Method 79

8.2.3 Sample Mean Method in Higher Dimensions 80

8.3 The Central Limit Theorem 81

8.4 Monte Carlo Errors and Standard Deviation 82

8.5 Nonuniform Probability Distributions 84

8.5.1 The Inverse Transform Method 84

8.5.2 The Acceptance-Rejection Method 86

8.6 Simulation 16: Midpoint and Monte Carlo Approximations 86

8.7 Simulation 17: Nonuniform Probability Distributions 87

9 The Metropolis Algorithm and the Ising Model 89

9.1 The Canonical Ensemble 89

9.2 Importance Sampling 90

9.3 The Ising Model 91

9.4 The Metropolis Algorithm 92

9.5 The Heat-Bath Algorithm 94

9.6 The Mean Field Approximation 94

9.6.1 Phase Diagram and Critical Temperature 94

9.6.2 Critical Exponents 96

9.7 Simulation of the Ising Model and Numerical Results 97

9.7.1 The Fortran Code 97

9.7.2 Some Numerical Results 99

9.8 Simulation 18: The Metropolis Algorithm and the Ising Model 101

9.9 Simulation 19: The Ferromagnetic Second Order Phase Transition 102

9.10 Simulation 20: The 2-Point Correlator 103

9.11 Simulation 21: Hysteresis and the First Order Phase Transition 104

Monte Carlo Simulations of Matrix Field Theory 105

10 Metropolis Algorithm for Yang-Mills Matrix Models 107

10.1 Dimensional Reduction 107

10.1.1 Yang-Mills Action 107

10.1.2 Chern-Simons Action: Myers Term 108

10.2 Metropolis Accept/Reject Step 112

10.3 Statistical Errors 113

10.4 Auto-Correlation Time 114

10.5 Code and Sample Calculation 115

References 117

11 Hybrid Monte Carlo Algorithm for Yang-Mills Matrix Models 119

11.1 The Yang-Mills Matrix Action 119

11.2 The Leap Frog Algorithm 120

11.3 Metropolis Algorithm 122

11.4 Gaussian Distribution 123

11.5 Physical Tests 123

11.6 Emergent Geometry: An Exotic Phase Transition 124

References 129

12 Hybrid Monte Carlo Algorithm for Noucommutative Phi-Four 131

12.1 The Matrix Scalar Action 131

12.2 The Leap Frog Algorithm 132

12.3 Hybrid Monte Carlo Algorithm 132

12.4 Optimization 132

12.4.1 Partial Optimization 132

12.4.2 Full Optimization 134

12.5 The Non-Uniform Order: Another Exotic Phase 134

12.5.1 Phase Structure 134

12.5.2 Sample Simulations 135

References 139

13 Lattice HMC Simulations of φ4/2: A Lattice Example 141

13.1 Model and Phase Structure 141

13.2 The HM Algorithm 145

13.3 Renormalization and Continuum Limit 147

13.4 HMC Simulation Calculation of the Critical Line 149

References 151

14 (Multi-Trace) Quartic Matrix Models 153

14.1 The Pure Real Quartic Matrix Model 153

14.2 The Multi-Trace Matrix Model 154

14.3 Model and Algorithm 156

14.4 The Disorder-to-Non-Uniform-Order Transition 158

14.5 Other Suitable Algorithms 160

14.5.1 Over-Relaxation Algorithm 160

14.5.2 Heat-Bath Algorithm 161

References 162

15 The Remez Algorithm and the Conjugate Gradient Method 163

15.1 Minimax Approximations 163

15.1.1 Minimax Polynomial Approximation and Chebyshev Polynomials 163

15.1.2 Minimax Rational Approximation and Remez Algorithm 168

15.1.3 The Code "AlgRemez" 171

15.2 Conjugate Gradient Method 171

15.2.3 Construction 171

15.2.2 The Conjugate Gradient Method as a Krylov Space Solver 175

15.2.3 The Multi-Mass Conjugate Gradient Method 177

References 179

16 Monte Carlo Simulation of Fermion Determinants 181

16.1 The Dirac Operator 181

16.2 Pseudo-Fermious and Rational Approximations 185

16.3 More on The Conjugate-Gradient 187

16.3.1 Multiplication by M' and (M') 187

16.3.2 The Fermionic Force 190

16.4 The Rational Hybrid Monte Carlo Algorithm 192

16.4.1 Statement 192

16.4.2 Preliminary Tests 193

16.5 Other Related Topics 197

References 199

17 U(1) Gauge Theory on the Lattice: Another Lattice Example 201

17.1 Continuum Considerations 201

17.2 Lattice Regularization 203

17.2.1 Lattice Fermions and Gauge Fields 203

17.2.2 Quenched Approximation 205

17.2.3 Wilson Loop, Creutz Ratio and Other Observables 206

17.3 Monte Carlo Simulation of Pure U(1) Gauge Theory 209

17.3.1 The Metropolis Algorithm 209

17.3.2 Some Numerical Results 212

17.3.3 Coulomb and Confinement Phases 215

References 216

18 Codes 217

9.1 Metropolis-ym.f 219

9.2 Hybrid-ym.f 225

9.3 Hybrid-scalar-fuzzy.f 232

9.4 Phi-four-on-lattice.f 242

9.5 Metropolis-scalar-multitrace.f 249

9.6 Romez.f 256

9.7 Conjugate-gradient.f 258

9.8 Hybrid-supersymmetric-ym.f 261

9.9 U-one-on-the-lattice.f 279

Index 291